QuickSort

Parosh Aziz Abdulla

Uppsala University

September 12, 2012

Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 1 / 7

1 The Partition Procedure

2 Algorithm

3 Worst-Case Behavior

4 Best-Case Behavior

5 Randomized QuickSort

Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 2 / 7

The Partition Procedure

Partition

Partition(A, p, r)

1 x ← A[r ]2 i ← p − 13 for j ← p to r − 14 do if A[j] ≤ x5 then i ← i + 16 exchange A[i ]↔ A[j]7 exchange A[i + 1]↔ A[r ]8 return i + 1

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i

jp r

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ip j r

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ip j r

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ip j r

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p i j r

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p i j r

2 1 3 9 8 6 7 5

p i r

2 1 3 5 8 6 7 9

p i r

Partition(A, 1, 8)

Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 3 / 7

The Partition Procedure

Partition

Partition(A, p, r)

1 x ← A[r ]2 i ← p − 13 for j ← p to r − 14 do if A[j] ≤ x5 then i ← i + 16 exchange A[i ]↔ A[j]7 exchange A[i + 1]↔ A[r ]8 return i + 1

2 9 8 1 3 6 7 5

i

jp r

2 9 8 1 3 6 7 5

ip j r

2 9 8 1 3 6 7 5

ip j r

2 9 8 1 3 6 7 5

ip j r

2 1 8 9 3 6 7 5

p i j r

2 1 3 9 8 6 7 5

p i j r

2 1 3 9 8 6 7 5

p i j r

2 1 3 9 8 6 7 5

p i r

2 1 3 5 8 6 7 9

p i r

Partition(A, 1, 8)

Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 3 / 7

The Partition Procedure

Partition

Partition(A, p, r)

1 x ← A[r ]2 i ← p − 13 for j ← p to r − 14 do if A[j] ≤ x5 then i ← i + 16 exchange A[i ]↔ A[j]7 exchange A[i + 1]↔ A[r ]8 return i + 1

2 9 8 1 3 6 7 5

i

jp r

2 9 8 1 3 6 7 5

ip j r

2 9 8 1 3 6 7 5

ip j r

2 9 8 1 3 6 7 5

ip j r

2 1 8 9 3 6 7 5

p i j r

2 1 3 9 8 6 7 5

p i j r

2 1 3 9 8 6 7 5

p i j r

2 1 3 9 8 6 7 5

p i r

2 1 3 5 8 6 7 9

p i r

Partition(A, 1, 8)

Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 3 / 7

The Partition Procedure

Partition

Partition(A, p, r)

1 x ← A[r ]2 i ← p − 13 for j ← p to r − 14 do if A[j] ≤ x5 then i ← i + 16 exchange A[i ]↔ A[j]7 exchange A[i + 1]↔ A[r ]8 return i + 1

2 9 8 1 3 6 7 5

i

jp r

2 9 8 1 3 6 7 5

ip j r

2 9 8 1 3 6 7 5

ip j r

2 9 8 1 3 6 7 5

ip j r

2 1 8 9 3 6 7 5

p i j r

2 1 3 9 8 6 7 5

p i j r

2 1 3 9 8 6 7 5

p i j r

2 1 3 9 8 6 7 5

p i r

2 1 3 5 8 6 7 9

p i r

Partition(A, 1, 8)

Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 3 / 7

The Partition Procedure

Partition

Partition(A, p, r)

1 x ← A[r ]2 i ← p − 13 for j ← p to r − 14 do if A[j] ≤ x5 then i ← i + 16 exchange A[i ]↔ A[j]7 exchange A[i + 1]↔ A[r ]8 return i + 1

2 9 8 1 3 6 7 5

i

jp r

2 9 8 1 3 6 7 5

ip j r

2 9 8 1 3 6 7 5

ip j r

2 9 8 1 3 6 7 5

ip j r

2 1 8 9 3 6 7 5

p i j r

2 1 3 9 8 6 7 5

p i j r

2 1 3 9 8 6 7 5

p i j r

2 1 3 9 8 6 7 5

p i r

2 1 3 5 8 6 7 9

p i r

Partition(A, 1, 8)

Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 3 / 7

The Partition Procedure

Partition

Partition(A, p, r)

1 x ← A[r ]2 i ← p − 13 for j ← p to r − 14 do if A[j] ≤ x5 then i ← i + 16 exchange A[i ]↔ A[j]7 exchange A[i + 1]↔ A[r ]8 return i + 1

2 9 8 1 3 6 7 5

i

jp r

2 9 8 1 3 6 7 5

ip j r

2 9 8 1 3 6 7 5

ip j r

2 9 8 1 3 6 7 5

ip j r

2 1 8 9 3 6 7 5

p i j r

2 1 3 9 8 6 7 5

p i j r

2 1 3 9 8 6 7 5

p i j r

2 1 3 9 8 6 7 5

p i r

2 1 3 5 8 6 7 9

p i r

Partition(A, 1, 8)

Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 3 / 7

The Partition Procedure

Partition

Partition(A, p, r)

1 x ← A[r ]2 i ← p − 13 for j ← p to r − 14 do if A[j] ≤ x5 then i ← i + 16 exchange A[i ]↔ A[j]7 exchange A[i + 1]↔ A[r ]8 return i + 1

2 9 8 1 3 6 7 5

i

jp r

2 9 8 1 3 6 7 5

ip j r

2 9 8 1 3 6 7 5

ip j r

2 9 8 1 3 6 7 5

ip j r

2 1 8 9 3 6 7 5

p i j r

2 1 3 9 8 6 7 5

p i j r

2 1 3 9 8 6 7 5

p i j r

2 1 3 9 8 6 7 5

p i r

2 1 3 5 8 6 7 9

p i r

Partition(A, 1, 8)

Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 3 / 7

The Partition Procedure

Partition

Partition(A, p, r)

1 x ← A[r ]2 i ← p − 13 for j ← p to r − 14 do if A[j] ≤ x5 then i ← i + 16 exchange A[i ]↔ A[j]7 exchange A[i + 1]↔ A[r ]8 return i + 1

2 9 8 1 3 6 7 5

i

jp r

2 9 8 1 3 6 7 5

ip j r

2 9 8 1 3 6 7 5

ip j r

2 9 8 1 3 6 7 5

ip j r

2 1 8 9 3 6 7 5

p i j r

2 1 3 9 8 6 7 5

p i j r

2 1 3 9 8 6 7 5

p i j r

2 1 3 9 8 6 7 5

p i r

2 1 3 5 8 6 7 9

p i r

Partition(A, 1, 8)

Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 3 / 7

The Partition Procedure

Partition

Partition(A, p, r)

1 x ← A[r ]2 i ← p − 13 for j ← p to r − 14 do if A[j] ≤ x5 then i ← i + 16 exchange A[i ]↔ A[j]7 exchange A[i + 1]↔ A[r ]8 return i + 1

2 9 8 1 3 6 7 5

i

jp r

2 9 8 1 3 6 7 5

ip j r

2 9 8 1 3 6 7 5

ip j r

2 9 8 1 3 6 7 5

ip j r

2 1 8 9 3 6 7 5

p i j r

2 1 3 9 8 6 7 5

p i j r

2 1 3 9 8 6 7 5

p i j r

2 1 3 9 8 6 7 5

p i r

2 1 3 5 8 6 7 9

p i r

Partition(A, 1, 8)

Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 3 / 7

Algorithm

Quick Sort

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 10)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 10)QuickSort(A, 5, 6)QuickSort(A, 5, 4)QuickSort(A, 6, 6)QuickSort(A, 8, 10)QuickSort(A, 8, 7)QuickSort(A, 9, 10)QuickSort(A, 9, 9)QuickSort(A, 11, 10)

Partition(A, 1, 10)Partition(A, 1, 3)Partition(A, 5, 10)Partition(A, 5, 6)Partition(A, 8, 10)Partition(A, 9, 10)(A,1,10)(A,1,10)

(A,1,3)(A,1,3)

(A,1,1)(A,1,1) (A,3,3)(A,3,3)

(A,5,10)(A,5,10)

(A,5,6)(A,5,6)

(A,5,4)(A,5,4) (A,6,6)(A,6,6)

(A,8,10)(A,8,10)

(A,8,7)(A,8,7) (A,9,10)(A,9,10)

(A,9,9)(A,9,9) (A,11,10)(A,11,10)

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7

Algorithm

Quick Sort

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 10)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 10)QuickSort(A, 5, 6)QuickSort(A, 5, 4)QuickSort(A, 6, 6)QuickSort(A, 8, 10)QuickSort(A, 8, 7)QuickSort(A, 9, 10)QuickSort(A, 9, 9)QuickSort(A, 11, 10)

Partition(A, 1, 10)Partition(A, 1, 3)Partition(A, 5, 10)Partition(A, 5, 6)Partition(A, 8, 10)Partition(A, 9, 10)(A,1,10)(A,1,10)

(A,1,3)(A,1,3)

(A,1,1)(A,1,1) (A,3,3)(A,3,3)

(A,5,10)(A,5,10)

(A,5,6)(A,5,6)

(A,5,4)(A,5,4) (A,6,6)(A,6,6)

(A,8,10)(A,8,10)

(A,8,7)(A,8,7) (A,9,10)(A,9,10)

(A,9,9)(A,9,9) (A,11,10)(A,11,10)

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7

Algorithm

Quick Sort

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 10)

QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 10)QuickSort(A, 5, 6)QuickSort(A, 5, 4)QuickSort(A, 6, 6)QuickSort(A, 8, 10)QuickSort(A, 8, 7)QuickSort(A, 9, 10)QuickSort(A, 9, 9)QuickSort(A, 11, 10)

Partition(A, 1, 10)Partition(A, 1, 3)Partition(A, 5, 10)Partition(A, 5, 6)Partition(A, 8, 10)Partition(A, 9, 10)

(A,1,10)

(A,1,10)

(A,1,3)(A,1,3)

(A,1,1)(A,1,1) (A,3,3)(A,3,3)

(A,5,10)(A,5,10)

(A,5,6)(A,5,6)

(A,5,4)(A,5,4) (A,6,6)(A,6,6)

(A,8,10)(A,8,10)

(A,8,7)(A,8,7) (A,9,10)(A,9,10)

(A,9,9)(A,9,9) (A,11,10)(A,11,10)

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7

Algorithm

Quick Sort

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 10)

QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 10)QuickSort(A, 5, 6)QuickSort(A, 5, 4)QuickSort(A, 6, 6)QuickSort(A, 8, 10)QuickSort(A, 8, 7)QuickSort(A, 9, 10)QuickSort(A, 9, 9)QuickSort(A, 11, 10)

Partition(A, 1, 10)

Partition(A, 1, 3)Partition(A, 5, 10)Partition(A, 5, 6)Partition(A, 8, 10)Partition(A, 9, 10)

(A,1,10)

(A,1,10)

(A,1,3)(A,1,3)

(A,1,1)(A,1,1) (A,3,3)(A,3,3)

(A,5,10)(A,5,10)

(A,5,6)(A,5,6)

(A,5,4)(A,5,4) (A,6,6)(A,6,6)

(A,8,10)(A,8,10)

(A,8,7)(A,8,7) (A,9,10)(A,9,10)

(A,9,9)(A,9,9) (A,11,10)(A,11,10)

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7

Algorithm

Quick Sort

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 10)

QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 10)QuickSort(A, 5, 6)QuickSort(A, 5, 4)QuickSort(A, 6, 6)QuickSort(A, 8, 10)QuickSort(A, 8, 7)QuickSort(A, 9, 10)QuickSort(A, 9, 9)QuickSort(A, 11, 10)

Partition(A, 1, 10)

Partition(A, 1, 3)Partition(A, 5, 10)Partition(A, 5, 6)Partition(A, 8, 10)Partition(A, 9, 10)

(A,1,10)

(A,1,10)

(A,1,3)(A,1,3)

(A,1,1)(A,1,1) (A,3,3)(A,3,3)

(A,5,10)(A,5,10)

(A,5,6)(A,5,6)

(A,5,4)(A,5,4) (A,6,6)(A,6,6)

(A,8,10)(A,8,10)

(A,8,7)(A,8,7) (A,9,10)(A,9,10)

(A,9,9)(A,9,9) (A,11,10)(A,11,10)

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7

Algorithm

Quick Sort

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 10)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 10)QuickSort(A, 5, 6)QuickSort(A, 5, 4)QuickSort(A, 6, 6)QuickSort(A, 8, 10)QuickSort(A, 8, 7)QuickSort(A, 9, 10)QuickSort(A, 9, 9)QuickSort(A, 11, 10)

Partition(A, 1, 10)Partition(A, 1, 3)Partition(A, 5, 10)Partition(A, 5, 6)Partition(A, 8, 10)Partition(A, 9, 10)

(A,1,10)

(A,1,10)

(A,1,3)

(A,1,3)

(A,1,1)(A,1,1) (A,3,3)(A,3,3)

(A,5,10)

(A,5,10)

(A,5,6)(A,5,6)

(A,5,4)(A,5,4) (A,6,6)(A,6,6)

(A,8,10)(A,8,10)

(A,8,7)(A,8,7) (A,9,10)(A,9,10)

(A,9,9)(A,9,9) (A,11,10)(A,11,10)

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7

Algorithm

Quick Sort

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 10)

QuickSort(A, 1, 3)

QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 10)QuickSort(A, 5, 6)QuickSort(A, 5, 4)QuickSort(A, 6, 6)QuickSort(A, 8, 10)QuickSort(A, 8, 7)QuickSort(A, 9, 10)QuickSort(A, 9, 9)QuickSort(A, 11, 10)

Partition(A, 1, 10)Partition(A, 1, 3)Partition(A, 5, 10)Partition(A, 5, 6)Partition(A, 8, 10)Partition(A, 9, 10)

(A,1,10)

(A,1,10)

(A,1,3)

(A,1,3)

(A,1,1)(A,1,1) (A,3,3)(A,3,3)

(A,5,10)

(A,5,10)

(A,5,6)(A,5,6)

(A,5,4)(A,5,4) (A,6,6)(A,6,6)

(A,8,10)(A,8,10)

(A,8,7)(A,8,7) (A,9,10)(A,9,10)

(A,9,9)(A,9,9) (A,11,10)(A,11,10)

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7

Algorithm

Quick Sort

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 10)

QuickSort(A, 1, 3)

QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 10)QuickSort(A, 5, 6)QuickSort(A, 5, 4)QuickSort(A, 6, 6)QuickSort(A, 8, 10)QuickSort(A, 8, 7)QuickSort(A, 9, 10)QuickSort(A, 9, 9)QuickSort(A, 11, 10)

Partition(A, 1, 10)

Partition(A, 1, 3)

Partition(A, 5, 10)Partition(A, 5, 6)Partition(A, 8, 10)Partition(A, 9, 10)

(A,1,10)

(A,1,10)

(A,1,3)

(A,1,3)

(A,1,1)(A,1,1) (A,3,3)(A,3,3)

(A,5,10)

(A,5,10)

(A,5,6)(A,5,6)

(A,5,4)(A,5,4) (A,6,6)(A,6,6)

(A,8,10)(A,8,10)

(A,8,7)(A,8,7) (A,9,10)(A,9,10)

(A,9,9)(A,9,9) (A,11,10)(A,11,10)

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7

Algorithm

Quick Sort

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 10)

QuickSort(A, 1, 3)

QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 10)QuickSort(A, 5, 6)QuickSort(A, 5, 4)QuickSort(A, 6, 6)QuickSort(A, 8, 10)QuickSort(A, 8, 7)QuickSort(A, 9, 10)QuickSort(A, 9, 9)QuickSort(A, 11, 10)

Partition(A, 1, 10)

Partition(A, 1, 3)

Partition(A, 5, 10)Partition(A, 5, 6)Partition(A, 8, 10)Partition(A, 9, 10)

(A,1,10)

(A,1,10)

(A,1,3)

(A,1,3)

(A,1,1)(A,1,1) (A,3,3)(A,3,3)

(A,5,10)

(A,5,10)

(A,5,6)(A,5,6)

(A,5,4)(A,5,4) (A,6,6)(A,6,6)

(A,8,10)(A,8,10)

(A,8,7)(A,8,7) (A,9,10)(A,9,10)

(A,9,9)(A,9,9) (A,11,10)(A,11,10)

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7

Algorithm

Quick Sort

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 10)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 10)QuickSort(A, 5, 6)QuickSort(A, 5, 4)QuickSort(A, 6, 6)QuickSort(A, 8, 10)QuickSort(A, 8, 7)QuickSort(A, 9, 10)QuickSort(A, 9, 9)QuickSort(A, 11, 10)

Partition(A, 1, 10)Partition(A, 1, 3)Partition(A, 5, 10)Partition(A, 5, 6)Partition(A, 8, 10)Partition(A, 9, 10)

(A,1,10)

(A,1,10)

(A,1,3)

(A,1,3)

(A,1,1)

(A,1,1)

(A,3,3)

(A,3,3)

(A,5,10)

(A,5,10)

(A,5,6)(A,5,6)

(A,5,4)(A,5,4) (A,6,6)(A,6,6)

(A,8,10)(A,8,10)

(A,8,7)(A,8,7) (A,9,10)(A,9,10)

(A,9,9)(A,9,9) (A,11,10)(A,11,10)

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7

Algorithm

Quick Sort

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 10)QuickSort(A, 1, 3)

QuickSort(A, 1, 1)

QuickSort(A, 3, 3)QuickSort(A, 5, 10)QuickSort(A, 5, 6)QuickSort(A, 5, 4)QuickSort(A, 6, 6)QuickSort(A, 8, 10)QuickSort(A, 8, 7)QuickSort(A, 9, 10)QuickSort(A, 9, 9)QuickSort(A, 11, 10)

Partition(A, 1, 10)Partition(A, 1, 3)Partition(A, 5, 10)Partition(A, 5, 6)Partition(A, 8, 10)Partition(A, 9, 10)

(A,1,10)

(A,1,10)

(A,1,3)

(A,1,3)

(A,1,1)

(A,1,1)

(A,3,3)

(A,3,3)

(A,5,10)

(A,5,10)

(A,5,6)(A,5,6)

(A,5,4)(A,5,4) (A,6,6)(A,6,6)

(A,8,10)(A,8,10)

(A,8,7)(A,8,7) (A,9,10)(A,9,10)

(A,9,9)(A,9,9) (A,11,10)(A,11,10)

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7

Algorithm

Quick Sort

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 10)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 10)QuickSort(A, 5, 6)QuickSort(A, 5, 4)QuickSort(A, 6, 6)QuickSort(A, 8, 10)QuickSort(A, 8, 7)QuickSort(A, 9, 10)QuickSort(A, 9, 9)QuickSort(A, 11, 10)

Partition(A, 1, 10)Partition(A, 1, 3)Partition(A, 5, 10)Partition(A, 5, 6)Partition(A, 8, 10)Partition(A, 9, 10)

(A,1,10)

(A,1,10)

(A,1,3)

(A,1,3)

(A,1,1)

(A,1,1)

(A,3,3)

(A,3,3)

(A,5,10)

(A,5,10)

(A,5,6)(A,5,6)

(A,5,4)(A,5,4) (A,6,6)(A,6,6)

(A,8,10)(A,8,10)

(A,8,7)(A,8,7) (A,9,10)(A,9,10)

(A,9,9)(A,9,9) (A,11,10)(A,11,10)

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7

Algorithm

Quick Sort

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 10)QuickSort(A, 1, 3)QuickSort(A, 1, 1)

QuickSort(A, 3, 3)

QuickSort(A, 5, 10)QuickSort(A, 5, 6)QuickSort(A, 5, 4)QuickSort(A, 6, 6)QuickSort(A, 8, 10)QuickSort(A, 8, 7)QuickSort(A, 9, 10)QuickSort(A, 9, 9)QuickSort(A, 11, 10)

Partition(A, 1, 10)Partition(A, 1, 3)Partition(A, 5, 10)Partition(A, 5, 6)Partition(A, 8, 10)Partition(A, 9, 10)

(A,1,10)

(A,1,10)

(A,1,3)

(A,1,3)

(A,1,1)

(A,1,1)

(A,3,3)

(A,3,3)

(A,5,10)

(A,5,10)

(A,5,6)(A,5,6)

(A,5,4)(A,5,4) (A,6,6)(A,6,6)

(A,8,10)(A,8,10)

(A,8,7)(A,8,7) (A,9,10)(A,9,10)

(A,9,9)(A,9,9) (A,11,10)(A,11,10)

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7

Algorithm

Quick Sort

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 10)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 10)QuickSort(A, 5, 6)QuickSort(A, 5, 4)QuickSort(A, 6, 6)QuickSort(A, 8, 10)QuickSort(A, 8, 7)QuickSort(A, 9, 10)QuickSort(A, 9, 9)QuickSort(A, 11, 10)

Partition(A, 1, 10)Partition(A, 1, 3)Partition(A, 5, 10)Partition(A, 5, 6)Partition(A, 8, 10)Partition(A, 9, 10)

(A,1,10)

(A,1,10)

(A,1,3)

(A,1,3)

(A,1,1)

(A,1,1) (A,3,3)

(A,3,3)

(A,5,10)

(A,5,10)

(A,5,6)(A,5,6)

(A,5,4)(A,5,4) (A,6,6)(A,6,6)

(A,8,10)(A,8,10)

(A,8,7)(A,8,7) (A,9,10)(A,9,10)

(A,9,9)(A,9,9) (A,11,10)(A,11,10)

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7

Algorithm

Quick Sort

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 10)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)

QuickSort(A, 5, 10)

QuickSort(A, 5, 6)QuickSort(A, 5, 4)QuickSort(A, 6, 6)QuickSort(A, 8, 10)QuickSort(A, 8, 7)QuickSort(A, 9, 10)QuickSort(A, 9, 9)QuickSort(A, 11, 10)

Partition(A, 1, 10)Partition(A, 1, 3)Partition(A, 5, 10)Partition(A, 5, 6)Partition(A, 8, 10)Partition(A, 9, 10)

(A,1,10)

(A,1,10)

(A,1,3)

(A,1,3)

(A,1,1)

(A,1,1) (A,3,3)

(A,3,3)

(A,5,10)

(A,5,10)

(A,5,6)(A,5,6)

(A,5,4)(A,5,4) (A,6,6)(A,6,6)

(A,8,10)(A,8,10)

(A,8,7)(A,8,7) (A,9,10)(A,9,10)

(A,9,9)(A,9,9) (A,11,10)(A,11,10)

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7

Algorithm

Quick Sort

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 10)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)

QuickSort(A, 5, 10)

QuickSort(A, 5, 6)QuickSort(A, 5, 4)QuickSort(A, 6, 6)QuickSort(A, 8, 10)QuickSort(A, 8, 7)QuickSort(A, 9, 10)QuickSort(A, 9, 9)QuickSort(A, 11, 10)

Partition(A, 1, 10)Partition(A, 1, 3)

Partition(A, 5, 10)

Partition(A, 5, 6)Partition(A, 8, 10)Partition(A, 9, 10)

(A,1,10)

(A,1,10)

(A,1,3)

(A,1,3)

(A,1,1)

(A,1,1) (A,3,3)

(A,3,3)

(A,5,10)

(A,5,10)

(A,5,6)(A,5,6)

(A,5,4)(A,5,4) (A,6,6)(A,6,6)

(A,8,10)(A,8,10)

(A,8,7)(A,8,7) (A,9,10)(A,9,10)

(A,9,9)(A,9,9) (A,11,10)(A,11,10)

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10

Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7

Algorithm

Quick Sort

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 10)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)

QuickSort(A, 5, 10)

QuickSort(A, 5, 6)QuickSort(A, 5, 4)QuickSort(A, 6, 6)QuickSort(A, 8, 10)QuickSort(A, 8, 7)QuickSort(A, 9, 10)QuickSort(A, 9, 9)QuickSort(A, 11, 10)

Partition(A, 1, 10)Partition(A, 1, 3)

Partition(A, 5, 10)

Partition(A, 5, 6)Partition(A, 8, 10)Partition(A, 9, 10)

(A,1,10)

(A,1,10)

(A,1,3)

(A,1,3)

(A,1,1)

(A,1,1) (A,3,3)

(A,3,3)

(A,5,10)

(A,5,10)

(A,5,6)(A,5,6)

(A,5,4)(A,5,4) (A,6,6)(A,6,6)

(A,8,10)(A,8,10)

(A,8,7)(A,8,7) (A,9,10)(A,9,10)

(A,9,9)(A,9,9) (A,11,10)(A,11,10)

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10

Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7

Algorithm

Quick Sort

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 10)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 10)QuickSort(A, 5, 6)QuickSort(A, 5, 4)QuickSort(A, 6, 6)QuickSort(A, 8, 10)QuickSort(A, 8, 7)QuickSort(A, 9, 10)QuickSort(A, 9, 9)QuickSort(A, 11, 10)

Partition(A, 1, 10)Partition(A, 1, 3)Partition(A, 5, 10)Partition(A, 5, 6)Partition(A, 8, 10)Partition(A, 9, 10)

(A,1,10)

(A,1,10)

(A,1,3)

(A,1,3)

(A,1,1)

(A,1,1) (A,3,3)

(A,3,3)

(A,5,10)

(A,5,10)

(A,5,6)

(A,5,6)

(A,5,4)(A,5,4) (A,6,6)(A,6,6)

(A,8,10)

(A,8,10)

(A,8,7)(A,8,7) (A,9,10)(A,9,10)

(A,9,9)(A,9,9) (A,11,10)(A,11,10)

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10

Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7

Algorithm

Quick Sort

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 10)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 10)

QuickSort(A, 5, 6)

QuickSort(A, 5, 4)QuickSort(A, 6, 6)QuickSort(A, 8, 10)QuickSort(A, 8, 7)QuickSort(A, 9, 10)QuickSort(A, 9, 9)QuickSort(A, 11, 10)

Partition(A, 1, 10)Partition(A, 1, 3)Partition(A, 5, 10)Partition(A, 5, 6)Partition(A, 8, 10)Partition(A, 9, 10)

(A,1,10)

(A,1,10)

(A,1,3)

(A,1,3)

(A,1,1)

(A,1,1) (A,3,3)

(A,3,3)

(A,5,10)

(A,5,10)

(A,5,6)

(A,5,6)

(A,5,4)(A,5,4) (A,6,6)(A,6,6)

(A,8,10)

(A,8,10)

(A,8,7)(A,8,7) (A,9,10)(A,9,10)

(A,9,9)(A,9,9) (A,11,10)(A,11,10)

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10

Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7

Algorithm

Quick Sort

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 10)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 10)

QuickSort(A, 5, 6)

QuickSort(A, 5, 4)QuickSort(A, 6, 6)QuickSort(A, 8, 10)QuickSort(A, 8, 7)QuickSort(A, 9, 10)QuickSort(A, 9, 9)QuickSort(A, 11, 10)

Partition(A, 1, 10)Partition(A, 1, 3)Partition(A, 5, 10)

Partition(A, 5, 6)

Partition(A, 8, 10)Partition(A, 9, 10)

(A,1,10)

(A,1,10)

(A,1,3)

(A,1,3)

(A,1,1)

(A,1,1) (A,3,3)

(A,3,3)

(A,5,10)

(A,5,10)

(A,5,6)

(A,5,6)

(A,5,4)(A,5,4) (A,6,6)(A,6,6)

(A,8,10)

(A,8,10)

(A,8,7)(A,8,7) (A,9,10)(A,9,10)

(A,9,9)(A,9,9) (A,11,10)(A,11,10)

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7

Algorithm

Quick Sort

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 10)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 10)

QuickSort(A, 5, 6)

QuickSort(A, 5, 4)QuickSort(A, 6, 6)QuickSort(A, 8, 10)QuickSort(A, 8, 7)QuickSort(A, 9, 10)QuickSort(A, 9, 9)QuickSort(A, 11, 10)

Partition(A, 1, 10)Partition(A, 1, 3)Partition(A, 5, 10)

Partition(A, 5, 6)

Partition(A, 8, 10)Partition(A, 9, 10)

(A,1,10)

(A,1,10)

(A,1,3)

(A,1,3)

(A,1,1)

(A,1,1) (A,3,3)

(A,3,3)

(A,5,10)

(A,5,10)

(A,5,6)

(A,5,6)

(A,5,4)(A,5,4) (A,6,6)(A,6,6)

(A,8,10)

(A,8,10)

(A,8,7)(A,8,7) (A,9,10)(A,9,10)

(A,9,9)(A,9,9) (A,11,10)(A,11,10)

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7

Algorithm

Quick Sort

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 10)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 10)QuickSort(A, 5, 6)QuickSort(A, 5, 4)QuickSort(A, 6, 6)QuickSort(A, 8, 10)QuickSort(A, 8, 7)QuickSort(A, 9, 10)QuickSort(A, 9, 9)QuickSort(A, 11, 10)

Partition(A, 1, 10)Partition(A, 1, 3)Partition(A, 5, 10)Partition(A, 5, 6)Partition(A, 8, 10)Partition(A, 9, 10)

(A,1,10)

(A,1,10)

(A,1,3)

(A,1,3)

(A,1,1)

(A,1,1) (A,3,3)

(A,3,3)

(A,5,10)

(A,5,10)

(A,5,6)

(A,5,6)

(A,5,4)

(A,5,4)

(A,6,6)

(A,6,6)

(A,8,10)

(A,8,10)

(A,8,7)(A,8,7) (A,9,10)(A,9,10)

(A,9,9)(A,9,9) (A,11,10)(A,11,10)

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7

Algorithm

Quick Sort

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 10)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 10)QuickSort(A, 5, 6)

QuickSort(A, 5, 4)

QuickSort(A, 6, 6)QuickSort(A, 8, 10)QuickSort(A, 8, 7)QuickSort(A, 9, 10)QuickSort(A, 9, 9)QuickSort(A, 11, 10)

Partition(A, 1, 10)Partition(A, 1, 3)Partition(A, 5, 10)Partition(A, 5, 6)Partition(A, 8, 10)Partition(A, 9, 10)

(A,1,10)

(A,1,10)

(A,1,3)

(A,1,3)

(A,1,1)

(A,1,1) (A,3,3)

(A,3,3)

(A,5,10)

(A,5,10)

(A,5,6)

(A,5,6)

(A,5,4)

(A,5,4)

(A,6,6)

(A,6,6)

(A,8,10)

(A,8,10)

(A,8,7)(A,8,7) (A,9,10)(A,9,10)

(A,9,9)(A,9,9) (A,11,10)(A,11,10)

9

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7

Algorithm

Quick Sort

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 10)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 10)QuickSort(A, 5, 6)QuickSort(A, 5, 4)QuickSort(A, 6, 6)QuickSort(A, 8, 10)QuickSort(A, 8, 7)QuickSort(A, 9, 10)QuickSort(A, 9, 9)QuickSort(A, 11, 10)

Partition(A, 1, 10)Partition(A, 1, 3)Partition(A, 5, 10)Partition(A, 5, 6)Partition(A, 8, 10)Partition(A, 9, 10)

(A,1,10)

(A,1,10)

(A,1,3)

(A,1,3)

(A,1,1)

(A,1,1) (A,3,3)

(A,3,3)

(A,5,10)

(A,5,10)

(A,5,6)

(A,5,6)

(A,5,4)

(A,5,4)

(A,6,6)

(A,6,6)

(A,8,10)

(A,8,10)

(A,8,7)(A,8,7) (A,9,10)(A,9,10)

(A,9,9)(A,9,9) (A,11,10)(A,11,10)

9

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10

Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7

Algorithm

Quick Sort

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 10)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 10)QuickSort(A, 5, 6)QuickSort(A, 5, 4)

QuickSort(A, 6, 6)

QuickSort(A, 8, 10)QuickSort(A, 8, 7)QuickSort(A, 9, 10)QuickSort(A, 9, 9)QuickSort(A, 11, 10)

Partition(A, 1, 10)Partition(A, 1, 3)Partition(A, 5, 10)Partition(A, 5, 6)Partition(A, 8, 10)Partition(A, 9, 10)

(A,1,10)

(A,1,10)

(A,1,3)

(A,1,3)

(A,1,1)

(A,1,1) (A,3,3)

(A,3,3)

(A,5,10)

(A,5,10)

(A,5,6)

(A,5,6)

(A,5,4)

(A,5,4)

(A,6,6)

(A,6,6)

(A,8,10)

(A,8,10)

(A,8,7)(A,8,7) (A,9,10)(A,9,10)

(A,9,9)(A,9,9) (A,11,10)(A,11,10)

9

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10

Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7

Algorithm

Quick Sort

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 10)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 10)QuickSort(A, 5, 6)QuickSort(A, 5, 4)QuickSort(A, 6, 6)QuickSort(A, 8, 10)QuickSort(A, 8, 7)QuickSort(A, 9, 10)QuickSort(A, 9, 9)QuickSort(A, 11, 10)

Partition(A, 1, 10)Partition(A, 1, 3)Partition(A, 5, 10)Partition(A, 5, 6)Partition(A, 8, 10)Partition(A, 9, 10)

(A,1,10)

(A,1,10)

(A,1,3)

(A,1,3)

(A,1,1)

(A,1,1) (A,3,3)

(A,3,3)

(A,5,10)

(A,5,10)

(A,5,6)

(A,5,6)

(A,5,4)

(A,5,4) (A,6,6)

(A,6,6)

(A,8,10)

(A,8,10)

(A,8,7)(A,8,7) (A,9,10)(A,9,10)

(A,9,9)(A,9,9) (A,11,10)(A,11,10)

9

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10

Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7

Algorithm

Quick Sort

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 10)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 10)QuickSort(A, 5, 6)QuickSort(A, 5, 4)QuickSort(A, 6, 6)

QuickSort(A, 8, 10)

QuickSort(A, 8, 7)QuickSort(A, 9, 10)QuickSort(A, 9, 9)QuickSort(A, 11, 10)

Partition(A, 1, 10)Partition(A, 1, 3)Partition(A, 5, 10)Partition(A, 5, 6)Partition(A, 8, 10)Partition(A, 9, 10)

(A,1,10)

(A,1,10)

(A,1,3)

(A,1,3)

(A,1,1)

(A,1,1) (A,3,3)

(A,3,3)

(A,5,10)

(A,5,10)

(A,5,6)

(A,5,6)

(A,5,4)

(A,5,4) (A,6,6)

(A,6,6)

(A,8,10)

(A,8,10)

(A,8,7)(A,8,7) (A,9,10)(A,9,10)

(A,9,9)(A,9,9) (A,11,10)(A,11,10)

9

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10

Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7

Algorithm

Quick Sort

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 10)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 10)QuickSort(A, 5, 6)QuickSort(A, 5, 4)QuickSort(A, 6, 6)

QuickSort(A, 8, 10)

QuickSort(A, 8, 7)QuickSort(A, 9, 10)QuickSort(A, 9, 9)QuickSort(A, 11, 10)

Partition(A, 1, 10)Partition(A, 1, 3)Partition(A, 5, 10)Partition(A, 5, 6)

Partition(A, 8, 10)

Partition(A, 9, 10)

(A,1,10)

(A,1,10)

(A,1,3)

(A,1,3)

(A,1,1)

(A,1,1) (A,3,3)

(A,3,3)

(A,5,10)

(A,5,10)

(A,5,6)

(A,5,6)

(A,5,4)

(A,5,4) (A,6,6)

(A,6,6)

(A,8,10)

(A,8,10)

(A,8,7)(A,8,7) (A,9,10)(A,9,10)

(A,9,9)(A,9,9) (A,11,10)(A,11,10)

9

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10

Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7

Algorithm

Quick Sort

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 10)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 10)QuickSort(A, 5, 6)QuickSort(A, 5, 4)QuickSort(A, 6, 6)

QuickSort(A, 8, 10)

QuickSort(A, 8, 7)QuickSort(A, 9, 10)QuickSort(A, 9, 9)QuickSort(A, 11, 10)

Partition(A, 1, 10)Partition(A, 1, 3)Partition(A, 5, 10)Partition(A, 5, 6)

Partition(A, 8, 10)

Partition(A, 9, 10)

(A,1,10)

(A,1,10)

(A,1,3)

(A,1,3)

(A,1,1)

(A,1,1) (A,3,3)

(A,3,3)

(A,5,10)

(A,5,10)

(A,5,6)

(A,5,6)

(A,5,4)

(A,5,4) (A,6,6)

(A,6,6)

(A,8,10)

(A,8,10)

(A,8,7)(A,8,7) (A,9,10)(A,9,10)

(A,9,9)(A,9,9) (A,11,10)(A,11,10)

9

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10

Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7

Algorithm

Quick Sort

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 10)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 10)QuickSort(A, 5, 6)QuickSort(A, 5, 4)QuickSort(A, 6, 6)QuickSort(A, 8, 10)QuickSort(A, 8, 7)QuickSort(A, 9, 10)QuickSort(A, 9, 9)QuickSort(A, 11, 10)

Partition(A, 1, 10)Partition(A, 1, 3)Partition(A, 5, 10)Partition(A, 5, 6)Partition(A, 8, 10)Partition(A, 9, 10)

(A,1,10)

(A,1,10)

(A,1,3)

(A,1,3)

(A,1,1)

(A,1,1) (A,3,3)

(A,3,3)

(A,5,10)

(A,5,10)

(A,5,6)

(A,5,6)

(A,5,4)

(A,5,4) (A,6,6)

(A,6,6)

(A,8,10)

(A,8,10)

(A,8,7)

(A,8,7)

(A,9,10)

(A,9,10)

(A,9,9)(A,9,9) (A,11,10)(A,11,10)

9

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7

Algorithm

Quick Sort

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 10)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 10)QuickSort(A, 5, 6)QuickSort(A, 5, 4)QuickSort(A, 6, 6)QuickSort(A, 8, 10)

QuickSort(A, 8, 7)

QuickSort(A, 9, 10)QuickSort(A, 9, 9)QuickSort(A, 11, 10)

Partition(A, 1, 10)Partition(A, 1, 3)Partition(A, 5, 10)Partition(A, 5, 6)Partition(A, 8, 10)Partition(A, 9, 10)

(A,1,10)

(A,1,10)

(A,1,3)

(A,1,3)

(A,1,1)

(A,1,1) (A,3,3)

(A,3,3)

(A,5,10)

(A,5,10)

(A,5,6)

(A,5,6)

(A,5,4)

(A,5,4) (A,6,6)

(A,6,6)

(A,8,10)

(A,8,10)

(A,8,7)

(A,8,7)

(A,9,10)

(A,9,10)

(A,9,9)(A,9,9) (A,11,10)(A,11,10)

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7

Algorithm

Quick Sort

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 10)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 10)QuickSort(A, 5, 6)QuickSort(A, 5, 4)QuickSort(A, 6, 6)QuickSort(A, 8, 10)QuickSort(A, 8, 7)QuickSort(A, 9, 10)QuickSort(A, 9, 9)QuickSort(A, 11, 10)

Partition(A, 1, 10)Partition(A, 1, 3)Partition(A, 5, 10)Partition(A, 5, 6)Partition(A, 8, 10)Partition(A, 9, 10)

(A,1,10)

(A,1,10)

(A,1,3)

(A,1,3)

(A,1,1)

(A,1,1) (A,3,3)

(A,3,3)

(A,5,10)

(A,5,10)

(A,5,6)

(A,5,6)

(A,5,4)

(A,5,4) (A,6,6)

(A,6,6)

(A,8,10)

(A,8,10)

(A,8,7)

(A,8,7)

(A,9,10)

(A,9,10)

(A,9,9)(A,9,9) (A,11,10)(A,11,10)

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7

Algorithm

Quick Sort

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 10)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 10)QuickSort(A, 5, 6)QuickSort(A, 5, 4)QuickSort(A, 6, 6)QuickSort(A, 8, 10)QuickSort(A, 8, 7)

QuickSort(A, 9, 10)

QuickSort(A, 9, 9)QuickSort(A, 11, 10)

Partition(A, 1, 10)Partition(A, 1, 3)Partition(A, 5, 10)Partition(A, 5, 6)Partition(A, 8, 10)Partition(A, 9, 10)

(A,1,10)

(A,1,10)

(A,1,3)

(A,1,3)

(A,1,1)

(A,1,1) (A,3,3)

(A,3,3)

(A,5,10)

(A,5,10)

(A,5,6)

(A,5,6)

(A,5,4)

(A,5,4) (A,6,6)

(A,6,6)

(A,8,10)

(A,8,10)

(A,8,7)

(A,8,7)

(A,9,10)

(A,9,10)

(A,9,9)(A,9,9) (A,11,10)(A,11,10)

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7

Algorithm

Quick Sort

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 10)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 10)QuickSort(A, 5, 6)QuickSort(A, 5, 4)QuickSort(A, 6, 6)QuickSort(A, 8, 10)QuickSort(A, 8, 7)

QuickSort(A, 9, 10)

QuickSort(A, 9, 9)QuickSort(A, 11, 10)

Partition(A, 1, 10)Partition(A, 1, 3)Partition(A, 5, 10)Partition(A, 5, 6)Partition(A, 8, 10)

Partition(A, 9, 10)(A,1,10)

(A,1,10)

(A,1,3)

(A,1,3)

(A,1,1)

(A,1,1) (A,3,3)

(A,3,3)

(A,5,10)

(A,5,10)

(A,5,6)

(A,5,6)

(A,5,4)

(A,5,4) (A,6,6)

(A,6,6)

(A,8,10)

(A,8,10)

(A,8,7)

(A,8,7)

(A,9,10)

(A,9,10)

(A,9,9)(A,9,9) (A,11,10)(A,11,10)

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7

Algorithm

Quick Sort

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 10)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 10)QuickSort(A, 5, 6)QuickSort(A, 5, 4)QuickSort(A, 6, 6)QuickSort(A, 8, 10)QuickSort(A, 8, 7)

QuickSort(A, 9, 10)

QuickSort(A, 9, 9)QuickSort(A, 11, 10)

Partition(A, 1, 10)Partition(A, 1, 3)Partition(A, 5, 10)Partition(A, 5, 6)Partition(A, 8, 10)

Partition(A, 9, 10)(A,1,10)

(A,1,10)

(A,1,3)

(A,1,3)

(A,1,1)

(A,1,1) (A,3,3)

(A,3,3)

(A,5,10)

(A,5,10)

(A,5,6)

(A,5,6)

(A,5,4)

(A,5,4) (A,6,6)

(A,6,6)

(A,8,10)

(A,8,10)

(A,8,7)

(A,8,7)

(A,9,10)

(A,9,10)

(A,9,9)(A,9,9) (A,11,10)(A,11,10)

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7

Algorithm

Quick Sort

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 10)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 10)QuickSort(A, 5, 6)QuickSort(A, 5, 4)QuickSort(A, 6, 6)QuickSort(A, 8, 10)QuickSort(A, 8, 7)QuickSort(A, 9, 10)QuickSort(A, 9, 9)QuickSort(A, 11, 10)

Partition(A, 1, 10)Partition(A, 1, 3)Partition(A, 5, 10)Partition(A, 5, 6)Partition(A, 8, 10)Partition(A, 9, 10)

(A,1,10)

(A,1,10)

(A,1,3)

(A,1,3)

(A,1,1)

(A,1,1) (A,3,3)

(A,3,3)

(A,5,10)

(A,5,10)

(A,5,6)

(A,5,6)

(A,5,4)

(A,5,4) (A,6,6)

(A,6,6)

(A,8,10)

(A,8,10)

(A,8,7)

(A,8,7)

(A,9,10)

(A,9,10)

(A,9,9)

(A,9,9)

(A,11,10)

(A,11,10)

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7

Algorithm

Quick Sort

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 10)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 10)QuickSort(A, 5, 6)QuickSort(A, 5, 4)QuickSort(A, 6, 6)QuickSort(A, 8, 10)QuickSort(A, 8, 7)QuickSort(A, 9, 10)

QuickSort(A, 9, 9)

QuickSort(A, 11, 10)

Partition(A, 1, 10)Partition(A, 1, 3)Partition(A, 5, 10)Partition(A, 5, 6)Partition(A, 8, 10)Partition(A, 9, 10)

(A,1,10)

(A,1,10)

(A,1,3)

(A,1,3)

(A,1,1)

(A,1,1) (A,3,3)

(A,3,3)

(A,5,10)

(A,5,10)

(A,5,6)

(A,5,6)

(A,5,4)

(A,5,4) (A,6,6)

(A,6,6)

(A,8,10)

(A,8,10)

(A,8,7)

(A,8,7)

(A,9,10)

(A,9,10)

(A,9,9)

(A,9,9)

(A,11,10)

(A,11,10)

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7

Algorithm

Quick Sort

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 10)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 10)QuickSort(A, 5, 6)QuickSort(A, 5, 4)QuickSort(A, 6, 6)QuickSort(A, 8, 10)QuickSort(A, 8, 7)QuickSort(A, 9, 10)QuickSort(A, 9, 9)QuickSort(A, 11, 10)

Partition(A, 1, 10)Partition(A, 1, 3)Partition(A, 5, 10)Partition(A, 5, 6)Partition(A, 8, 10)Partition(A, 9, 10)

(A,1,10)

(A,1,10)

(A,1,3)

(A,1,3)

(A,1,1)

(A,1,1) (A,3,3)

(A,3,3)

(A,5,10)

(A,5,10)

(A,5,6)

(A,5,6)

(A,5,4)

(A,5,4) (A,6,6)

(A,6,6)

(A,8,10)

(A,8,10)

(A,8,7)

(A,8,7)

(A,9,10)

(A,9,10)

(A,9,9)

(A,9,9) (A,11,10)(A,11,10)

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7

Algorithm

Quick Sort

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 10)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 10)QuickSort(A, 5, 6)QuickSort(A, 5, 4)QuickSort(A, 6, 6)QuickSort(A, 8, 10)QuickSort(A, 8, 7)QuickSort(A, 9, 10)QuickSort(A, 9, 9)

QuickSort(A, 11, 10)

Partition(A, 1, 10)Partition(A, 1, 3)Partition(A, 5, 10)Partition(A, 5, 6)Partition(A, 8, 10)Partition(A, 9, 10)

(A,1,10)

(A,1,10)

(A,1,3)

(A,1,3)

(A,1,1)

(A,1,1) (A,3,3)

(A,3,3)

(A,5,10)

(A,5,10)

(A,5,6)

(A,5,6)

(A,5,4)

(A,5,4) (A,6,6)

(A,6,6)

(A,8,10)

(A,8,10)

(A,8,7)

(A,8,7)

(A,9,10)

(A,9,10)

(A,9,9)

(A,9,9) (A,11,10)(A,11,10)

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7

Algorithm

Quick Sort

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 10)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 10)QuickSort(A, 5, 6)QuickSort(A, 5, 4)QuickSort(A, 6, 6)QuickSort(A, 8, 10)QuickSort(A, 8, 7)QuickSort(A, 9, 10)QuickSort(A, 9, 9)QuickSort(A, 11, 10)

Partition(A, 1, 10)Partition(A, 1, 3)Partition(A, 5, 10)Partition(A, 5, 6)Partition(A, 8, 10)Partition(A, 9, 10)(A,1,10)

(A,1,10)

(A,1,3)

(A,1,3)

(A,1,1)

(A,1,1) (A,3,3)

(A,3,3)

(A,5,10)

(A,5,10)

(A,5,6)

(A,5,6)

(A,5,4)

(A,5,4) (A,6,6)

(A,6,6)

(A,8,10)

(A,8,10)

(A,8,7)

(A,8,7) (A,9,10)

(A,9,10)

(A,9,9)

(A,9,9) (A,11,10)

(A,11,10)

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7

Worst-Case Behavior

QuickSort: Worst-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)

Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)

(A,1,8)(A,1,8)

(A,1,7)(A,1,7) (A,9,8)(A,9,8)

(A,1,6)(A,1,6) (A,8,7)(A,8,7)

(A,1,5)(A,1,5) (A,7,6)(A,7,6)

(A,1,4)(A,1,4) (A,6,5)(A,6,5)

(A,1,3)(A,1,3) (A,5,4)(A,5,4)

(A,1,2)(A,1,2) (A,4,3)(A,4,3)

(A,1,1)(A,1,1) (A,3,2)(A,3,2)

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7

Worst-Case Behavior

QuickSort: Worst-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)

Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)

(A,1,8)(A,1,8)

(A,1,7)(A,1,7) (A,9,8)(A,9,8)

(A,1,6)(A,1,6) (A,8,7)(A,8,7)

(A,1,5)(A,1,5) (A,7,6)(A,7,6)

(A,1,4)(A,1,4) (A,6,5)(A,6,5)

(A,1,3)(A,1,3) (A,5,4)(A,5,4)

(A,1,2)(A,1,2) (A,4,3)(A,4,3)

(A,1,1)(A,1,1) (A,3,2)(A,3,2)

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7

Worst-Case Behavior

QuickSort: Worst-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)

Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)

(A,1,8)

(A,1,8)

(A,1,7)(A,1,7) (A,9,8)(A,9,8)

(A,1,6)(A,1,6) (A,8,7)(A,8,7)

(A,1,5)(A,1,5) (A,7,6)(A,7,6)

(A,1,4)(A,1,4) (A,6,5)(A,6,5)

(A,1,3)(A,1,3) (A,5,4)(A,5,4)

(A,1,2)(A,1,2) (A,4,3)(A,4,3)

(A,1,1)(A,1,1) (A,3,2)(A,3,2)

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7

Worst-Case Behavior

QuickSort: Worst-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 8)

QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)

Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)

(A,1,8)

(A,1,8)

(A,1,7)(A,1,7) (A,9,8)(A,9,8)

(A,1,6)(A,1,6) (A,8,7)(A,8,7)

(A,1,5)(A,1,5) (A,7,6)(A,7,6)

(A,1,4)(A,1,4) (A,6,5)(A,6,5)

(A,1,3)(A,1,3) (A,5,4)(A,5,4)

(A,1,2)(A,1,2) (A,4,3)(A,4,3)

(A,1,1)(A,1,1) (A,3,2)(A,3,2)

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7

Worst-Case Behavior

QuickSort: Worst-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 8)

QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)

Partition(A, 1, 8)

Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)

(A,1,8)

(A,1,8)

(A,1,7)(A,1,7) (A,9,8)(A,9,8)

(A,1,6)(A,1,6) (A,8,7)(A,8,7)

(A,1,5)(A,1,5) (A,7,6)(A,7,6)

(A,1,4)(A,1,4) (A,6,5)(A,6,5)

(A,1,3)(A,1,3) (A,5,4)(A,5,4)

(A,1,2)(A,1,2) (A,4,3)(A,4,3)

(A,1,1)(A,1,1) (A,3,2)(A,3,2)

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7

Worst-Case Behavior

QuickSort: Worst-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 8)

QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)

Partition(A, 1, 8)

Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)

(A,1,8)

(A,1,8)

(A,1,7)(A,1,7) (A,9,8)(A,9,8)

(A,1,6)(A,1,6) (A,8,7)(A,8,7)

(A,1,5)(A,1,5) (A,7,6)(A,7,6)

(A,1,4)(A,1,4) (A,6,5)(A,6,5)

(A,1,3)(A,1,3) (A,5,4)(A,5,4)

(A,1,2)(A,1,2) (A,4,3)(A,4,3)

(A,1,1)(A,1,1) (A,3,2)(A,3,2)

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7

Worst-Case Behavior

QuickSort: Worst-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)

Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)

(A,1,8)

(A,1,8)

(A,1,7)

(A,1,7) (A,9,8)

(A,9,8)

(A,1,6)(A,1,6) (A,8,7)(A,8,7)

(A,1,5)(A,1,5) (A,7,6)(A,7,6)

(A,1,4)(A,1,4) (A,6,5)(A,6,5)

(A,1,3)(A,1,3) (A,5,4)(A,5,4)

(A,1,2)(A,1,2) (A,4,3)(A,4,3)

(A,1,1)(A,1,1) (A,3,2)(A,3,2)

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7

Worst-Case Behavior

QuickSort: Worst-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 8)QuickSort(A, 9, 8)

QuickSort(A, 1, 7)

QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)

Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)

(A,1,8)

(A,1,8)

(A,1,7)

(A,1,7) (A,9,8)

(A,9,8)

(A,1,6)(A,1,6) (A,8,7)(A,8,7)

(A,1,5)(A,1,5) (A,7,6)(A,7,6)

(A,1,4)(A,1,4) (A,6,5)(A,6,5)

(A,1,3)(A,1,3) (A,5,4)(A,5,4)

(A,1,2)(A,1,2) (A,4,3)(A,4,3)

(A,1,1)(A,1,1) (A,3,2)(A,3,2)

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7

Worst-Case Behavior

QuickSort: Worst-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 8)QuickSort(A, 9, 8)

QuickSort(A, 1, 7)

QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)

Partition(A, 1, 8)

Partition(A, 1, 7)

Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)

(A,1,8)

(A,1,8)

(A,1,7)

(A,1,7) (A,9,8)

(A,9,8)

(A,1,6)(A,1,6) (A,8,7)(A,8,7)

(A,1,5)(A,1,5) (A,7,6)(A,7,6)

(A,1,4)(A,1,4) (A,6,5)(A,6,5)

(A,1,3)(A,1,3) (A,5,4)(A,5,4)

(A,1,2)(A,1,2) (A,4,3)(A,4,3)

(A,1,1)(A,1,1) (A,3,2)(A,3,2)

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7

Worst-Case Behavior

QuickSort: Worst-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 8)QuickSort(A, 9, 8)

QuickSort(A, 1, 7)

QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)

Partition(A, 1, 8)

Partition(A, 1, 7)

Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)

(A,1,8)

(A,1,8)

(A,1,7)

(A,1,7) (A,9,8)

(A,9,8)

(A,1,6)(A,1,6) (A,8,7)(A,8,7)

(A,1,5)(A,1,5) (A,7,6)(A,7,6)

(A,1,4)(A,1,4) (A,6,5)(A,6,5)

(A,1,3)(A,1,3) (A,5,4)(A,5,4)

(A,1,2)(A,1,2) (A,4,3)(A,4,3)

(A,1,1)(A,1,1) (A,3,2)(A,3,2)

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7

Worst-Case Behavior

QuickSort: Worst-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)

Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)

(A,1,8)

(A,1,8)

(A,1,7)

(A,1,7) (A,9,8)

(A,9,8)

(A,1,6)

(A,1,6) (A,8,7)

(A,8,7)

(A,1,5)(A,1,5) (A,7,6)(A,7,6)

(A,1,4)(A,1,4) (A,6,5)(A,6,5)

(A,1,3)(A,1,3) (A,5,4)(A,5,4)

(A,1,2)(A,1,2) (A,4,3)(A,4,3)

(A,1,1)(A,1,1) (A,3,2)(A,3,2)

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7

Worst-Case Behavior

QuickSort: Worst-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)

QuickSort(A, 1, 6)

QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)

Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)

(A,1,8)

(A,1,8)

(A,1,7)

(A,1,7) (A,9,8)

(A,9,8)

(A,1,6)

(A,1,6) (A,8,7)

(A,8,7)

(A,1,5)(A,1,5) (A,7,6)(A,7,6)

(A,1,4)(A,1,4) (A,6,5)(A,6,5)

(A,1,3)(A,1,3) (A,5,4)(A,5,4)

(A,1,2)(A,1,2) (A,4,3)(A,4,3)

(A,1,1)(A,1,1) (A,3,2)(A,3,2)

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7

Worst-Case Behavior

QuickSort: Worst-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)

QuickSort(A, 1, 6)

QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)

Partition(A, 1, 8)Partition(A, 1, 7)

Partition(A, 1, 6)

Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)

(A,1,8)

(A,1,8)

(A,1,7)

(A,1,7) (A,9,8)

(A,9,8)

(A,1,6)

(A,1,6) (A,8,7)

(A,8,7)

(A,1,5)(A,1,5) (A,7,6)(A,7,6)

(A,1,4)(A,1,4) (A,6,5)(A,6,5)

(A,1,3)(A,1,3) (A,5,4)(A,5,4)

(A,1,2)(A,1,2) (A,4,3)(A,4,3)

(A,1,1)(A,1,1) (A,3,2)(A,3,2)

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7

Worst-Case Behavior

QuickSort: Worst-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)

QuickSort(A, 1, 6)

QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)

Partition(A, 1, 8)Partition(A, 1, 7)

Partition(A, 1, 6)

Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)

(A,1,8)

(A,1,8)

(A,1,7)

(A,1,7) (A,9,8)

(A,9,8)

(A,1,6)

(A,1,6) (A,8,7)

(A,8,7)

(A,1,5)(A,1,5) (A,7,6)(A,7,6)

(A,1,4)(A,1,4) (A,6,5)(A,6,5)

(A,1,3)(A,1,3) (A,5,4)(A,5,4)

(A,1,2)(A,1,2) (A,4,3)(A,4,3)

(A,1,1)(A,1,1) (A,3,2)(A,3,2)

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7

Worst-Case Behavior

QuickSort: Worst-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)

Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)

(A,1,8)

(A,1,8)

(A,1,7)

(A,1,7) (A,9,8)

(A,9,8)

(A,1,6)

(A,1,6) (A,8,7)

(A,8,7)

(A,1,5)

(A,1,5) (A,7,6)

(A,7,6)

(A,1,4)(A,1,4) (A,6,5)(A,6,5)

(A,1,3)(A,1,3) (A,5,4)(A,5,4)

(A,1,2)(A,1,2) (A,4,3)(A,4,3)

(A,1,1)(A,1,1) (A,3,2)(A,3,2)

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7

Worst-Case Behavior

QuickSort: Worst-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)

QuickSort(A, 1, 5)

QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)

Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)

(A,1,8)

(A,1,8)

(A,1,7)

(A,1,7) (A,9,8)

(A,9,8)

(A,1,6)

(A,1,6) (A,8,7)

(A,8,7)

(A,1,5)

(A,1,5) (A,7,6)

(A,7,6)

(A,1,4)(A,1,4) (A,6,5)(A,6,5)

(A,1,3)(A,1,3) (A,5,4)(A,5,4)

(A,1,2)(A,1,2) (A,4,3)(A,4,3)

(A,1,1)(A,1,1) (A,3,2)(A,3,2)

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7

Worst-Case Behavior

QuickSort: Worst-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)

QuickSort(A, 1, 5)

QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)

Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)

Partition(A, 1, 5)

Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)

(A,1,8)

(A,1,8)

(A,1,7)

(A,1,7) (A,9,8)

(A,9,8)

(A,1,6)

(A,1,6) (A,8,7)

(A,8,7)

(A,1,5)

(A,1,5) (A,7,6)

(A,7,6)

(A,1,4)(A,1,4) (A,6,5)(A,6,5)

(A,1,3)(A,1,3) (A,5,4)(A,5,4)

(A,1,2)(A,1,2) (A,4,3)(A,4,3)

(A,1,1)(A,1,1) (A,3,2)(A,3,2)

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7

Worst-Case Behavior

QuickSort: Worst-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)

QuickSort(A, 1, 5)

QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)

Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)

Partition(A, 1, 5)

Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)

(A,1,8)

(A,1,8)

(A,1,7)

(A,1,7) (A,9,8)

(A,9,8)

(A,1,6)

(A,1,6) (A,8,7)

(A,8,7)

(A,1,5)

(A,1,5) (A,7,6)

(A,7,6)

(A,1,4)(A,1,4) (A,6,5)(A,6,5)

(A,1,3)(A,1,3) (A,5,4)(A,5,4)

(A,1,2)(A,1,2) (A,4,3)(A,4,3)

(A,1,1)(A,1,1) (A,3,2)(A,3,2)

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7

Worst-Case Behavior

QuickSort: Worst-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)

Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)

(A,1,8)

(A,1,8)

(A,1,7)

(A,1,7) (A,9,8)

(A,9,8)

(A,1,6)

(A,1,6) (A,8,7)

(A,8,7)

(A,1,5)

(A,1,5) (A,7,6)

(A,7,6)

(A,1,4)

(A,1,4) (A,6,5)

(A,6,5)

(A,1,3)(A,1,3) (A,5,4)(A,5,4)

(A,1,2)(A,1,2) (A,4,3)(A,4,3)

(A,1,1)(A,1,1) (A,3,2)(A,3,2)

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7

Worst-Case Behavior

QuickSort: Worst-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)

QuickSort(A, 1, 4)

QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)

Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)

(A,1,8)

(A,1,8)

(A,1,7)

(A,1,7) (A,9,8)

(A,9,8)

(A,1,6)

(A,1,6) (A,8,7)

(A,8,7)

(A,1,5)

(A,1,5) (A,7,6)

(A,7,6)

(A,1,4)

(A,1,4) (A,6,5)

(A,6,5)

(A,1,3)(A,1,3) (A,5,4)(A,5,4)

(A,1,2)(A,1,2) (A,4,3)(A,4,3)

(A,1,1)(A,1,1) (A,3,2)(A,3,2)

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7

Worst-Case Behavior

QuickSort: Worst-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)

QuickSort(A, 1, 4)

QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)

Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)

Partition(A, 1, 4)

Partition(A, 1, 3)Partition(A, 1, 2)

(A,1,8)

(A,1,8)

(A,1,7)

(A,1,7) (A,9,8)

(A,9,8)

(A,1,6)

(A,1,6) (A,8,7)

(A,8,7)

(A,1,5)

(A,1,5) (A,7,6)

(A,7,6)

(A,1,4)

(A,1,4) (A,6,5)

(A,6,5)

(A,1,3)(A,1,3) (A,5,4)(A,5,4)

(A,1,2)(A,1,2) (A,4,3)(A,4,3)

(A,1,1)(A,1,1) (A,3,2)(A,3,2)

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7

Worst-Case Behavior

QuickSort: Worst-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)

QuickSort(A, 1, 4)

QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)

Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)

Partition(A, 1, 4)

Partition(A, 1, 3)Partition(A, 1, 2)

(A,1,8)

(A,1,8)

(A,1,7)

(A,1,7) (A,9,8)

(A,9,8)

(A,1,6)

(A,1,6) (A,8,7)

(A,8,7)

(A,1,5)

(A,1,5) (A,7,6)

(A,7,6)

(A,1,4)

(A,1,4) (A,6,5)

(A,6,5)

(A,1,3)(A,1,3) (A,5,4)(A,5,4)

(A,1,2)(A,1,2) (A,4,3)(A,4,3)

(A,1,1)(A,1,1) (A,3,2)(A,3,2)

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7

Worst-Case Behavior

QuickSort: Worst-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)

Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)

(A,1,8)

(A,1,8)

(A,1,7)

(A,1,7) (A,9,8)

(A,9,8)

(A,1,6)

(A,1,6) (A,8,7)

(A,8,7)

(A,1,5)

(A,1,5) (A,7,6)

(A,7,6)

(A,1,4)

(A,1,4) (A,6,5)

(A,6,5)

(A,1,3)

(A,1,3) (A,5,4)

(A,5,4)

(A,1,2)(A,1,2) (A,4,3)(A,4,3)

(A,1,1)(A,1,1) (A,3,2)(A,3,2)

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7

Worst-Case Behavior

QuickSort: Worst-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)

QuickSort(A, 1, 3)

QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)

Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)

(A,1,8)

(A,1,8)

(A,1,7)

(A,1,7) (A,9,8)

(A,9,8)

(A,1,6)

(A,1,6) (A,8,7)

(A,8,7)

(A,1,5)

(A,1,5) (A,7,6)

(A,7,6)

(A,1,4)

(A,1,4) (A,6,5)

(A,6,5)

(A,1,3)

(A,1,3) (A,5,4)

(A,5,4)

(A,1,2)(A,1,2) (A,4,3)(A,4,3)

(A,1,1)(A,1,1) (A,3,2)(A,3,2)

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7

Worst-Case Behavior

QuickSort: Worst-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)

QuickSort(A, 1, 3)

QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)

Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)

Partition(A, 1, 3)

Partition(A, 1, 2)

(A,1,8)

(A,1,8)

(A,1,7)

(A,1,7) (A,9,8)

(A,9,8)

(A,1,6)

(A,1,6) (A,8,7)

(A,8,7)

(A,1,5)

(A,1,5) (A,7,6)

(A,7,6)

(A,1,4)

(A,1,4) (A,6,5)

(A,6,5)

(A,1,3)

(A,1,3) (A,5,4)

(A,5,4)

(A,1,2)(A,1,2) (A,4,3)(A,4,3)

(A,1,1)(A,1,1) (A,3,2)(A,3,2)

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7

Worst-Case Behavior

QuickSort: Worst-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)

QuickSort(A, 1, 3)

QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)

Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)

Partition(A, 1, 3)

Partition(A, 1, 2)

(A,1,8)

(A,1,8)

(A,1,7)

(A,1,7) (A,9,8)

(A,9,8)

(A,1,6)

(A,1,6) (A,8,7)

(A,8,7)

(A,1,5)

(A,1,5) (A,7,6)

(A,7,6)

(A,1,4)

(A,1,4) (A,6,5)

(A,6,5)

(A,1,3)

(A,1,3) (A,5,4)

(A,5,4)

(A,1,2)(A,1,2) (A,4,3)(A,4,3)

(A,1,1)(A,1,1) (A,3,2)(A,3,2)

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7

Worst-Case Behavior

QuickSort: Worst-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)

Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)

(A,1,8)

(A,1,8)

(A,1,7)

(A,1,7) (A,9,8)

(A,9,8)

(A,1,6)

(A,1,6) (A,8,7)

(A,8,7)

(A,1,5)

(A,1,5) (A,7,6)

(A,7,6)

(A,1,4)

(A,1,4) (A,6,5)

(A,6,5)

(A,1,3)

(A,1,3) (A,5,4)

(A,5,4)

(A,1,2)

(A,1,2) (A,4,3)

(A,4,3)

(A,1,1)(A,1,1) (A,3,2)(A,3,2)

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7

Worst-Case Behavior

QuickSort: Worst-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)

QuickSort(A, 1, 2)

QuickSort(A, 3, 2)QuickSort(A, 1, 1)

Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)

(A,1,8)

(A,1,8)

(A,1,7)

(A,1,7) (A,9,8)

(A,9,8)

(A,1,6)

(A,1,6) (A,8,7)

(A,8,7)

(A,1,5)

(A,1,5) (A,7,6)

(A,7,6)

(A,1,4)

(A,1,4) (A,6,5)

(A,6,5)

(A,1,3)

(A,1,3) (A,5,4)

(A,5,4)

(A,1,2)

(A,1,2) (A,4,3)

(A,4,3)

(A,1,1)(A,1,1) (A,3,2)(A,3,2)

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7

Worst-Case Behavior

QuickSort: Worst-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)

QuickSort(A, 1, 2)

QuickSort(A, 3, 2)QuickSort(A, 1, 1)

Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)

Partition(A, 1, 2)

(A,1,8)

(A,1,8)

(A,1,7)

(A,1,7) (A,9,8)

(A,9,8)

(A,1,6)

(A,1,6) (A,8,7)

(A,8,7)

(A,1,5)

(A,1,5) (A,7,6)

(A,7,6)

(A,1,4)

(A,1,4) (A,6,5)

(A,6,5)

(A,1,3)

(A,1,3) (A,5,4)

(A,5,4)

(A,1,2)

(A,1,2) (A,4,3)

(A,4,3)

(A,1,1)(A,1,1) (A,3,2)(A,3,2)

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7

Worst-Case Behavior

QuickSort: Worst-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)

QuickSort(A, 1, 2)

QuickSort(A, 3, 2)QuickSort(A, 1, 1)

Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)

Partition(A, 1, 2)

(A,1,8)

(A,1,8)

(A,1,7)

(A,1,7) (A,9,8)

(A,9,8)

(A,1,6)

(A,1,6) (A,8,7)

(A,8,7)

(A,1,5)

(A,1,5) (A,7,6)

(A,7,6)

(A,1,4)

(A,1,4) (A,6,5)

(A,6,5)

(A,1,3)

(A,1,3) (A,5,4)

(A,5,4)

(A,1,2)

(A,1,2) (A,4,3)

(A,4,3)

(A,1,1)(A,1,1) (A,3,2)(A,3,2)

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7

Worst-Case Behavior

QuickSort: Worst-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)

Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)

(A,1,8)

(A,1,8)

(A,1,7)

(A,1,7) (A,9,8)

(A,9,8)

(A,1,6)

(A,1,6) (A,8,7)

(A,8,7)

(A,1,5)

(A,1,5) (A,7,6)

(A,7,6)

(A,1,4)

(A,1,4) (A,6,5)

(A,6,5)

(A,1,3)

(A,1,3) (A,5,4)

(A,5,4)

(A,1,2)

(A,1,2) (A,4,3)

(A,4,3)

(A,1,1)

(A,1,1) (A,3,2)

(A,3,2)

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7

Worst-Case Behavior

QuickSort: Worst-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)

QuickSort(A, 1, 1)

Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)

(A,1,8)

(A,1,8)

(A,1,7)

(A,1,7) (A,9,8)

(A,9,8)

(A,1,6)

(A,1,6) (A,8,7)

(A,8,7)

(A,1,5)

(A,1,5) (A,7,6)

(A,7,6)

(A,1,4)

(A,1,4) (A,6,5)

(A,6,5)

(A,1,3)

(A,1,3) (A,5,4)

(A,5,4)

(A,1,2)

(A,1,2) (A,4,3)

(A,4,3)

(A,1,1)

(A,1,1) (A,3,2)

(A,3,2)

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7

Worst-Case Behavior

QuickSort: Worst-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)

Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)

(A,1,8)

(A,1,8)

(A,1,7)

(A,1,7) (A,9,8)

(A,9,8)

(A,1,6)

(A,1,6) (A,8,7)

(A,8,7)

(A,1,5)

(A,1,5) (A,7,6)

(A,7,6)

(A,1,4)

(A,1,4) (A,6,5)

(A,6,5)

(A,1,3)

(A,1,3) (A,5,4)

(A,5,4)

(A,1,2)

(A,1,2) (A,4,3)

(A,4,3)

(A,1,1)

(A,1,1) (A,3,2)

(A,3,2)

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7

Worst-Case Behavior

QuickSort: Worst-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)

QuickSort(A, 3, 2)

QuickSort(A, 1, 1)

Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)

(A,1,8)

(A,1,8)

(A,1,7)

(A,1,7) (A,9,8)

(A,9,8)

(A,1,6)

(A,1,6) (A,8,7)

(A,8,7)

(A,1,5)

(A,1,5) (A,7,6)

(A,7,6)

(A,1,4)

(A,1,4) (A,6,5)

(A,6,5)

(A,1,3)

(A,1,3) (A,5,4)

(A,5,4)

(A,1,2)

(A,1,2) (A,4,3)

(A,4,3)

(A,1,1)

(A,1,1) (A,3,2)

(A,3,2)

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7

Worst-Case Behavior

QuickSort: Worst-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)

Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)

(A,1,8)

(A,1,8)

(A,1,7)

(A,1,7) (A,9,8)

(A,9,8)

(A,1,6)

(A,1,6) (A,8,7)

(A,8,7)

(A,1,5)

(A,1,5) (A,7,6)

(A,7,6)

(A,1,4)

(A,1,4) (A,6,5)

(A,6,5)

(A,1,3)

(A,1,3) (A,5,4)

(A,5,4)

(A,1,2)

(A,1,2) (A,4,3)

(A,4,3)

(A,1,1)

(A,1,1)

(A,3,2)

(A,3,2)

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7

Worst-Case Behavior

QuickSort: Worst-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)

QuickSort(A, 4, 3)

QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)

Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)

(A,1,8)

(A,1,8)

(A,1,7)

(A,1,7) (A,9,8)

(A,9,8)

(A,1,6)

(A,1,6) (A,8,7)

(A,8,7)

(A,1,5)

(A,1,5) (A,7,6)

(A,7,6)

(A,1,4)

(A,1,4) (A,6,5)

(A,6,5)

(A,1,3)

(A,1,3) (A,5,4)

(A,5,4)

(A,1,2)

(A,1,2) (A,4,3)

(A,4,3)

(A,1,1)

(A,1,1)

(A,3,2)

(A,3,2)

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7

Worst-Case Behavior

QuickSort: Worst-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)

Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)

(A,1,8)

(A,1,8)

(A,1,7)

(A,1,7) (A,9,8)

(A,9,8)

(A,1,6)

(A,1,6) (A,8,7)

(A,8,7)

(A,1,5)

(A,1,5) (A,7,6)

(A,7,6)

(A,1,4)

(A,1,4) (A,6,5)

(A,6,5)

(A,1,3)

(A,1,3) (A,5,4)

(A,5,4)

(A,1,2)

(A,1,2)

(A,4,3)

(A,4,3)

(A,1,1)

(A,1,1)

(A,3,2)

(A,3,2)

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7

Worst-Case Behavior

QuickSort: Worst-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)

QuickSort(A, 5, 4)

QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)

Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)

(A,1,8)

(A,1,8)

(A,1,7)

(A,1,7) (A,9,8)

(A,9,8)

(A,1,6)

(A,1,6) (A,8,7)

(A,8,7)

(A,1,5)

(A,1,5) (A,7,6)

(A,7,6)

(A,1,4)

(A,1,4) (A,6,5)

(A,6,5)

(A,1,3)

(A,1,3) (A,5,4)

(A,5,4)

(A,1,2)

(A,1,2)

(A,4,3)

(A,4,3)

(A,1,1)

(A,1,1)

(A,3,2)

(A,3,2)

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7

Worst-Case Behavior

QuickSort: Worst-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)

Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)

(A,1,8)

(A,1,8)

(A,1,7)

(A,1,7) (A,9,8)

(A,9,8)

(A,1,6)

(A,1,6) (A,8,7)

(A,8,7)

(A,1,5)

(A,1,5) (A,7,6)

(A,7,6)

(A,1,4)

(A,1,4) (A,6,5)

(A,6,5)

(A,1,3)

(A,1,3)

(A,5,4)

(A,5,4)

(A,1,2)

(A,1,2)

(A,4,3)

(A,4,3)

(A,1,1)

(A,1,1)

(A,3,2)

(A,3,2)

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7

Worst-Case Behavior

QuickSort: Worst-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)

QuickSort(A, 6, 5)

QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)

Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)

(A,1,8)

(A,1,8)

(A,1,7)

(A,1,7) (A,9,8)

(A,9,8)

(A,1,6)

(A,1,6) (A,8,7)

(A,8,7)

(A,1,5)

(A,1,5) (A,7,6)

(A,7,6)

(A,1,4)

(A,1,4) (A,6,5)

(A,6,5)

(A,1,3)

(A,1,3)

(A,5,4)

(A,5,4)

(A,1,2)

(A,1,2)

(A,4,3)

(A,4,3)

(A,1,1)

(A,1,1)

(A,3,2)

(A,3,2)

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7

Worst-Case Behavior

QuickSort: Worst-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)

Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)

(A,1,8)

(A,1,8)

(A,1,7)

(A,1,7) (A,9,8)

(A,9,8)

(A,1,6)

(A,1,6) (A,8,7)

(A,8,7)

(A,1,5)

(A,1,5) (A,7,6)

(A,7,6)

(A,1,4)

(A,1,4)

(A,6,5)

(A,6,5)

(A,1,3)

(A,1,3)

(A,5,4)

(A,5,4)

(A,1,2)

(A,1,2)

(A,4,3)

(A,4,3)

(A,1,1)

(A,1,1)

(A,3,2)

(A,3,2)

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7

Worst-Case Behavior

QuickSort: Worst-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)

QuickSort(A, 7, 6)

QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)

Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)

(A,1,8)

(A,1,8)

(A,1,7)

(A,1,7) (A,9,8)

(A,9,8)

(A,1,6)

(A,1,6) (A,8,7)

(A,8,7)

(A,1,5)

(A,1,5) (A,7,6)

(A,7,6)

(A,1,4)

(A,1,4)

(A,6,5)

(A,6,5)

(A,1,3)

(A,1,3)

(A,5,4)

(A,5,4)

(A,1,2)

(A,1,2)

(A,4,3)

(A,4,3)

(A,1,1)

(A,1,1)

(A,3,2)

(A,3,2)

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7

Worst-Case Behavior

QuickSort: Worst-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)

Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)

(A,1,8)

(A,1,8)

(A,1,7)

(A,1,7) (A,9,8)

(A,9,8)

(A,1,6)

(A,1,6) (A,8,7)

(A,8,7)

(A,1,5)

(A,1,5)

(A,7,6)

(A,7,6)

(A,1,4)

(A,1,4)

(A,6,5)

(A,6,5)

(A,1,3)

(A,1,3)

(A,5,4)

(A,5,4)

(A,1,2)

(A,1,2)

(A,4,3)

(A,4,3)

(A,1,1)

(A,1,1)

(A,3,2)

(A,3,2)

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7

Worst-Case Behavior

QuickSort: Worst-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)

QuickSort(A, 8, 7)

QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)

Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)

(A,1,8)

(A,1,8)

(A,1,7)

(A,1,7) (A,9,8)

(A,9,8)

(A,1,6)

(A,1,6) (A,8,7)

(A,8,7)

(A,1,5)

(A,1,5)

(A,7,6)

(A,7,6)

(A,1,4)

(A,1,4)

(A,6,5)

(A,6,5)

(A,1,3)

(A,1,3)

(A,5,4)

(A,5,4)

(A,1,2)

(A,1,2)

(A,4,3)

(A,4,3)

(A,1,1)

(A,1,1)

(A,3,2)

(A,3,2)

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7

Worst-Case Behavior

QuickSort: Worst-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)

Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)

(A,1,8)

(A,1,8)

(A,1,7)

(A,1,7) (A,9,8)

(A,9,8)

(A,1,6)

(A,1,6)

(A,8,7)

(A,8,7)

(A,1,5)

(A,1,5)

(A,7,6)

(A,7,6)

(A,1,4)

(A,1,4)

(A,6,5)

(A,6,5)

(A,1,3)

(A,1,3)

(A,5,4)

(A,5,4)

(A,1,2)

(A,1,2)

(A,4,3)

(A,4,3)

(A,1,1)

(A,1,1)

(A,3,2)

(A,3,2)

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7

Worst-Case Behavior

QuickSort: Worst-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 8)

QuickSort(A, 9, 8)

QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)

Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)

(A,1,8)

(A,1,8)

(A,1,7)

(A,1,7) (A,9,8)

(A,9,8)

(A,1,6)

(A,1,6)

(A,8,7)

(A,8,7)

(A,1,5)

(A,1,5)

(A,7,6)

(A,7,6)

(A,1,4)

(A,1,4)

(A,6,5)

(A,6,5)

(A,1,3)

(A,1,3)

(A,5,4)

(A,5,4)

(A,1,2)

(A,1,2)

(A,4,3)

(A,4,3)

(A,1,1)

(A,1,1)

(A,3,2)

(A,3,2)

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7

Worst-Case Behavior

QuickSort: Worst-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)

Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)

(A,1,8)

(A,1,8)

(A,1,7)

(A,1,7)

(A,9,8)

(A,9,8)

(A,1,6)

(A,1,6)

(A,8,7)

(A,8,7)

(A,1,5)

(A,1,5)

(A,7,6)

(A,7,6)

(A,1,4)

(A,1,4)

(A,6,5)

(A,6,5)

(A,1,3)

(A,1,3)

(A,5,4)

(A,5,4)

(A,1,2)

(A,1,2)

(A,4,3)

(A,4,3)

(A,1,1)

(A,1,1)

(A,3,2)

(A,3,2)

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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7

Best-Case Behavior

QuickSort: Best-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 15)QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)

Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)

3

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13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

(A,1,15)(A,1,15)

(A,1,7)(A,1,7)

(A,1,3)(A,1,3)

(A,1,1)(A,1,1) (A,3,3)(A,3,3)

(A,5,7)(A,5,7)

(A,5,5)(A,5,5) (A,7,7)(A,7,7)

(A,9,15)(A,9,15)

(A,9,11)(A,9,11)

(A,9,9)(A,9,9) (A,11,11)(A,11,11)

(A,13,15)(A,13,15)

(A,13,13)(A,13,13) (A,15,15)(A,15,15)

Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7

Best-Case Behavior

QuickSort: Best-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 15)QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)

Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

(A,1,15)(A,1,15)

(A,1,7)(A,1,7)

(A,1,3)(A,1,3)

(A,1,1)(A,1,1) (A,3,3)(A,3,3)

(A,5,7)(A,5,7)

(A,5,5)(A,5,5) (A,7,7)(A,7,7)

(A,9,15)(A,9,15)

(A,9,11)(A,9,11)

(A,9,9)(A,9,9) (A,11,11)(A,11,11)

(A,13,15)(A,13,15)

(A,13,13)(A,13,13) (A,15,15)(A,15,15)

Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7

Best-Case Behavior

QuickSort: Best-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 15)QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)

Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

(A,1,15)

(A,1,15)

(A,1,7)(A,1,7)

(A,1,3)(A,1,3)

(A,1,1)(A,1,1) (A,3,3)(A,3,3)

(A,5,7)(A,5,7)

(A,5,5)(A,5,5) (A,7,7)(A,7,7)

(A,9,15)(A,9,15)

(A,9,11)(A,9,11)

(A,9,9)(A,9,9) (A,11,11)(A,11,11)

(A,13,15)(A,13,15)

(A,13,13)(A,13,13) (A,15,15)(A,15,15)

Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7

Best-Case Behavior

QuickSort: Best-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 15)

QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)

Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

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12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

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13

11

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15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

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13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

(A,1,15)

(A,1,15)

(A,1,7)(A,1,7)

(A,1,3)(A,1,3)

(A,1,1)(A,1,1) (A,3,3)(A,3,3)

(A,5,7)(A,5,7)

(A,5,5)(A,5,5) (A,7,7)(A,7,7)

(A,9,15)(A,9,15)

(A,9,11)(A,9,11)

(A,9,9)(A,9,9) (A,11,11)(A,11,11)

(A,13,15)(A,13,15)

(A,13,13)(A,13,13) (A,15,15)(A,15,15)

Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7

Best-Case Behavior

QuickSort: Best-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 15)

QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)

Partition(A, 1, 15)

Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

(A,1,15)

(A,1,15)

(A,1,7)(A,1,7)

(A,1,3)(A,1,3)

(A,1,1)(A,1,1) (A,3,3)(A,3,3)

(A,5,7)(A,5,7)

(A,5,5)(A,5,5) (A,7,7)(A,7,7)

(A,9,15)(A,9,15)

(A,9,11)(A,9,11)

(A,9,9)(A,9,9) (A,11,11)(A,11,11)

(A,13,15)(A,13,15)

(A,13,13)(A,13,13) (A,15,15)(A,15,15)

Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7

Best-Case Behavior

QuickSort: Best-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 15)

QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)

Partition(A, 1, 15)

Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

(A,1,15)

(A,1,15)

(A,1,7)(A,1,7)

(A,1,3)(A,1,3)

(A,1,1)(A,1,1) (A,3,3)(A,3,3)

(A,5,7)(A,5,7)

(A,5,5)(A,5,5) (A,7,7)(A,7,7)

(A,9,15)(A,9,15)

(A,9,11)(A,9,11)

(A,9,9)(A,9,9) (A,11,11)(A,11,11)

(A,13,15)(A,13,15)

(A,13,13)(A,13,13) (A,15,15)(A,15,15)

Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7

Best-Case Behavior

QuickSort: Best-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 15)QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)

Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

(A,1,15)

(A,1,15)

(A,1,7)

(A,1,7)

(A,1,3)(A,1,3)

(A,1,1)(A,1,1) (A,3,3)(A,3,3)

(A,5,7)(A,5,7)

(A,5,5)(A,5,5) (A,7,7)(A,7,7)

(A,9,15)

(A,9,15)

(A,9,11)(A,9,11)

(A,9,9)(A,9,9) (A,11,11)(A,11,11)

(A,13,15)(A,13,15)

(A,13,13)(A,13,13) (A,15,15)(A,15,15)

Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7

Best-Case Behavior

QuickSort: Best-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 15)

QuickSort(A, 1, 7)

QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)

Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

(A,1,15)

(A,1,15)

(A,1,7)

(A,1,7)

(A,1,3)(A,1,3)

(A,1,1)(A,1,1) (A,3,3)(A,3,3)

(A,5,7)(A,5,7)

(A,5,5)(A,5,5) (A,7,7)(A,7,7)

(A,9,15)

(A,9,15)

(A,9,11)(A,9,11)

(A,9,9)(A,9,9) (A,11,11)(A,11,11)

(A,13,15)(A,13,15)

(A,13,13)(A,13,13) (A,15,15)(A,15,15)

Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7

Best-Case Behavior

QuickSort: Best-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 15)

QuickSort(A, 1, 7)

QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)

Partition(A, 1, 15)

Partition(A, 1, 7)

Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

(A,1,15)

(A,1,15)

(A,1,7)

(A,1,7)

(A,1,3)(A,1,3)

(A,1,1)(A,1,1) (A,3,3)(A,3,3)

(A,5,7)(A,5,7)

(A,5,5)(A,5,5) (A,7,7)(A,7,7)

(A,9,15)

(A,9,15)

(A,9,11)(A,9,11)

(A,9,9)(A,9,9) (A,11,11)(A,11,11)

(A,13,15)(A,13,15)

(A,13,13)(A,13,13) (A,15,15)(A,15,15)

Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7

Best-Case Behavior

QuickSort: Best-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 15)

QuickSort(A, 1, 7)

QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)

Partition(A, 1, 15)

Partition(A, 1, 7)

Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

(A,1,15)

(A,1,15)

(A,1,7)

(A,1,7)

(A,1,3)(A,1,3)

(A,1,1)(A,1,1) (A,3,3)(A,3,3)

(A,5,7)(A,5,7)

(A,5,5)(A,5,5) (A,7,7)(A,7,7)

(A,9,15)

(A,9,15)

(A,9,11)(A,9,11)

(A,9,9)(A,9,9) (A,11,11)(A,11,11)

(A,13,15)(A,13,15)

(A,13,13)(A,13,13) (A,15,15)(A,15,15)

Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7

Best-Case Behavior

QuickSort: Best-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 15)QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)

Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

(A,1,15)

(A,1,15)

(A,1,7)

(A,1,7)

(A,1,3)

(A,1,3)

(A,1,1)(A,1,1) (A,3,3)(A,3,3)

(A,5,7)

(A,5,7)

(A,5,5)(A,5,5) (A,7,7)(A,7,7)

(A,9,15)

(A,9,15)

(A,9,11)(A,9,11)

(A,9,9)(A,9,9) (A,11,11)(A,11,11)

(A,13,15)(A,13,15)

(A,13,13)(A,13,13) (A,15,15)(A,15,15)

Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7

Best-Case Behavior

QuickSort: Best-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 15)QuickSort(A, 1, 7)

QuickSort(A, 1, 3)

QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)

Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

(A,1,15)

(A,1,15)

(A,1,7)

(A,1,7)

(A,1,3)

(A,1,3)

(A,1,1)(A,1,1) (A,3,3)(A,3,3)

(A,5,7)

(A,5,7)

(A,5,5)(A,5,5) (A,7,7)(A,7,7)

(A,9,15)

(A,9,15)

(A,9,11)(A,9,11)

(A,9,9)(A,9,9) (A,11,11)(A,11,11)

(A,13,15)(A,13,15)

(A,13,13)(A,13,13) (A,15,15)(A,15,15)

Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7

Best-Case Behavior

QuickSort: Best-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 15)QuickSort(A, 1, 7)

QuickSort(A, 1, 3)

QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)

Partition(A, 1, 15)Partition(A, 1, 7)

Partition(A, 1, 3)

Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

(A,1,15)

(A,1,15)

(A,1,7)

(A,1,7)

(A,1,3)

(A,1,3)

(A,1,1)(A,1,1) (A,3,3)(A,3,3)

(A,5,7)

(A,5,7)

(A,5,5)(A,5,5) (A,7,7)(A,7,7)

(A,9,15)

(A,9,15)

(A,9,11)(A,9,11)

(A,9,9)(A,9,9) (A,11,11)(A,11,11)

(A,13,15)(A,13,15)

(A,13,13)(A,13,13) (A,15,15)(A,15,15)

Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7

Best-Case Behavior

QuickSort: Best-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 15)QuickSort(A, 1, 7)

QuickSort(A, 1, 3)

QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)

Partition(A, 1, 15)Partition(A, 1, 7)

Partition(A, 1, 3)

Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

(A,1,15)

(A,1,15)

(A,1,7)

(A,1,7)

(A,1,3)

(A,1,3)

(A,1,1)(A,1,1) (A,3,3)(A,3,3)

(A,5,7)

(A,5,7)

(A,5,5)(A,5,5) (A,7,7)(A,7,7)

(A,9,15)

(A,9,15)

(A,9,11)(A,9,11)

(A,9,9)(A,9,9) (A,11,11)(A,11,11)

(A,13,15)(A,13,15)

(A,13,13)(A,13,13) (A,15,15)(A,15,15)

Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7

Best-Case Behavior

QuickSort: Best-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 15)QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)

Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

(A,1,15)

(A,1,15)

(A,1,7)

(A,1,7)

(A,1,3)

(A,1,3)

(A,1,1)

(A,1,1) (A,3,3)

(A,3,3)

(A,5,7)

(A,5,7)

(A,5,5)(A,5,5) (A,7,7)(A,7,7)

(A,9,15)

(A,9,15)

(A,9,11)(A,9,11)

(A,9,9)(A,9,9) (A,11,11)(A,11,11)

(A,13,15)(A,13,15)

(A,13,13)(A,13,13) (A,15,15)(A,15,15)

Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7

Best-Case Behavior

QuickSort: Best-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 15)QuickSort(A, 1, 7)QuickSort(A, 1, 3)

QuickSort(A, 1, 1)

QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)

Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

(A,1,15)

(A,1,15)

(A,1,7)

(A,1,7)

(A,1,3)

(A,1,3)

(A,1,1)

(A,1,1) (A,3,3)

(A,3,3)

(A,5,7)

(A,5,7)

(A,5,5)(A,5,5) (A,7,7)(A,7,7)

(A,9,15)

(A,9,15)

(A,9,11)(A,9,11)

(A,9,9)(A,9,9) (A,11,11)(A,11,11)

(A,13,15)(A,13,15)

(A,13,13)(A,13,13) (A,15,15)(A,15,15)

Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7

Best-Case Behavior

QuickSort: Best-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 15)QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)

Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

(A,1,15)

(A,1,15)

(A,1,7)

(A,1,7)

(A,1,3)

(A,1,3)

(A,1,1)

(A,1,1) (A,3,3)

(A,3,3)

(A,5,7)

(A,5,7)

(A,5,5)(A,5,5) (A,7,7)(A,7,7)

(A,9,15)

(A,9,15)

(A,9,11)(A,9,11)

(A,9,9)(A,9,9) (A,11,11)(A,11,11)

(A,13,15)(A,13,15)

(A,13,13)(A,13,13) (A,15,15)(A,15,15)

Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7

Best-Case Behavior

QuickSort: Best-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 15)QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)

QuickSort(A, 3, 3)

QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)

Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

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13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

(A,1,15)

(A,1,15)

(A,1,7)

(A,1,7)

(A,1,3)

(A,1,3)

(A,1,1)

(A,1,1) (A,3,3)

(A,3,3)

(A,5,7)

(A,5,7)

(A,5,5)(A,5,5) (A,7,7)(A,7,7)

(A,9,15)

(A,9,15)

(A,9,11)(A,9,11)

(A,9,9)(A,9,9) (A,11,11)(A,11,11)

(A,13,15)(A,13,15)

(A,13,13)(A,13,13) (A,15,15)(A,15,15)

Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7

Best-Case Behavior

QuickSort: Best-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 15)QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)

Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

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13

11

14

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15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

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14

12

15

3

1

1

2

6

3

2

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7

5

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4

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8

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11

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9

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15

3

1

1

2

2

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4

7

5

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6

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8

8

11

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9

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15

13

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12

15

3

1

1

2

2

3

4

4

7

5

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6

6

7

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8

11

9

9

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13

10

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12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

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11

14

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15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

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8

11

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12

15

1

1

2

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3

3

4

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7

5

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6

6

7

8

8

11

9

9

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15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

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13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

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11

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15

13

10

14

12

15

1

1

2

2

3

3

4

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5

5

6

6

7

7

8

8

11

9

9

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11

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13

10

14

12

15

1

1

2

2

3

3

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5

5

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6

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8

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11

9

9

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13

10

14

12

15

1

1

2

2

3

3

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5

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6

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7

8

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11

9

9

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13

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15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

(A,1,15)

(A,1,15)

(A,1,7)

(A,1,7)

(A,1,3)

(A,1,3)

(A,1,1)

(A,1,1)

(A,3,3)

(A,3,3)

(A,5,7)

(A,5,7)

(A,5,5)(A,5,5) (A,7,7)(A,7,7)

(A,9,15)

(A,9,15)

(A,9,11)(A,9,11)

(A,9,9)(A,9,9) (A,11,11)(A,11,11)

(A,13,15)(A,13,15)

(A,13,13)(A,13,13) (A,15,15)(A,15,15)

Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7

Best-Case Behavior

QuickSort: Best-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 15)QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)

QuickSort(A, 5, 7)

QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)

Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

(A,1,15)

(A,1,15)

(A,1,7)

(A,1,7)

(A,1,3)

(A,1,3)

(A,1,1)

(A,1,1)

(A,3,3)

(A,3,3)

(A,5,7)

(A,5,7)

(A,5,5)(A,5,5) (A,7,7)(A,7,7)

(A,9,15)

(A,9,15)

(A,9,11)(A,9,11)

(A,9,9)(A,9,9) (A,11,11)(A,11,11)

(A,13,15)(A,13,15)

(A,13,13)(A,13,13) (A,15,15)(A,15,15)

Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7

Best-Case Behavior

QuickSort: Best-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 15)QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)

QuickSort(A, 5, 7)

QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)

Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)

Partition(A, 5, 7)

Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

(A,1,15)

(A,1,15)

(A,1,7)

(A,1,7)

(A,1,3)

(A,1,3)

(A,1,1)

(A,1,1)

(A,3,3)

(A,3,3)

(A,5,7)

(A,5,7)

(A,5,5)(A,5,5) (A,7,7)(A,7,7)

(A,9,15)

(A,9,15)

(A,9,11)(A,9,11)

(A,9,9)(A,9,9) (A,11,11)(A,11,11)

(A,13,15)(A,13,15)

(A,13,13)(A,13,13) (A,15,15)(A,15,15)

Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7

Best-Case Behavior

QuickSort: Best-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 15)QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)

QuickSort(A, 5, 7)

QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)

Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)

Partition(A, 5, 7)

Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

(A,1,15)

(A,1,15)

(A,1,7)

(A,1,7)

(A,1,3)

(A,1,3)

(A,1,1)

(A,1,1)

(A,3,3)

(A,3,3)

(A,5,7)

(A,5,7)

(A,5,5)(A,5,5) (A,7,7)(A,7,7)

(A,9,15)

(A,9,15)

(A,9,11)(A,9,11)

(A,9,9)(A,9,9) (A,11,11)(A,11,11)

(A,13,15)(A,13,15)

(A,13,13)(A,13,13) (A,15,15)(A,15,15)

Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7

Best-Case Behavior

QuickSort: Best-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 15)QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)

Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

(A,1,15)

(A,1,15)

(A,1,7)

(A,1,7)

(A,1,3)

(A,1,3)

(A,1,1)

(A,1,1)

(A,3,3)

(A,3,3)

(A,5,7)

(A,5,7)

(A,5,5)

(A,5,5) (A,7,7)

(A,7,7)

(A,9,15)

(A,9,15)

(A,9,11)(A,9,11)

(A,9,9)(A,9,9) (A,11,11)(A,11,11)

(A,13,15)(A,13,15)

(A,13,13)(A,13,13) (A,15,15)(A,15,15)

Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7

Best-Case Behavior

QuickSort: Best-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 15)QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)

QuickSort(A, 5, 5)

QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)

Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

(A,1,15)

(A,1,15)

(A,1,7)

(A,1,7)

(A,1,3)

(A,1,3)

(A,1,1)

(A,1,1)

(A,3,3)

(A,3,3)

(A,5,7)

(A,5,7)

(A,5,5)

(A,5,5) (A,7,7)

(A,7,7)

(A,9,15)

(A,9,15)

(A,9,11)(A,9,11)

(A,9,9)(A,9,9) (A,11,11)(A,11,11)

(A,13,15)(A,13,15)

(A,13,13)(A,13,13) (A,15,15)(A,15,15)

Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7

Best-Case Behavior

QuickSort: Best-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 15)QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)

Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

(A,1,15)

(A,1,15)

(A,1,7)

(A,1,7)

(A,1,3)

(A,1,3)

(A,1,1)

(A,1,1)

(A,3,3)

(A,3,3)

(A,5,7)

(A,5,7)

(A,5,5)

(A,5,5) (A,7,7)

(A,7,7)

(A,9,15)

(A,9,15)

(A,9,11)(A,9,11)

(A,9,9)(A,9,9) (A,11,11)(A,11,11)

(A,13,15)(A,13,15)

(A,13,13)(A,13,13) (A,15,15)(A,15,15)

Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7

Best-Case Behavior

QuickSort: Best-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 15)QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)

QuickSort(A, 7, 7)

QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)

Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

(A,1,15)

(A,1,15)

(A,1,7)

(A,1,7)

(A,1,3)

(A,1,3)

(A,1,1)

(A,1,1)

(A,3,3)

(A,3,3)

(A,5,7)

(A,5,7)

(A,5,5)

(A,5,5) (A,7,7)

(A,7,7)

(A,9,15)

(A,9,15)

(A,9,11)(A,9,11)

(A,9,9)(A,9,9) (A,11,11)(A,11,11)

(A,13,15)(A,13,15)

(A,13,13)(A,13,13) (A,15,15)(A,15,15)

Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7

Best-Case Behavior

QuickSort: Best-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 15)QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)

Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

(A,1,15)

(A,1,15)

(A,1,7)

(A,1,7)

(A,1,3)

(A,1,3)

(A,1,1)

(A,1,1)

(A,3,3)

(A,3,3)

(A,5,7)

(A,5,7)

(A,5,5)

(A,5,5)

(A,7,7)

(A,7,7)

(A,9,15)

(A,9,15)

(A,9,11)(A,9,11)

(A,9,9)(A,9,9) (A,11,11)(A,11,11)

(A,13,15)(A,13,15)

(A,13,13)(A,13,13) (A,15,15)(A,15,15)

Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7

Best-Case Behavior

QuickSort: Best-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 15)QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)

QuickSort(A, 9, 15)

QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)

Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

(A,1,15)

(A,1,15)

(A,1,7)

(A,1,7)

(A,1,3)

(A,1,3)

(A,1,1)

(A,1,1)

(A,3,3)

(A,3,3)

(A,5,7)

(A,5,7)

(A,5,5)

(A,5,5)

(A,7,7)

(A,7,7)

(A,9,15)

(A,9,15)

(A,9,11)(A,9,11)

(A,9,9)(A,9,9) (A,11,11)(A,11,11)

(A,13,15)(A,13,15)

(A,13,13)(A,13,13) (A,15,15)(A,15,15)

Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7

Best-Case Behavior

QuickSort: Best-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 15)QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)

QuickSort(A, 9, 15)

QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)

Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)

Partition(A, 9, 15)

Partition(A, 9, 11)Partition(A, 13, 15)

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

(A,1,15)

(A,1,15)

(A,1,7)

(A,1,7)

(A,1,3)

(A,1,3)

(A,1,1)

(A,1,1)

(A,3,3)

(A,3,3)

(A,5,7)

(A,5,7)

(A,5,5)

(A,5,5)

(A,7,7)

(A,7,7)

(A,9,15)

(A,9,15)

(A,9,11)(A,9,11)

(A,9,9)(A,9,9) (A,11,11)(A,11,11)

(A,13,15)(A,13,15)

(A,13,13)(A,13,13) (A,15,15)(A,15,15)

Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7

Best-Case Behavior

QuickSort: Best-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 15)QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)

QuickSort(A, 9, 15)

QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)

Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)

Partition(A, 9, 15)

Partition(A, 9, 11)Partition(A, 13, 15)

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

(A,1,15)

(A,1,15)

(A,1,7)

(A,1,7)

(A,1,3)

(A,1,3)

(A,1,1)

(A,1,1)

(A,3,3)

(A,3,3)

(A,5,7)

(A,5,7)

(A,5,5)

(A,5,5)

(A,7,7)

(A,7,7)

(A,9,15)

(A,9,15)

(A,9,11)(A,9,11)

(A,9,9)(A,9,9) (A,11,11)(A,11,11)

(A,13,15)(A,13,15)

(A,13,13)(A,13,13) (A,15,15)(A,15,15)

Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7

Best-Case Behavior

QuickSort: Best-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 15)QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)

Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

(A,1,15)

(A,1,15)

(A,1,7)

(A,1,7)

(A,1,3)

(A,1,3)

(A,1,1)

(A,1,1)

(A,3,3)

(A,3,3)

(A,5,7)

(A,5,7)

(A,5,5)

(A,5,5)

(A,7,7)

(A,7,7)

(A,9,15)

(A,9,15)

(A,9,11)

(A,9,11)

(A,9,9)(A,9,9) (A,11,11)(A,11,11)

(A,13,15)

(A,13,15)

(A,13,13)(A,13,13) (A,15,15)(A,15,15)

Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7

Best-Case Behavior

QuickSort: Best-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 15)QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)

QuickSort(A, 9, 11)

QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)

Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

(A,1,15)

(A,1,15)

(A,1,7)

(A,1,7)

(A,1,3)

(A,1,3)

(A,1,1)

(A,1,1)

(A,3,3)

(A,3,3)

(A,5,7)

(A,5,7)

(A,5,5)

(A,5,5)

(A,7,7)

(A,7,7)

(A,9,15)

(A,9,15)

(A,9,11)

(A,9,11)

(A,9,9)(A,9,9) (A,11,11)(A,11,11)

(A,13,15)

(A,13,15)

(A,13,13)(A,13,13) (A,15,15)(A,15,15)

Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7

Best-Case Behavior

QuickSort: Best-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 15)QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)

QuickSort(A, 9, 11)

QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)

Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)

Partition(A, 9, 11)

Partition(A, 13, 15)

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

(A,1,15)

(A,1,15)

(A,1,7)

(A,1,7)

(A,1,3)

(A,1,3)

(A,1,1)

(A,1,1)

(A,3,3)

(A,3,3)

(A,5,7)

(A,5,7)

(A,5,5)

(A,5,5)

(A,7,7)

(A,7,7)

(A,9,15)

(A,9,15)

(A,9,11)

(A,9,11)

(A,9,9)(A,9,9) (A,11,11)(A,11,11)

(A,13,15)

(A,13,15)

(A,13,13)(A,13,13) (A,15,15)(A,15,15)

Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7

Best-Case Behavior

QuickSort: Best-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 15)QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)

QuickSort(A, 9, 11)

QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)

Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)

Partition(A, 9, 11)

Partition(A, 13, 15)

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

(A,1,15)

(A,1,15)

(A,1,7)

(A,1,7)

(A,1,3)

(A,1,3)

(A,1,1)

(A,1,1)

(A,3,3)

(A,3,3)

(A,5,7)

(A,5,7)

(A,5,5)

(A,5,5)

(A,7,7)

(A,7,7)

(A,9,15)

(A,9,15)

(A,9,11)

(A,9,11)

(A,9,9)(A,9,9) (A,11,11)(A,11,11)

(A,13,15)

(A,13,15)

(A,13,13)(A,13,13) (A,15,15)(A,15,15)

Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7

Best-Case Behavior

QuickSort: Best-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 15)QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)

Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

(A,1,15)

(A,1,15)

(A,1,7)

(A,1,7)

(A,1,3)

(A,1,3)

(A,1,1)

(A,1,1)

(A,3,3)

(A,3,3)

(A,5,7)

(A,5,7)

(A,5,5)

(A,5,5)

(A,7,7)

(A,7,7)

(A,9,15)

(A,9,15)

(A,9,11)

(A,9,11)

(A,9,9)

(A,9,9) (A,11,11)

(A,11,11)

(A,13,15)

(A,13,15)

(A,13,13)(A,13,13) (A,15,15)(A,15,15)

Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7

Best-Case Behavior

QuickSort: Best-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 15)QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)

QuickSort(A, 9, 9)

QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)

Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

(A,1,15)

(A,1,15)

(A,1,7)

(A,1,7)

(A,1,3)

(A,1,3)

(A,1,1)

(A,1,1)

(A,3,3)

(A,3,3)

(A,5,7)

(A,5,7)

(A,5,5)

(A,5,5)

(A,7,7)

(A,7,7)

(A,9,15)

(A,9,15)

(A,9,11)

(A,9,11)

(A,9,9)

(A,9,9) (A,11,11)

(A,11,11)

(A,13,15)

(A,13,15)

(A,13,13)(A,13,13) (A,15,15)(A,15,15)

Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7

Best-Case Behavior

QuickSort: Best-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 15)QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)

Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

(A,1,15)

(A,1,15)

(A,1,7)

(A,1,7)

(A,1,3)

(A,1,3)

(A,1,1)

(A,1,1)

(A,3,3)

(A,3,3)

(A,5,7)

(A,5,7)

(A,5,5)

(A,5,5)

(A,7,7)

(A,7,7)

(A,9,15)

(A,9,15)

(A,9,11)

(A,9,11)

(A,9,9)

(A,9,9) (A,11,11)

(A,11,11)

(A,13,15)

(A,13,15)

(A,13,13)(A,13,13) (A,15,15)(A,15,15)

Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7

Best-Case Behavior

QuickSort: Best-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 15)QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)

QuickSort(A, 11, 11)

QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)

Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

(A,1,15)

(A,1,15)

(A,1,7)

(A,1,7)

(A,1,3)

(A,1,3)

(A,1,1)

(A,1,1)

(A,3,3)

(A,3,3)

(A,5,7)

(A,5,7)

(A,5,5)

(A,5,5)

(A,7,7)

(A,7,7)

(A,9,15)

(A,9,15)

(A,9,11)

(A,9,11)

(A,9,9)

(A,9,9) (A,11,11)

(A,11,11)

(A,13,15)

(A,13,15)

(A,13,13)(A,13,13) (A,15,15)(A,15,15)

Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7

Best-Case Behavior

QuickSort: Best-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 15)QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)

Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

(A,1,15)

(A,1,15)

(A,1,7)

(A,1,7)

(A,1,3)

(A,1,3)

(A,1,1)

(A,1,1)

(A,3,3)

(A,3,3)

(A,5,7)

(A,5,7)

(A,5,5)

(A,5,5)

(A,7,7)

(A,7,7)

(A,9,15)

(A,9,15)

(A,9,11)

(A,9,11)

(A,9,9)

(A,9,9)

(A,11,11)

(A,11,11)

(A,13,15)

(A,13,15)

(A,13,13)(A,13,13) (A,15,15)(A,15,15)

Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7

Best-Case Behavior

QuickSort: Best-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 15)QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)

QuickSort(A, 13, 15)

QuickSort(A, 13, 13)QuickSort(A, 15, 15)

Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

(A,1,15)

(A,1,15)

(A,1,7)

(A,1,7)

(A,1,3)

(A,1,3)

(A,1,1)

(A,1,1)

(A,3,3)

(A,3,3)

(A,5,7)

(A,5,7)

(A,5,5)

(A,5,5)

(A,7,7)

(A,7,7)

(A,9,15)

(A,9,15)

(A,9,11)

(A,9,11)

(A,9,9)

(A,9,9)

(A,11,11)

(A,11,11)

(A,13,15)

(A,13,15)

(A,13,13)(A,13,13) (A,15,15)(A,15,15)

Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7

Best-Case Behavior

QuickSort: Best-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 15)QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)

QuickSort(A, 13, 15)

QuickSort(A, 13, 13)QuickSort(A, 15, 15)

Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)

Partition(A, 13, 15)

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

(A,1,15)

(A,1,15)

(A,1,7)

(A,1,7)

(A,1,3)

(A,1,3)

(A,1,1)

(A,1,1)

(A,3,3)

(A,3,3)

(A,5,7)

(A,5,7)

(A,5,5)

(A,5,5)

(A,7,7)

(A,7,7)

(A,9,15)

(A,9,15)

(A,9,11)

(A,9,11)

(A,9,9)

(A,9,9)

(A,11,11)

(A,11,11)

(A,13,15)

(A,13,15)

(A,13,13)(A,13,13) (A,15,15)(A,15,15)

Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7

Best-Case Behavior

QuickSort: Best-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 15)QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)

QuickSort(A, 13, 15)

QuickSort(A, 13, 13)QuickSort(A, 15, 15)

Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)

Partition(A, 13, 15)

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

(A,1,15)

(A,1,15)

(A,1,7)

(A,1,7)

(A,1,3)

(A,1,3)

(A,1,1)

(A,1,1)

(A,3,3)

(A,3,3)

(A,5,7)

(A,5,7)

(A,5,5)

(A,5,5)

(A,7,7)

(A,7,7)

(A,9,15)

(A,9,15)

(A,9,11)

(A,9,11)

(A,9,9)

(A,9,9)

(A,11,11)

(A,11,11)

(A,13,15)

(A,13,15)

(A,13,13)(A,13,13) (A,15,15)(A,15,15)

Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7

Best-Case Behavior

QuickSort: Best-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 15)QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)

Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

(A,1,15)

(A,1,15)

(A,1,7)

(A,1,7)

(A,1,3)

(A,1,3)

(A,1,1)

(A,1,1)

(A,3,3)

(A,3,3)

(A,5,7)

(A,5,7)

(A,5,5)

(A,5,5)

(A,7,7)

(A,7,7)

(A,9,15)

(A,9,15)

(A,9,11)

(A,9,11)

(A,9,9)

(A,9,9)

(A,11,11)

(A,11,11)

(A,13,15)

(A,13,15)

(A,13,13)

(A,13,13) (A,15,15)

(A,15,15)

Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7

Best-Case Behavior

QuickSort: Best-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 15)QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)

QuickSort(A, 13, 13)

QuickSort(A, 15, 15)

Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

(A,1,15)

(A,1,15)

(A,1,7)

(A,1,7)

(A,1,3)

(A,1,3)

(A,1,1)

(A,1,1)

(A,3,3)

(A,3,3)

(A,5,7)

(A,5,7)

(A,5,5)

(A,5,5)

(A,7,7)

(A,7,7)

(A,9,15)

(A,9,15)

(A,9,11)

(A,9,11)

(A,9,9)

(A,9,9)

(A,11,11)

(A,11,11)

(A,13,15)

(A,13,15)

(A,13,13)

(A,13,13) (A,15,15)

(A,15,15)

Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7

Best-Case Behavior

QuickSort: Best-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 15)QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)

Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

(A,1,15)

(A,1,15)

(A,1,7)

(A,1,7)

(A,1,3)

(A,1,3)

(A,1,1)

(A,1,1)

(A,3,3)

(A,3,3)

(A,5,7)

(A,5,7)

(A,5,5)

(A,5,5)

(A,7,7)

(A,7,7)

(A,9,15)

(A,9,15)

(A,9,11)

(A,9,11)

(A,9,9)

(A,9,9)

(A,11,11)

(A,11,11)

(A,13,15)

(A,13,15)

(A,13,13)

(A,13,13) (A,15,15)

(A,15,15)

Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7

Best-Case Behavior

QuickSort: Best-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 15)QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)

QuickSort(A, 15, 15)

Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

(A,1,15)

(A,1,15)

(A,1,7)

(A,1,7)

(A,1,3)

(A,1,3)

(A,1,1)

(A,1,1)

(A,3,3)

(A,3,3)

(A,5,7)

(A,5,7)

(A,5,5)

(A,5,5)

(A,7,7)

(A,7,7)

(A,9,15)

(A,9,15)

(A,9,11)

(A,9,11)

(A,9,9)

(A,9,9)

(A,11,11)

(A,11,11)

(A,13,15)

(A,13,15)

(A,13,13)

(A,13,13) (A,15,15)

(A,15,15)

Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7

Best-Case Behavior

QuickSort: Best-Case Behavior

QuickSort(A, p, r)

1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)

QuickSort(A, 1, 15)QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)

Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

5

3

11

4

2

5

12

6

7

7

6

8

4

9

9

10

13

11

14

12

15

13

10

14

8

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

6

3

2

4

7

5

5

6

4

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

3

1

1

2

2

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

7

5

5

6

6

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

13

11

14

12

15

13

10

14

12

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

11

9

9

10

10

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

15

13

13

14

14

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

(A,1,15)

(A,1,15)

(A,1,7)

(A,1,7)

(A,1,3)

(A,1,3)

(A,1,1)

(A,1,1)

(A,3,3)

(A,3,3)

(A,5,7)

(A,5,7)

(A,5,5)

(A,5,5)

(A,7,7)

(A,7,7)

(A,9,15)

(A,9,15)

(A,9,11)

(A,9,11)

(A,9,9)

(A,9,9)

(A,11,11)

(A,11,11)

(A,13,15)

(A,13,15)

(A,13,13)

(A,13,13)

(A,15,15)

(A,15,15)

Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7

Randomized QuickSort

Randomized Quick Sort

RandomizedPartition(A, p, r)

1 i ← Random(p, r)2 exchange A[r ]↔ A[i ]3 return Partition(A, p, r)

RandomizedQuickSort(A, p, r)

1 if p < r2 then q ← RandomizedPartition(A, p, r)3 RandomizedQuickSort(A, p, q − 1)4 RandomizedQuickSort(A, q + 1, r)

Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 7 / 7