Upload
claire-sims
View
220
Download
0
Embed Size (px)
DESCRIPTION
The Heap A binary heap is a nearly complete binary tree Implemented as an array A Two similar attributes: – length[A] is the size (number of slots) in A – heap-size[A] is the number of elements in A – Thus, heap-size[A] length[A] – Also, no element past A[heap-size[A]] is an element Jeff Chastine
Citation preview
Heapsort
A minimalist's approach
Jeff Chastine
Heapsort
• Like MERGESORT, it runs in O(n lg n)• Unlike MERGESORT, it sorts in place• Based off of a “heap”, which has several uses• The word “heap” doesn’t refer to memory
management
Jeff Chastine
The Heap
• A binary heap is a nearly complete binary tree• Implemented as an array A• Two similar attributes:– length[A] is the size (number of slots) in A– heap-size[A] is the number of elements in A– Thus, heap-size[A] length[A]– Also, no element past A[heap-size[A]] is an
element
Jeff Chastine
The Heap
• Can be a min-heap or a max-heap
87
51 67
55 434125
21 17 33 35
87
87 51 67 25 41 55 43 21 17 33 35Jeff Chastine
Simple Functions
PARENT(i)return (i/2)
LEFT(i)return (2i)
RIGHT(i)return (2i + 1)
Jeff Chastine
Properties
• Max-heap property:– A[PARENT(i)] A[i]
• Min-heap property:– A[PARENT(i)] A[i]
• Max-heaps are used for sorting• Min-heaps are used for priority queues (later)• We define the height of a node to be the longest
path from the node to a leaf.• The height of the tree is (lg n)
Jeff Chastine
MAX-HEAPIFY
• This is the heart of the algorithm• Determines if an individual node is smaller
than its children• Parent swaps with largest child if that child is
larger• Calls itself recursively• Runs in O(lg n) or O(h)
Jeff Chastine
HEAPIFYMAX-HEAPIFY (A, i)l ← LEFT (i)r ← RIGHT(i)if l ≤ heap-size[A] and A[l] > A[i]then largest ← lelse largest ← iif r ≤ heap-size[A] and A[r]>A[largest]then largest ← rif largest ≠ ithen exchange A[i] with A[largest]MAX-HEAPIFY (A, largest)
Jeff Chastine
16
4 10
9 314 7
2 8 1
Jeff Chastine
16
14 10
9 34 7
2 8 1
Jeff Chastine
16
14 10
9 38 7
2 4 1
Jeff Chastine
Of Note
• The children’s subtrees each have size at most 2n/3 – when the last row is exactly ½ full
• Therefore, the running time is:
T (n) = T(2n/3) + (1) = O(lg n)
Jeff Chastine
BUILD-HEAP• Use MAX-HEAPIFY in bottom up manner• Why does the loop start at length[A]/2?• At the start of each loop, each node i is the
root of a max-heap!
BUILD-HEAP (A)heap-size[A] ← length[A]for i ← length[A]/2 downto 1
do MAX-HEAPIFY(A, i)
Jeff Chastine
Analysis of Building a Heap
• Since each call to MAX-HEAPIFY costs O(lg n) and there are O(n) calls, this is O(n lg n)...
• Can derive a tighter bound: do all nodes take log n time?
• Has at most n/2h+1 nodes at any height (the more the height, the less nodes there are)
• It takes O(h) time to insert a node of height h.
Jeff Chastine
• Thus, the running time is 2n = O(n)
Sum up the work at each level
n
hh
n
hh
hnhOn lg
0
lg
01 2
)(2
The number ofnodes at height h
Multiplied by their height
Jeff Chastine
Height h islogarithmic
HEAPSORTHEAPSORT (A)
BUILD-HEAP(A)for i ← length[A] downto 2
do exchange A[1] with A[i]heap-size[A] ← heap-size[A] - 1MAX-HEAPIFY(A, 1)
Jeff Chastine
16
14 10
9 38 7
2 4 1
Jeff Chastine
1
14 10
9 38 7
2 4 16
Swap A[1] A[i]Jeff Chastine
14
8 10
9 34 7
2 1 16
heap-size heap-size – 1MAX-HEAPIFY(A, 1) Jeff Chastine
1
8 10
9 34 7
2 14 16
Swap A[1] A[i]Jeff Chastine
10
8 9
1 34 7
2 14 16
heap-size heap-size – 1MAX-HEAPIFY(A, 1) Jeff Chastine
2
8 9
1 34 7
10 14 16
Swap A[1] A[i]Jeff Chastine
9
8 3
1 24 7
10 14 16
heap-size heap-size – 1MAX-HEAPIFY(A, 1) Jeff Chastine
2
8 3
1 94 7
10 14 16
Swap A[1] A[i]Jeff Chastine
8
7 3
1 94 2
10 14 16
heap-size heap-size – 1MAX-HEAPIFY(A, 1) Jeff Chastine
1
7 3
8 94 2
10 14 16
Swap A[1] A[i]Jeff Chastine
7
4 3
8 91 2
10 14 16
heap-size heap-size – 1MAX-HEAPIFY(A, 1) Jeff Chastine
2
4 3
8 91 7
10 14 16
Swap A[1] A[i]Jeff Chastine
4
2 3
8 91 7
10 14 16
heap-size heap-size – 1MAX-HEAPIFY(A, 1) Jeff Chastine
1
2 3
8 94 7
10 14 16
Swap A[1] A[i]Jeff Chastine
3
2 1
8 94 7
10 14 16
heap-size heap-size – 1MAX-HEAPIFY(A, 1) Jeff Chastine
1
2 3
8 94 7
10 14 16
Swap A[1] A[i]Jeff Chastine
2
1 3
8 94 7
10 14 16
heap-size heap-size – 1MAX-HEAPIFY(A, 1) Jeff Chastine
1
2 3
8 94 7
10 14 16
Swap A[1] A[i]Jeff Chastine
1
2 3
8 94 7
10 14 16
heap-size heap-size – 1MAX-HEAPIFY(A, 1) Jeff Chastine
Priority Queues
• A priority queue is a heap that uses a key• Common in operating systems (processes)• Supports HEAP-MAXIMUM, EXTRACT-MAX,
INCREASE-KEY, INSERT
HEAP-MAXIMUM (A) 1 return A[1]
Jeff Chastine
HEAP-EXTRACT-MAX (A) 1 if heap-size[A] < 12 then error “heap underflow”3 max A[1]4 A[1] A[heap-size[A]]5 heap-size[A] heap-size[A] – 16 MAX-HEAPIFY (A, 1)7 return max
Jeff Chastine
HEAP-INCREASE-KEY(A, i, key)1 if key < A[i]2 then error “new key smaller than current”3 A[i] key4 while i > 1 and A[PARENT(i)] < A[i]5 do exchange A[i] A[PARENT(i)]6 i PARENT(i)
Note: runs in O(lg n) Jeff Chastine
MAX-HEAP-INSERT (A, key)1 heap-size[A] heap-size[A] + 12 A[heap-size[A]] -3 HEAP-INCREASE-KEY(A, heap-size[A], key)
Jeff Chastine