2-4 tree

•k=2

•Each node has 2,3,or 4 children

Delete

15 30

14 30 40 5016 179

delete(14,T).

1 2 3 4 5 6 7 8

7 9

5 14

3

Delete

30 40 5016 179

delete(14,T).

1 2 3 4 5 6 7 8

7 9

5 14

303

Delete

30 40 5016 179

delete(17,T).

1 2 3 4 5 6 7 8

7 9

5 14

303

Delete

30 40 509

delete(17,T).

1 2 3 4 5 6 7 8

7 9

5 14

30

16

3

Delete

30 40 509

delete(16,T).

1 2 3 4 5 6 7 8

7 9

5 14

30

16

3

Delete

30 40 509

delete(16,T).

1 2 3 4 5 6 7 8

7 9

5 14

303

Borrow

30 40 509

delete(16,T).

1 2 3 4 5 6 7 8

7 9

5 14

303

Borrow

30 40 509

delete(16,T).

1 2 3 4 5 6 7 8

5 9

303 7

30 40 509

delete(9,T).

1 2 3 4 5 6 7 8

5 9

303 7

30 40 50

delete(9,T).

1 2 3 4 5 6 7 8

5 9

303 7

30 40 50

delete(9,T).

1 2 3 4 5 6 7 8

5 9

3

Fusion

7 30

30 40 50

delete(9,T).

1 2 3 4 5 6 7 8

3

Fusion

7 30

5

Delete -- definitionRemove the key.

If it is the only key in the node remove the node, and let v be the parent that looses a child, otherwise return

(*) If v has one child, and v is the root discard v.

Otherwise (v is not a root), if v has a sibling w of degree 3 or 4, borrow a child from w to v and terminate.

Otherwise, fuse v with its sibling to a degree 3 node and repeat (*) with the parent of v.

2 )Orthogonal range searching

Range trees

Reporting )1-D)

Given a set of points S on the line, preprocess them to build structure that allows efficient queries of the from:

Given an interval I=[x1,x2] find all points in S that are in the interval.

2 4 5 7 8 12 15 19

6 17

Reporting )1-D)

Build a balanced tree over the pointsConcatenate them in a list

7

7 19151282 4 5

2

4

5 8

12

15

2 4 5 7 8 12 15 19

6 17

query: O(log n+k)space: O(n)

Reporting )2-D)

Given a set of points S on the plane, preprocess them to build structure that allows efficient queries of the from:Given an rectangle R=[x1,x2][y1,y2] find all points in S that are in the rectangle.

P1

P2

P3

P4

P5 P8

P6

P7

Range Trees )2-D)

P4P3P2P1

P3

P2

P4

P1

P1

P2

P3

P4

Maintain the points in a balanced search tree ordered by x-coor. In each internal node maintain the points in its subtree in a balanced search tree ordered by y

Range Trees )Contd.)

P4P3P2P1

P3

P2

P4

P1

P1

P2

P3

P4

P8P7P6P5

P5 P8

P6

P7

Range Trees )Contd.)

P4P3P2P1

P3

P2

P4

P1

P1

P2

P3

P4

P8P7P6P5

P5 P8

P6

P7

Query processing

•Search by the first dimension gives us O)logn) trees which together contain the

output.

•We search each of these trees to get the answer

Analysis )2-D)

•Space O)nlog n)

•Query O)log2 n+k)

95הוספת מפתח

. (90, 85, 80, 75(שוב צריך לפצל, הפעם את העלה במקומו יהיו שני עלים בעץ, בשמאלי יהיו המפתחות

.(95, 90, 85(ובימני המפתחות ( 80, 75(

אבל עכשיו צריך להוסיף מפתח (ומצביע( להורה. . (85, 75, 60, 50, 25(אמורים להיות בו המפתחות:

אך יש מקום לארבעה מפתחות בלבד. מה עושים? מפצלים (אך לא בדיוק כמו מקודם(

(המשך(95הוספת מפתח אחרי שכבר פיצלנו את העלה, נפצל גם את הרמה הבאה ונארגן

( נשלח עוד רמה כלפי 60את המצביעים. את המפתח האמצעי(מעלה

60

(סיום(95הוספת מפתח

לא הייתה רמה נוספת, לכן נוצר שורש חדש (מותר שלא יהיה ½ מלא לפחות(