AVL tree self-adjusting tree

Lai Ah Fur

AVL tree discovers: Adelson-Velskii and Landis balanced Binary search tree the depth of the tree: O(lg N) definition: An AVL tree is a binary search

tree with the additional balance property that, for any node in the tree, the height of the left and right subtrees can be differ by at most 1.

Balance factor (BF) 如果 node T 之 the height of left subtree

is Hl, the height of right subtree is Hr, then the BF of node T is Hl- Hr

高度平衡之 BST: BF(t)<2 …for any node t AVL tree: BF(t)<2 Full binary tree: BF=? complete binary tree: BF=?

51 73

9563

78

31

51 73

9563

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31

51 78

95

63

73

2

1

1

0

1

Right rotate

LL 型

10583

9563

78

10583

9563

78

100

-2

-1

1 10578

95

63 83 100

Left rotate

RR 型

插入 6678

63

66

63

66

78

2

-1

0

0

0 0

第一種 LR 型

插入 65

6455 7336

51 70 90

9563

78

6555

64

36

51

63 78

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95

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6555

64

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51

63 73 90

9570

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65

第二種 LR 型

2

-1

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-1 0

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插入 7578

63 95

90

55 73

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65

51

36

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70 95

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55

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51 75

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63 78

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90

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第三種 LR 型

2

-1

-1

-1

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插入 90

96

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90 96

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78 96

第一種 RL 型

-2

1

0

插入 80

51

6378

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83 105

79 200

51

6378

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85 105

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第二種 RL 型

-2

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-1

插入 88

51

6378

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83 105

79 20063

7883

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8851 100

85 10579

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第三種 RL 型

-2

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Del 7

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RR

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Del 71

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RL

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8 10311

1. 刪除 7

或是

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9

2. 刪除 5

Del 5

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RL

Insertion

case 1: an insertion into the left subtree of the left child of X

case 2: an insertion into the right subtree of the left child of X

case 3: an insertion into the left subtree of the right child of X

case 4: an insertion into the right subtree of the right child of X

case 1: single rotation left subtree 較高

.A single rotation switches the role of the parent and the child while maintaining the search order

.rotate binary tree with left child static BinaryNode withLeftChild (BinaryNode k2) { BinaryNode k1=k2.left; k2.left=k1.right; k1.right=k2; return k1; }

K2

K1

CB

A

K2

K1

CBA

case 1 (single L-L rotation, 單一左左迴轉 )

single rotation

+2

+1

0

0

case 1

128 16

1410462

1

12

816

1410

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62

1

k2

k1

BA

c A

k1K2

B C

Insert “1”

case 2: double rotation (double L-R rotation)

double rotate binary tree node: first left child with its right child; then, node k3 with new left child.

static BinaryNode doubleWithLeftChild(BinaryNode k3) { k3.left=withRightChild(k3.left); return withLeftChild(k3); }

case 2: double rotationK3

K1

D

AB

K2

C

K1

DC

K3

K2

A B

K3

K2

D

A B

K1 C

+2

-1

0

case 2: double rotation

128 16

1410462

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816

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54

2k2

k1

B

A

D

A

k1

K2

B D

Insert “5”

k3

CC

K3

case 3:double rotation (double R-L rotation)

double rotate binary tree node: first right child with its left child; then, node k1 with new right child.

static BinaryNode doubleWithRightChild(BinaryNode k1){ k1.right=withLeftChild(k1.right); return withRightChild(k1);}

case 3

K3

K1

DB

K2

C

K1

DC

K3

K2

A B

K1

K2

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B

C

A K3A

-2

+1

0

R-L type

Insert 85 or 92

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+1/-1

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90 99

92

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907085 95

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case 4: single rotation right subtree 較高

rotate binary tree with right child static BinaryNode withRightChild (BinaryNode k1){ BinaryNode k2=k1.right; k1.right=k2.left; k2.left=k1; return k2;}

case 4 (single R-R rotation, 單一右右迴轉 )

K2

K1

CB

A

K2

K1

CBA

single rotation

-2

-1

0

R-R type

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9560

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50Insert 50

-10-1

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-1

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0 0

exercise Insert the following data into the

empty AVL tree, 90 80 70 60 50 40

exercise Insert the following data into the

empty AVL tree, “Mar,May,Nov,Aug,Apr,Jane,Dec,July,Feb,June,Oct,Sept”

3-1May

Nov

Apr Mar+1

-1

Jan

+2

Insert Jan

Aug

2

May

Nov

Aug

Apr

+1

+2

Insert Aug, Apr

Mar

1

Mar

Nov

-1

-2

Insert Mar, May, Nov

May

3-2

May

Nov

Apr Jan

Mar

Aug

4-1

May

Nov

Aug

Apr

Mar

July

Feb

Jan

Dec-1

+1

-2

Insert Dec, July, Feb

4-2

May

Nov

Aug

Apr

Mar

July

Dec

Jan

Feb

5-1

May

Nov

Apr

Mar

July

Dec

Jan

Feb

Aug

June

-1

-1

-1

+2

Insert June

5-2

May

Nov

Apr

Mar

JulyDec

Jan

FebAug June

6

May

NovApr

July

Dec

FebAug

June

Jan

Mar

OctInsert Oct

-1

-2

The Answer:7

Apr

Dec

FebAug

June

Jan

Insert Sept

July

May

Nov

Mar

Sept

Oct

JanDec Mar

Aug Feb July Nov

Apr June May Oct

Sept

-1+1

-1 -1

-1

0

+1 0

0 00

-1

After delete 40,…60

4030

80

70 90

65 75

Double R-L rotations

After delete 85,…80

4030

85

70 90

65 75

Double L-Rrotations

After delete 40,…80

40 85

90Single R-Rrotations

After delete 85,…80

40 85

20Single L-Lrotations

After delete 85,…

Single L-Lrotations

8040 85

30 50

How a newly arriving element enters… If a newly arriving element

endangers the tree balance, how to rectify the problem immediately?

By restructuring the tree locally (the AVL method) or by re-creating the tree (the DSW method)

self-adjusting tree But, not all elements are used with

the same frequency. 在低層之 node若 infrequently accessed, 則對於程式效能影響不大

The strategy in self-adjusting tree is to restructure trees only by moving up the tree those elements that are used more often, creating a kind of “priority tree”

Self-restructuring tree Proposed by Brian Allen and Ian

Munro and James Bitner Strategy:

Single rotation: rotate a child about its parent if an element in a child is accessed unless it is the root

Moving to the root: repeat the child-parent rotation until the element being accessed is in the root

splaying A modification of the “move to the root” Apply single rotation in pairs in an order

depending on the links between the child, parent and grandparent. (node R is accessed) Case 1:node R’s parent is the root Case 2:homogeneous configuration: node R is the

left child of its parent Q and Q is the left child of its parent P, or R and Q are both right children.

Case 3: heterogeneous configuration: node R the right child of its parent Q and Q is the left child of its parent P, or R is the left child of Q and Q is the right child of P

3 cases of splayingAccessing node R

Examples of splaying

Restructuring a tree with splaying (a-c) after accessing T and (c-d) then R

Algorithm of splayingSplaying(P,Q,R) while R is not the root if the R’s parent is the root perform a singular splay, rotate R about its parent; else if R is in homogeneous configuration with its predecessor perform a homoegeous splay, first rotate Q about P and then R about Q; else if R is in heterogeneous configuration with its predecessor perform a heterogeneous splay, first rotate R about Q and then about P;

The problem of splaying Splaying is a strategy focusing upon the

elements rather than the shape of the tree. It may perform well in situation in which some elements are used much more frequently than others

If the elements near the root are accessed with about the same frequency as elements on the lowest levels, then splaying may not be the best choice.

semisplaying A modification that requires only one

rotation for a homogeneous splay and continues splaying with the parent of the accessed node.

Example of semisplaying

(a)-(c) accessing T and restructuring the tree with semisplaying(c ) -(d)accessing T again