AVL tree self-adjusting tree

1. AVL treeself-adjusting tree Lai Ah Fur

2. 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.

3. 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=?

4. 2 0 78 78 63 1 1 63 95 63 95 51 78 1 Right rotate 51 73 51 73 73 95 31 31 LL型

5. -2 78 78 -1 95 63 95 63 95 78 105 1 105 83 105 83 63 83 100 100 Left rotate RR型

6. 78 63 66 插入66 2 0 66 -1 0 0 63 78 0 第一種LR型

7. 78 78 70 70 95 63 95 63 78 90 63 73 90 73 95 51 70 51 64 51 64 36 90 55 65 36 55 64 73 65 36 55 65 插入65 2 0 -1 0 -1 1 -1 1 -1 0 0 0 第二種LR型

8. 70 78 78 63 78 63 95 70 95 51 65 73 95 51 70 90 63 73 90 36 55 75 90 36 55 65 73 75 51 65 75 36 55 插入75 2 -1 1 -1 -1 第三種LR型

9. 78 78 90 96 90 78 96 90 96 插入90 -2 1 0 第一種RL型

10. 83 78 78 78 95 63 95 63 83 105 63 85 79 95 51 79 105 51 83 100 200 51 80 80 85 105 79 85 100 200 100 200 80 插入80 -2 1 1 -1 第二種RL型

11. 83 78 78 95 63 95 105 105 63 85 79 51 83 88 100 200 79 85 100 200 51 88 插入88 -2 0 1 0 1 -1 0 -1 1 -1 第三種RL型

12. 5 5 5 5 2 2 8 8 2 2 8 8 10 10 1 1 3 3 10 10 1 1 7 7 3 3 5 9 9 11 11 9 9 11 11 2 10 11 1 8 3 9 1.刪除7 RR Del 7 或是 5 Del 7 RL 2 9 10 1 8 3 11

13. 3 5 2 10 2 10 11 1 8 11 1 8 3 9 9 8 2 10 11 1 9 3 2.刪除5 Del 5 RL

14. 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

15. 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 • staticBinaryNode withLeftChild (BinaryNode k2) • { BinaryNode k1=k2.left; k2.left=k1.right; k1.right=k2; return k1; }

16. K1 single rotation K2 A B C case 1(single L-L rotation,單一左左迴轉) +2 K2 0 +1 K1 0 C B A

17. case 1 12 12 k2 k1 8 16 4 16 K2 k1 14 A 8 14 4 c 10 2 10 6 2 6 1 A B B C 1 Insert “1”

18. 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); }

19. K2 K3 K1 A B C D case 2: double rotation +2 0 K3 K3 K1 K2 -1 D D C K2 K1 A B C B A

20. case 2: double rotation 12 12 k3 K2 8 16 6 16 k1 14 8 14 4 D 10 k1 4 K3 k2 10 5 2 6 2 A D B C A 5 B C Insert “5”

21. 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); }

22. K2 K3 K1 A B C D case 3 0 -2 K1 K1 K2 K3 +1 A A K2 K3 B D B C D C

23. 80 90 90 70 80 95 85 95 99 92 70 85 99 92 R-L type -2 80 +1 0 95 70 99 90 +1/-1 85 92 Insert 85 or 92

24. case 4: single rotation rightsubtree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; }

25. single rotation case 4 (single R-R rotation,單一右右迴轉) -2 K1 K2 0 A K1 K2 -1 C B A B C

26. -1 80 0 -1 95 70 0 60 50 Insert 50 R-R type -1 80 0 0 95 60 0 0 50 70

27. exercise • Insert the following data into the empty AVL tree, 90 80 70 60 50 40

28. exercise • Insert the following data into the empty AVL tree, “Mar,May,Nov,Aug,Apr,Jane,Dec,July,Feb,June,Oct,Sept”

29. -2 1 2 3-1 +2 Mar May May -1 -1 +2 May Aug Nov Mar Nov Nov +1 Apr Mar Aug +1 Insert Mar, May, Nov Insert Aug, Apr Jan Apr Insert Jan Mar 4-1 3-2 May -2 May Aug Mar Nov Nov Apr Jan +1 Aug Dec July -1 Insert Dec, July, Feb Apr Jan Feb

30. Mar 4-2 5-1 +2 Mar May Aug -1 May Dec Nov Apr Dec Aug -1 Nov Jan Jan Apr Feb July -1 Feb July Insert June June 5-2 6 Mar Jan Jan May Dec Mar -2 Dec July Nov Aug May Feb July Aug Feb June -1 Nov Apr June Apr Oct Insert Oct

31. 7 Jan Dec Mar Aug Feb July Nov Apr June May Oct Sept Insert Sept The Answer:

32. -1 Jan +1 -1 Dec Mar -1 +1 0 -1 Aug Feb July Nov 0 0 0 -1 Apr June May Oct 0 Sept

33. 60 40 80 30 70 90 65 75 After delete 40,… Double R-L rotations

34. 80 40 85 30 70 90 65 75 After delete 85,… Double L-R rotations

35. 80 40 85 90 After delete 40,… Single R-R rotations

36. 80 40 85 20 After delete 85,… Single L-L rotations

37. 80 40 85 30 50 After delete 85,… Single L-L rotations

38. 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)

39. 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”

40. 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

41. 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

42. 3 cases of splaying Accessing node R

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

44. Algorithm of splaying Splaying(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;

45. 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.

46. semisplaying • A modification that requires only one rotation for a homogeneous splay and continues splaying with the parent of the accessed node.

47. Example of semisplaying (a)-(c) accessing T and restructuring the tree with semisplaying (c ) -(d)accessing T again