Balanced Binary Search Tree

1. Balanced Binary Search Tree 황승원 Fall 2011 CSE, POSTECH

2. height is O(log n), where n is the number of elements in the tree AVL (Adelson-Velsky and Landis) trees red-black trees get, put, and remove take O(log n) time Balanced Binary Search Trees

3. Indexed AVL trees Indexed red-black trees Indexed operations alsotake O(log n) time Balanced Binary Search Trees

4. weight balanced binary search trees 2-3 & 2-3-4 trees AA trees B-trees BBST etc. Balanced Search Trees

5. AVL Tree Definition • Binary tree. • If T is a nonempty binary tree with TL and TR as its left and right subtrees, then T is an AVL tree iff • TL and TR are AVL trees, and • |hL – hR|  1 where hL and hR are the heights of TL and TR, respectively

6. This is an AVL tree. Balance Factors -1 1 1 -1 0 1 0 0 -1 0 0 0 0

7. The height of an AVL tree that has n nodes is at most 1.44 log2 (n+2). Nh=Nh-1 + Nh-2+1 – reminds you anything? Fn=Fn-1 + Fn-2 Nh=Fh+2-1 when Height (by induction)

8. -1 10 1 1 7 40 -1 0 1 0 45 3 8 30 0 -1 0 0 0 60 35 1 20 5 0 25 AVL Search Tree

9. put(9) -1 10 0 1 1 7 40 -1 0 1 -1 0 45 3 8 30 0 -1 0 0 0 0 60 35 1 9 20 5 0 25

10. put(29) -1 10 1 1 7 40 -1 0 1 0 45 3 8 30 0 -1 0 0 0 -2 60 35 1 20 5 0 -1 RR imbalance => new node is in right subtree of right subtree of blue node (node with bf = -2) 25 0 29

11. put(29) -1 10 1 1 7 40 -1 0 1 0 45 3 8 30 0 0 0 0 0 60 35 1 25 5 0 0 20 29 RR rotation.

12. RR LL RL LR AVL Rotations

13. Imbalance Types • After an insertion, when the balance factor of node A is –2 or 2, the node A is one of the following four imbalance types • LL: new node is in the left subtree of the left subtree of A • LR: new node is in the right subtree of the left subtree of A • RR: new node is in the right subtree of the right subtree of A • RL: new node is in the left subtree of the right subtree of A

14. Rotation Definition • To switch children and parents among two or three adjacent nodes to restore balance of a tree. • A rotation may change the depth of some nodes, but does not change their relative ordering.

15. AVL Rotations • To balance an unbalanced AVL tree (after an insertion), we may need to perform one of the following rotations: LL, RR, LR, RL

16. An LL Rotation

17. An LR Rotation

18. RR and RL? • Symmetric!

19. -1 10 1 1 7 40 0 0 1 -1 3 8 30 45 0 0 0 -1 0 1 5 20 35 60 0 25 29 Inserting into an AVL Search Tree Insert(29) • Where is 29 going to be inserted into? • After the insertion, is the tree still an AVL search tree? (i.e., still balanced?)

20. -1 10 1 1 7 40 0 0 1 -1 3 8 30 45 0 0 0 0 1 5 20 35 60 -2 -1 25 0 29 Inserting into an AVL Search Tree Insert(29) • What are the balance factors for 20, 25, 29? • RR imbalance  new node is in the right subtree of right subtree of node 20 (node with bf = -2)

21. -1 10 1 1 7 40 0 0 1 -1 3 8 30 45 0 0 0 0 0 1 5 25 35 60 0 0 20 29 After RR Rotation • What would the left subtree of 30 look like after an RR rotation? • After the RR rotation, is the resulting tree an AVL search tree now?

22. Deletion from an AVL Search Tree • Deletion of a node may also produce an imbalance • Imbalance incurred by deletion is classified intothe types R0, R1, R-1, L0, L1, and L-1 • Rotation is also needed for rebalancing • Read the observations after deleting a node from an AVL search tree • Read Section 11.2.6 for Deletion from an AVL search tree

23. An R1 (decided by B) Rotation=LL

24. An R-1 Rotation=LR

25. An R0 Rotation~LL

26. Complexity of Dictionary Operations:Search, Insert, and Delete Data Structure Worst Case Expected Hash Table O(n) O(1) Binary SearchTree O(n) O(logn) BalancedBinary Search Tree O(logn) O(logn) • n is the number of elements in dictionary.

27. Complexity of Other OperationsAscend, Search(rank), and Delete(rank) Data Structure ascend get and remove Hash Table O(D+nlogn) O(D+nlogn) O(n) Indexed BST O(n) IndexedBalanced BST O(n) O(log n) • D is the number of buckets.

28. Colored Nodes Definition Binary search tree. Each node is colored red or black. Root and all external nodes are black. No root-to-external-node path has two consecutive red nodes. All root-to-external-node paths have the same number of black nodes Red Black Trees

29. 10 7 40 45 3 8 30 60 35 1 20 5 25 Example Red Black Tree

30. Properties • The height of a red black tree that has n (internal) nodes is between log2(n+1) and 2log2(n+1).

31. Lemma 16.2 • r : rank(=# black nodes-1) of root

32. Insert • New pair is placed in a new node, which is inserted into the red-black tree. • New node color options. • Black node => one root-to-external-node path has an extra black node (black pointer). • Hard to remedy. • Red node => one root-to-external-node path may have two consecutive red nodes (pointers). • May be remedied by color flips and/or a rotation.

33. gp pp d p c a b Classification Of 2 Red Nodes/Pointers • XYz • X => relationship between gp and pp. • pp left child of gp => X = L. • Y => relationship between pp and p. • p right child of pp => Y = R. • z = b (black) if d = null or a black node. • z = r (red) if d is a red node. LLb

34. XYr • Color flip. gp gp pp pp d d p p c c a b a b • Continue rebalancing if changing gp to red causes problems

35. LLb • Rotate. gp y z pp x z y d p x a b c d c a b • Done! • Same as LL rotation of AVL tree.

36. LRb • Rotate. gp y z pp x z x d a b c d a p y b c • Done! • Same as LR rotation of AVL tree. • RRb and RLb are symmetric.

37. Red-black Tree Animation • http://www.ece.uc.edu/~franco/C321/html/RedBlack/redblack.html • java.util.TreeMap => red black tree

38. Coming Up Next • READING: Ch 16.1~2 • NEXT: Graphs (Ch 17.1~7)