1. CSE 2331/5331 Topic 10: Balanced search trees Rotate operation Red-black tree Augmenting data struct.

2. CSE 2331/5331 Set Operations Maximum Extract-Max Insert Increase-key Search Delete Successor Predecessor

3. Motivation CSE 2331/5331

4. Balanced Search Tree CSE 2331/5331

5. Un-balanced Search Tree CSE 2331/5331

6. Goal: CSE 2331/5331

7. Rotation Operation A key operation in maintaining the “balance” of a binary search tree Locally rotate subtree, while maintain binary search tree property CSE 2331/5331 x y AB C y x A BC right rotation left rotation

8. Right Rotation Examples Right rotation at 9. CSE 2331/5331 Right rotation at 7 ?Right rotation at 13 ? Does it change Binary search tree property?

9. Left Rotation Examples Left rotation at node 3. CSE 2331/5331 what if we then right-rotate at node 6 ? left and right rotations are inverse of each other.

10. Recall: Transplant CSE 2331/5331

11. Implementation of Right Rotation CSE 2331/5331

12. Implementation of Left Rotation CSE 2331/5331

13. Rotation does not change binary search tree property! CSE 2331/5331 x y AB C y x A BC right rotation left rotation

14. Rotation to Re-balance Apply rotation to node 5 CSE 2331/5331 Reduce height to 3!

15. Rotation to Re-balance Apply rotation at node 5 CSE 2331/5331 Height not reduced! Need a double-rotation (first at 16, then at 5) In general, how and when?

16. Red-Black Trees Compared to an ordinary binary tree, a red-black binary tree: A leaf node is NIL Hence all nodes are either NIL or have exactly two children! Each node will have a color, either red or black CSE 2331/5331

17. Definition A Red-Black tree is a binary tree with the following properties: Every node is either red or black Root is black Every leaf is NIL and is black If a node is red, then both its children are black For each node, all simple paths from this node to its decedent leaves contain same number of black nodes CSE 2331/5331

18. An Example Double nodes are black. CSE 2331/5331 No two consecutive red nodes.

19. Skipping Leaves CSE 2331/5331

20. Is This a Red-Black Tree? CSE 2331/5331

21. Exercise Color the following tree to make a valid red-black tree CSE 2331/5331

22. Given a tree node x size(x): the total number of internal nodes contained in the subtree rooted at x bh(x): the number of black nodes on the path from x to leaf (not counting x) CSE 2331/5331

23. Balancing Property of RB-tree CSE 2331/5331

24. Proof Cont. CSE 2331/5331

25. Balancing Property of RB-tree CSE 2331/5331

26. Given a tree node x size(x): the total number of internal nodes contained in the subtree rooted at x bh(x): the number of black nodes on the path from x to leaf (not counting x) CSE 2331/5331

27. Balancing Property of RB-tree CSE 2331/5331

28. Balancing Property of RB-tree CSE 2331/5331

29. Implication of the Theorem CSE 2331/5331 How to maintain RB-tree under Operations?

30. CSE 2331/5331 Set Operations Maximum Extract-Max Insert Increase-key Search Delete Successor Predecessor Only need to consider Insert / delete

31. Insertion Example CSE 2331/5331 Insert 24? Insert 2?Insert 36?

32. Red-Black Tree Insert CSE 2331/5331

33. Insert Fixup: Case 3 (z is the new node) CSE 2331/5331

34. Example: Case 3 CSE 2331/5331 Insert 12?

35. Implementation of Fixup Case 3 CSE 2331/5331

36. Remarks CSE 2331/5331

37. Insert Fixup: Case 2 CSE 2331/5331

38. Example: Case 2 CSE 2331/5331 Insert 14?

39. Implementation of Case 2 CSE 2331/5331

40. Insert Fixup: Case 1: (z is new node) The parent and uncle of z are red: Color the parent and uncle of z both black Color the grandparent of z red Continue with the grandparent of z CSE 2331/5331

41. Sibling CSE 2331/5331

42. Implementation of Case 1 CSE 2331/5331 When does this procedure terminate?

43. Red-black Tree Insert Fixup CSE 2331/5331

44. Example CSE 2331/5331 Insert 21?

45. Augmenting Data Structure CSE 2331/5331

46. Balanced Binary Search Tree Maximum Extract-Max Insert Increase-key Search Delete Successor Predecessor Also support Select operation ?

47. Augment Tree Structure Select ( T, k ) Goal: Augment the binary search tree data structure so as to support Select ( T, k ) efficiently Ordinary binary search tree O(h) time for Select(T, k) Red-black tree (balanced search tree) O(lg n) time for Select(T, k) CSE 2331/5331

48. How To Augment Tree Structure? CSE 2331/5331

49. Example CSE 2331/5331 M9M9 C5C5 A1A1 F3F3 D1D1 H1H1 P3P3 T2T2 Q1Q1

50. How to Setup Size Information? CSE 2331/5331 Postorder traversal of the tree !

51. Augmented Binary Search Tree Let T be an augmented binary search tree OS-Select(x, k): Return the k-th smallest element in the subtree rooted at x OS-Select(T.root, k) returns the k-th smallest elements in the entire tree. CSE 2331/5331 OS-Select(T.root, 5) ?

52. Correctness? Running time? O(h) CSE 2331/5331

53. OS-Rank(T, x) Return the rank of the element x in the linear order determined by an inorder walk of T CSE 2331/5331

54. Example CSE 2331/5331 M9M9 C5C5 A1A1 F3F3 D1D1 H1H1 P3P3 T2T2 Q1Q1 OS-Rank(T, M) ? OS-Rank(T, D) ?

55. Correctness ? Time complexity? O(h) CSE 2331/5331

56. Need to maintain augmented information under dynamic operations Insert / delete Extract-Max can be implemented with delete CSE 2331/5331 Are we done ?

57. Example CSE 2331/5331 M9M9 C5C5 A1A1 F3F3 D1D1 H1H1 P3P3 T2T2 Q1Q1 Insert(J) ?

58. During the downward search, increase the size attribute of each node visited along the path from root to the final insert location. Time complexity: O(h) However, if we have to maintain balanced binary search tree, say Red-black tree Also need to adjust size attribute after rotation CSE 2331/5331

59. Left-Rotate y.size = x.size x.size = x.left.size + x.right.size + 1 O(1) time per rotation CSE 2331/5331

60. Right-rotate can be done similarly. Overall: Two phases: Update size for all nodes along the path from root to insertion location O(h) = O(lg n) time Update size for the Fixup stage involving O(1) rotations O(1) + O(lg n) = O(lg n) time O(h) = O(lg n) time to insert in a Red-Black tree Same asymptotic time complexity as the non-augmented version CSE 2331/5331

61. Delete Two phases: Decrement size in each node on the path from the root to the node to be deleted O(h) = O(lg n) time During Fixup (to maintain balanced binary search tree property), update size for O(1) rotations O(1) + O(lg n) time Overall: O(h) = O(lg n) time CSE 2331/5331

62. Summary Simple example of augmenting data structures In general, the augmented information can be quite complicated Can be a separate data structure! Need to consider how to maintain such information under dynamic changes CSE 2331/5331