Lecture 26

1. Lecture 26 • Welcome back! • Binary search trees • 15.1-5

2. Binary Search Trees • Dictionary Operations: • get(key) • put(key, value) • remove(key) • Additional operations: • ascend() • get(index) (indexed binary search tree) • remove(index) (indexed binary search tree)

3. n is number of elements in dictionary Complexity Of Dictionary Operationsget(), put() and remove()

4. D is number of buckets Complexity Of Other Operationsascend(), get(index), remove(index)

5. Definition Of Binary Search Tree A binary tree. Each node has a (key, value) pair. For every node x, all keys in the left subtree of x are smaller than that in x. For every node x, all keys in the right subtree of x are greater than that in x.

6. Example Binary Search Tree 20 10 40 6 15 30 25 2 8 Only keys are shown.

7. 20 10 40 6 15 30 25 2 8 The Operation ascend() Do an inorder traversal. O(n) time.

8. 20 10 40 6 15 30 25 2 8 The Operation get() 8 Get pair whose key is 8

9. The Operation get() 20 8 10 40 6 15 30 25 2 8 Get pair whose key is 8

10. The Operation get() 20 10 40 8 6 15 30 25 2 8 Get pair whose key is 8

11. The Operation get() 20 10 40 6 15 30 8 25 2 8 Get pair whose key is 8

12. The Operation get() 20 10 40 6 15 30 25 2 8 8! Get pair whose key is 8

13. The Operation get() 20 10 40 6 15 30 25 2 8 Complexity is O(height) = O(n), where n is number of nodes/elements.

14. The Operation put() 35 20 10 40 6 15 30 25 2 8 Put a pair whose key is 35.

15. The Operation put() 20 10 40 35 6 15 30 25 2 8 Put a pair whose key is 35.

16. The Operation put() 20 10 40 6 15 30 35 25 2 8 Put a pair whose key is 35.

17. The Operation put() 20 10 40 6 15 30 25 35 2 8 Put a pair whose key is 35.

18. The Operation put() 20 7 10 40 6 15 30 25 35 2 8 Put a pair whose key is 7.

19. The Operation put() 20 10 40 7 6 15 30 25 35 2 8 Put a pair whose key is 7.

20. The Operation put() 20 10 40 6 15 30 7 25 35 2 8 Put a pair whose key is 7.

21. The Operation put() 20 10 40 6 15 30 25 35 2 8 7 Put a pair whose key is 7.

22. The Operation put() 20 10 40 6 15 30 25 35 2 8 7 Put a pair whose key is 7.

23. The Operation put() 20 10 40 6 15 30 18 25 35 2 8 7 Put a pair whose key is 18.

24. The Operation put() 20 10 40 6 15 30 18 25 35 2 8 7 Complexity of put() is O(height).

25. The Operation remove() • Three cases: • Element is in a leaf. • Element is in a degree 1 node. • Element is in a degree 2 node.

26. Remove From A Leaf 20 10 40 6 15 30 18 25 35 2 8 7 Remove a leaf element. key = 7

27. Remove From A Leaf 20 10 40 6 15 30 18 25 35 2 8 Remove a leaf element. key = 7

28. Remove From A Leaf (contd.) 20 10 40 6 15 30 18 25 35 2 8 7 Remove a leaf element. key = 35

29. Remove From A Leaf (contd.) 20 10 40 6 15 30 18 25 2 8 7 Remove a leaf element. key = 35

30. Remove From A Degree 1 Node 20 10 40 6 15 30 18 25 35 2 8 7 Remove from a degree 1 node. key = 40

31. Remove From A Degree 1 Node 20 10 6 15 30 18 25 35 2 8 7 Remove from a degree 1 node. key = 40

32. Remove From A Degree 1 Node 20 10 6 15 30 18 25 35 2 8 7 Remove from a degree 1 node. key = 40

33. Remove From A Degree 1 Node (contd.) 20 10 40 6 15 30 18 25 35 2 8 7 Remove from a degree 1 node. key = 15

34. Remove From A Degree 1 Node (contd.) 20 10 40 6 30 18 25 35 2 8 7 Remove from a degree 1 node. key = 15

35. Remove From A Degree 1 Node (contd.) 20 10 40 6 30 18 25 35 2 8 7 Remove from a degree 1 node. key = 15

36. Remove From A Degree 2 Node 20 10 40 6 15 30 18 25 35 2 8 7 Remove from a degree 2 node. key = 10

37. Remove From A Degree 2 Node 20 40 6 15 30 18 25 35 2 8 7 How to patch tree back together?

38. Remove From A Degree 2 Node 20 40 6 15 30 18 25 35 2 8 7 Replace with largest key in left subtree (or smallest in right subtree).

39. Remove From A Degree 2 Node 20 8 40 6 15 30 18 25 35 2 8 7 Largest (smallest) key in left (right) subtree must be in a leaf or degree 1 node.

40. Remove From A Degree 2 Node 20 8 40 6 15 30 18 25 35 2 7 Take the 1 (possibly empty) subtree of the replacement node and promote it.

41. Remove From A Degree 2 Node 20 8 40 6 15 30 18 25 35 2 7 Done

42. Another Remove From A Degree 2 Node 20 10 40 6 15 30 18 25 35 2 8 7 Remove from a degree 2 node. key = 20

43. Remove From A Degree 2 Node 10 40 6 15 30 18 25 35 2 8 7 Replace with largest in left subtree.

44. Remove From A Degree 2 Node 18 10 40 6 15 30 25 35 2 8 7 No left child to promote this time.

45. Remove From A Degree 2 Node 18 10 40 6 15 30 25 35 2 8 7 Complexity is O(height).

46. Indexed Binary Search Tree • Binary search tree • Each node has an additional field • leftSize = number of nodes in its left subtree

47. Example Indexed Binary Search Tree 7 20 4 3 10 40 1 0 1 6 15 30 0 0 0 0 1 18 25 35 2 8 0 7 leftSize values are in red

48. leftSize And Rank Rank of an element is its position in inorder (inorder = ascending key order). [2,6,7,8,10,15,18,20,25,30,35,40] rank(2) = 0 rank(15) = 5 rank(20) = 7 leftSize(x) = rank(x)with respect to elements in subtree rooted at x

49. leftSize And Rank 7 20 4 3 10 40 1 0 1 6 15 30 0 0 0 0 1 18 25 35 2 8 0 7 sorted list = [2,6,7,8,10,15,18,20,25,30,35,40]

50. 7 20 4 3 10 40 1 0 1 6 15 30 0 0 0 0 1 18 25 35 2 8 0 7 sorted list = [2,6,7,8,10,15,18,20,25,30,35,40] get(index) And remove(index)