Multi-Way search Trees

1. Multi-Way search Trees • 2-3 Trees: a. Nodes may contain 1 or 2 items. b. A node with k items has k + 1 children c. All leaves are on same level.

2. Example • A 2-3 tree storing 18 items. 20 80 5 30 70 90 100 25 40 50 75 85 110 120 2 4 10 95

3. Updating • Insertion: • Find the appropriate leaf. If there is only one item, just add to leaf. • Insert(23); Insert(15) • If no room, move middle item to parent and split remaining two items among two children. • Insert(3);

4. Insertion • Insert(3); 20 80 5 30 70 90 100 40 50 75 85 110 120 2 3 4 10 15 23 25 95

5. Insert(3); • In mid air… 20 80 5 30 70 90 100 3 23 25 40 50 75 85 110 120 2 10 15 95 4

6. Done…. 20 80 3 5 30 70 90 100 4 23 25 40 50 75 85 110 120 2 10 15 95

7. Tree grows at the root… • Insert(45); 20 80 3 5 30 70 90 100 4 25 40 45 50 75 85 110 120 2 10 95

8. New root: 45 20 80 3 5 30 70 90 100 4 25 40 50 75 85 110 120 2 10 95

9. Delete • If item is not in a leaf exchange with in-order successor. • If leaf has another item, remove item. • Examples: Remove(110); • (Insert(110); Remove(100); ) • If leaf has only one item but sibling has two items: redistribute items. Remove(80);

10. Remove(80); • Step 1: Exchange 80 with in-order successor. 45 20 85 3 5 30 70 90 100 110 120 4 25 40 50 75 80 2 10 95

11. Remove(80); • Redistribute 45 20 85 3 5 30 70 95 110 120 4 25 40 50 75 90 2 10 100

12. Some more removals… • Remove(70); Swap(70, 75); Remove(70); “Merge” Empty node with sibling; Join parent with node; Now every node has k+1 children except that one node has 0 items and one child. Sibling can spare an item: redistribute. • 110

13. Delete(70) 45 20 85 3 5 30 75 95 110 120 4 25 40 50 90 2 10 100

14. New tree: • Delete(85) will “shrink” the tree. 45 20 95 3 5 30 85 110 120 4 25 40 50 90 100 2 10

15. Details • 1. Swap(85, 90) //inorder successor • 2. Remove(85) //empty node created • 3. Merge with sibling • 4. Drop item from parent// (50,90) empty Parent • 5. Merge empty node with sibling, drop item from parent (95) • 6. Parent empty, merge with sibling drop item. Parent (root) empty, remove root.

16. “Shorter” 2-3 Tree 20 45 3 5 30 95 110 50 90 120 4 25 40 100 2 10

17. Deletion Summary • If item k is present but not in a leaf, swap with inorder successor; • Delete item k from leaf L. • If L has no items: Fix(L); • Fix(Node N); • //All nodes have k items and k+1 children • // A node with 0 items and 1 child is possible, it will have to be fixed.

18. Deletion (continued) • If N is the root, delete it and return its child as the new root. • Example: Delete(8); 5 5 1 2 3 3 3 5 3 8 Return 3 5

19. Deletion (Continued) • If a sibling S of N has 2 items distribute items among N, S and the parent P; if N is internal, move the appropriate child from S to N. • Else bring an item from P into S; • If N is internal, make its (single) child the child of S; remove N. • If P has no items Fix(P) (recursive call)

20. (2,4) Trees • Size Property: nodes may have 1,2,3 items. • Every node, except leaves has size+1 children. • Depth property: all leaves have the same depth. • Insertion: If during the search for the leaf you encounter a “full” node, split it.

21. (2,4) Tree 3 8 10 45 60 70 90 100 50 55 25

22. Insert(38); Insert(38); 45 10 60 70 90 100 3 8 25 38 50 55

23. Insert(105) • Insert(105); 45 10 60 90 3 8 50 55 70 100 105 25 38

24. Removal • As with BS trees, we may place the node to be removed in a leaf. • If the leaf v has another item, done. • If not, we have an UNDERFLOW. • If a sibling of v has 2 or 3 items, transfer an item. • If v has 2 or 3 siblings we perform a transfer

25. Removal • If v has only one sibling with a single item we drop an item from the parent to the sibling, remove v. This may create an underflow at the parent. We “percolate” up the underflow. It may reach the root in which case the root will be discarded and the tree will “shrink”.

26. Delete(15) 35 20 60 6 15 40 50 70 80 90

27. Delete(15) 35 20 60 6 40 50 70 80 90

28. Continued • Drop item from parent 35 60 6 20 40 50 70 80 90

29. Fuse 35 60 6 20 40 50 70 80 90

30. Drop item from root • Remove root, return the child. 35 60 6 20 40 50 70 80 90

31. Summary • Both 2-3 trees and 2-4 trees make it very easy to maintain balance. • Insertion and deletion easier for 2-4 tree. • Cost is waste of space in each node. Also extra comparison inside each node. • Does not “extend” binary trees.

32. Red-Black Trees • Root property: Root is BLACK. • External Property: Every external node is BLACK (external nodes: null nodes) • Internal property: Children of a RED node are BLACK. • Depth property: All external nodes have the same BLACK depth.

33. A RedBlack tree.Black depth 3. 30 15 70 10 20 85 60 5 80 90 50 65 40 55

34. RedBlack Insertion

35. Red Black Trees, Insertion • Find proper external node. • Insert and color node red. • No black depth violation but may violate the red-black parent-child relationship. • Let: z be the inserted node, v its parent and u its grandparent. If v is red then u must be black.

36. Color adjustments. • Red child, red parent. Parent has a black sibling (Zig-Zag). a b u w v z Vl Zr Zl

37. Rotation • Z-middle key. Black height does not change! No more red-red. a b z u v w Zr Zl Vl

38. Color adjustment II a b u w v Vr z Zr Zl

39. Rotation II a • v-middle key b v u z w Zl Zr Vr

40. Recoloring • Red child, red parent. Parent has a red sibling. a b u w v z Vl Zr

41. Color adjustment • Red-red may move up… a b u w v z Vl Zr Zl

42. Red Black Tree • Insert 10 – root 10

43. Red Black Tree • Insert 10 – root (external nodes not shown) 10

44. Red Black Tree • Insert 85 10 85

45. Red Black Tree • Insert 15 10 85 15

46. Red Black Tree • Rotate – Change colors 15 10 85

47. Red Black Tree • Insert 70 15 10 85 70

48. Red Black Tree • Change Color 15 10 85 70

49. Red Black Tree • Insert 20 (sibling of parent is black) 15 10 85 70 20

50. Red Black Tree • Rotate 15 10 70 85 20