1. Lecture 16 Multiway Search Trees
Slides modified from © 2010 Goodrich, Tamassia & by Prof. Naveen Garg’s Lectures

2. Ordered Maps So far
List/Array
BST
AVL Tree

3. Lets try multiple children…30 25 40 50 75

4. Multiple keys at each node!30 50 25 40 75

5. How many keys do we need at a node to have d-way search tree?

6. Confusion Points 1. nodes 2. keys 3. children

7. Multiway search tree with degree d!
internal nodes have at least 2 children and at max d children
each node stores upto d-1 items

8. In-order traversal of multiway tree30 50 25 29 40 75 80 14 20 26 27

9. Search 12

10. Search 24

11. What is the time complexity of search in a d-way search tree?

12. O(h*log d)

13. 2-4 Tree a node can have 2 to 4 children
all leaf nodes are at the same level

14. Search complexity?
O(h)

15. Min & Max height (h)?

16. Minimum number of entries?
n≥2h+1-1 h≤log2(n+1)-1

17. Maximum number of entries?
n≤4h+1-1 h≥log4(n+1)-1

18. log4(n+1)-1 ≤ h ≤ log2(n+1)-1 1/2log2(n+1) ≤ h+1 ≤ log2(n+1) O(log n)
2-4 Tree Height
log4(n+1)-1 ≤ h ≤ log2(n+1)-1 1/2log2(n+1) ≤ h+1 ≤ log2(n+1) O(log n)

19. Insert

20. Case1: Node has empty space21 23 40 29 7 13 22 32 3 8 10 18 25 35 1 2 4 5 6 9 11 12 14 15 20 24 26 28 30 33 37 39

21. Case2: node is already full
split the node 29 7 13 22 32 3 8 10 18 25 35 1 2 4 5 6 9 11 12 14 15 20 21 23 24 26 28 30 33 37 39 40

22. Promote one key to the parent!7 13 22 32 3 8 10 18 25 35 1 2 4 5 6 9 11 12 14 15 20 21 23 24 33 37 39 40 26 28 29 30

23. Promote one key to the parent!
Split the parent! 13 22 32 3 8 10 18 25 28 35 1 2 9 11 12 14 15 20 21 23 24 33 37 39 40 4 5 6 7 26 29 30

24. Eventually, we may have to create a new root!5 13 22 32 3 8 10 18 25 28 35 1 2 9 11 12 14 15 20 21 23 24 33 37 39 40 4 6 7 26 29 30

25. Complexity of insert? o(log n)
O(log n) to find the node
O(log n) node splits in the worst
Node split takes constant time
o(log n)

26. Delete

27. Claim: We always delete key in the leaf node!13 5 22 3 8 10 18 25 28 1 2 9 11 12 14 15 20 23 24 4 6 7 26 29 30

28. Case 1: leaf node has at least 2 keys
Delete 21 13 5 22 3 8 10 18 25 28 1 2 9 11 12 14 15 20 21 23 24 4 6 7 26 29 30

29. Case 1: leaf node has at least 2 keys
Delete 25 13 5 22 3 8 10 18 25 28 1 2 9 11 12 14 15 20 23 24 4 6 7 26 29 30

30. Case 2: leaf node has only 1 key
Delete 20
borrow from sibling 13 5 22 3 8 10 18 24 28 1 2 9 11 12 14 15 20 23 4 6 7 26 29 30

31. What if sibling has only 1 key?
Delete 23
Merge siblings 13 5 22 3 8 10 15 24 28 1 2 9 11 12 14 18 23 4 6 7 26 29 30

32. What if the parent has only 1 key?
Delete 18
Cascaded merge 13 5 22 3 8 10 15 28 1 2 9 11 12 14 18 4 6 7 29 30 24 26

33. 2-4 Trees The height of a (2,4) tree is O(log n).
Search, insertion and deletion each take O(log n) .
Pretty fundamental to the ideas of advanced trees such as Red-Black trees.