1. Splay Trees CSE 331 Section 2 James Daly

2. Reminder Homework 2 is out Due Thursday in class Project 2 is out Covers tree sets Due next Friday at midnight

3. Review: Binary Tree Every node has at most 2 children Left and right child Variation: n-ary trees have at most n children

4. Review: Binary Search Tree For every node Left descendents smaller (l ≤ k) Right descendents bigger (r ≥ k) k <k>k

5. Review: AVL Trees Named for Adelson-Velskii & Landis Self-balancing binary search tree Goal: Keep BST in balance Ability to check / track balance

6. Review: AVL Trees For every node, the height of both subtrees differs by at most 1.

7. RotateRight(&t) // Rotates t down to the right // Brings up left child left ← t.left temp = left.right left.right ← t t.left ← temp t ← left

8. Left-Left / Right-Right Case D B a6a6 c5c5 e5e5 B D e5e5 c5c5 a6a6

9. Left-Right / Right-Left Case F B a5a5 e4e4 g5g5 D c5c5 D B a5a5 c5c5 F e4e4 g5g5 B a5a5 c5c5 e4e4 D F g5g5

10. Motivation Frequently we care more about how long it takes to do a string of operations than any one O(n) isn’t too bad – if we do it only once Want O(m log n) for m searches Recently used items are more likely to be called again Basis for caches

11. Splay Tree Not kept rigorously balanced When a node is accessed, rotate it to the top Next search for the same item will be very quick No need for extra information AVL Tree: Node height Red-Black Tree: Node color

12. Wrong way k1 k2 k3 k4 k5 A B C D E F

13. Wrong way k2 k3 k4 k5 k1 A B C D E F

14. Wrong way k2 k3 k4 k5 k1 A B C D E F

15. Wrong way k2 k3 k4 k5 k1 A B C D E F

16. Wrong way k2 k3 k4 k5 k1 A B C D E F

17. Problem K3 was pushed down almost as far as K2 came up. Easy to show that you could keep selecting bad nodes O(n 2 ) time total. Need a smarter way to do this

18. Splaying Find X If root, done If parent(X) = root, rotate up Otherwise, X has both parent and grandparent Two cases: Zig-Zag and Zig-Zig

19. Zig-Zag X P G A B C D X PG A C B D

20. Zig-Zig X P G A B C D X P G A B C D

21. Splaying Tends to reduce the height of the tree Many items will be half as deep as before Some items may be at most 2 deeper than before

22. Right way k1 k2 k3 k4 k5 A B C D E F

23. Right way k1 k2 k3 k4 k5 A C B D E F

24. Right way k1 k2 k3 k4 k5 A C B D E F

25. Insertion Insert X as normal for a BST Splay X to the top

26. Deletion Remove X as normal for a BST Splay its parent to the top