1. Tree Rotations & Splay Trees

2. BST Structure BST's only perform well when balanced But common cases lead to unbalanced trees

3. Restoring Balance How could this unbalanced tree be balanced?

4. Restoring Balance How could this unbalanced tree be balanced?

5. Rotation Parent node become child, child becomes parent – Right rotation : Everyone moves right (clockwise) – Left rotation : Counter clockwise

6. Rotation Description Describe rotations by parent – Rotate 38 right – Rotate 21 right

7. Child Issues Want to rotate 38 right (make 21 parent) – Where does 26 go?

8. Child Issues Right child of old child  left child of old parent

9. Right Rotation

10. Left Rotation

11. Multi Step Want to balance this tree:

12. Multi Step Want to balance this tree: – Rotate root right is just as bad:

13. Multi Step Want to balance zig-zag pattern

14. Multi Step Want to balance zig-zag pattern – Rotate child of root left

15. Multi Step Want to balance zig-zag pattern – Rotate child of root left – Rotate root right

16. Deciding What to Rotate Figuring out where to rotate is tough – Need more info than available locally in BST AVL, RB

17. Deciding What to Rotate Figuring out where to rotate is tough – Need more info than available locally in BST AVL, RB Alternative: – Keep recent nodes at top of tree – Do a pretty good job balancing

18. Splay Tree Splaying : move node to root using rotates Splay Tree : after any access, move node to root – O(1) to access same element again – Faster to access nearby values – Amortized O(logn) average access

19. How it works: Insert 1-6 in order BST: http://www.cs.usfca.edu/~galles/visualization /BST.html http://www.cs.usfca.edu/~galles/visualization /BST.html Insert 1-6 in order Splay Tree: http://www.cs.usfca.edu/~galles/visualization /SplayTree.html http://www.cs.usfca.edu/~galles/visualization /SplayTree.html

20. Splay Algorithm Rotating upwards to root does not always balance tree:

21. Find 1 Insert 1-6, Find 1:

22. Find 1 Insert 1-6, Find 1:

23. Zig-Zig and Zig-Zag Rotations done in 3 ways – Node is right below root: Zig (Rotate left or right)

24. Zig-Zig and Zig-Zag Rotations done in 3 ways – Path to node is left-left or right-right: Zig-Zig (Rotate grandparent, then rotate parent)

25. Zig-Zig and Zig-Zag Rotations done in 3 ways – Path to node is left-right or right-left: Zig-Zag (Rotate parent one way, then grand parent other way)

26. Sample: Splay 42

27. Sample: Splay 42 – It is left-left child : Zig-Zig

28. Sample: Splay 42 – It is left-left child : Zig-Zig Rotate 52 right

29. Sample: Splay 42 – It is left-left child : Zig-Zig Rotate 52 right, rotate 47 right

30. Sample: Splay 42 – Now it is a left-right child : Zig-Zag

31. Sample: Splay 42 – Now it is a left-right child : Zig-Zag 35 Rotates left

32. Sample: Splay 42 – Now it is a left-right child : Zig-Zag 35 Rotates left, 70 rotate right