Tree Rotations & Splay Trees

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 balancezig-zagpattern

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 • Insert 1-6 in order Splay Tree: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