Jim Anderson Comp 750, Fall 2009 Splay Trees - 1 Splay Trees In balanced tree schemes, explicit rules are followed to ensure balance. In splay trees, there.

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Jim Anderson Comp 750, Fall 2009 Splay Trees - 1 Splay Trees In balanced tree schemes, explicit rules are followed to ensure balance. In splay trees, there are no such rules. Search, insert, and delete operations are like in binary search trees, except at the end of each operation a special step called splaying is done. Splaying ensures that all operations take O(lg n) amortized time. First, a quick review of BST operations…

Jim Anderson Comp 750, Fall 2009 Splay Trees - 2 Splaying In splay trees, after performing an ordinary BST Search, Insert, or Delete, a splay operation is performed on some node x (as described later). The splay operation moves x to the root of the tree. The splay operation consists of sub-operations called zig-zig, zig-zag, and zig.

Jim Anderson Comp 750, Fall 2009 Splay Trees - 3 Zig-Zig T3T3 T4T4 T2T2 T1T1 z y x T1T1 T2T2 T3T3 T4T4 x y z (Symmetric case too) Note: x’s depth decreases by two. x has a grandparent

Jim Anderson Comp 750, Fall 2009 Splay Trees - 4 Zig-Zag T3T3 T4T4 T2T2 T1T1 z y x 10 T1T1 T2T2 T3T3 T4T4 x z (Symmetric case too) Note: x’s depth decreases by two. 30 y x has a grandparent

Jim Anderson Comp 750, Fall 2009 Splay Trees - 5 Zig T1T1 T2T2 T3T3 T4T4 x y (Symmetric case too) Note: x’s depth decreases by one. 30 w T3T3 T4T4 T2T2 T1T1 y x w x has no grandparent (so, y is the root) Note: w could be NIL

Jim Anderson Comp 750, Fall 2009 Splay Trees - 6 Complete Example Splay(78) 50 x y z zig-zag

Jim Anderson Comp 750, Fall 2009 Splay Trees - 7 Complete Example Splay(78) 50 zig-zag x y z

Jim Anderson Comp 750, Fall 2009 Splay Trees - 8 Complete Example Splay(78) 50 x y z zig-zag

Jim Anderson Comp 750, Fall 2009 Splay Trees - 9 Complete Example Splay(78) 50 zig-zag z y x

Jim Anderson Comp 750, Fall 2009 Splay Trees - 10 Complete Example Splay(78) 50 x y z zig-zag

Jim Anderson Comp 750, Fall 2009 Splay Trees - 11 Complete Example Splay(78) zig-zag z y x

Jim Anderson Comp 750, Fall 2009 Splay Trees - 12 Complete Example Splay(78) y x w zig

Jim Anderson Comp 750, Fall 2009 Splay Trees - 13 Complete Example Splay(78) x y w zig

Jim Anderson Result of splaying The result is a binary tree, with the left subtree having all keys less than the root, and the right subtree having keys greater than the root. Also, the final tree is “more balanced” than the original. However, if an operation near the root is done, the tree can become less balanced. Comp 750, Fall 2009 Splay Trees - 14

Jim Anderson Comp 750, Fall 2009 Splay Trees - 15 When to Splay Search:  Successful: Splay node where key was found.  Unsuccessful: Splay last-visited internal node (i.e., last node with a key). Insert:  Splay newly added node. Delete:  Splay parent of removed node (which is either the node with the deleted key or its successor). Note: All operations run in O(h) time, for a tree of height h.