Splay Trees CSE 331 Section 2 James Daly. Reminder Homework 2 is out Due Thursday in class Project 2 is out Covers tree sets Due next Friday at midnight.

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Presentation transcript:

Splay Trees CSE 331 Section 2 James Daly

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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