Splay Trees. (Splay Trees) Data Structures and Programming Spring / 27
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1 Splay Trees (Splay Trees) Data Structures and Programming Spring / 27
2 Basic Idea Invented by Sleator and Tarjan (1985) Blind rebalancing no height info kept! Worst-case time per operation is O(n) Worst-case amortized time is O(log n) Insert/find always rotates node to the root! Good locality: Most commonly accessed keys move high in tree become easier and easier to find Incorporate Move-to-Top strategy (Splay Trees) Data Structures and Programming Spring / 27
3 Idea (Splay Trees) Data Structures and Programming Spring / 27
4 A Simple Idea (That Does Not Work) Perform single rotations bottom up. There is a sequence of M operations requiring Ω(M N) time, so this idea is not quite good enough. (Splay Trees) Data Structures and Programming Spring / 27
5 Zig-Zag (Splay Trees) Data Structures and Programming Spring / 27
6 Zig-Zig Note: Rotate along edge {g, p}, then {p, n}. (Splay Trees) Data Structures and Programming Spring / 27
7 Why Splaying Helps Node n and its children are always helped (raised) Except for last step, nodes that are hurt by a zig-zag or zig-zig are later helped by a rotation higher up the tree! Result: shallow nodes may increase depth by one or two helped nodes decrease depth by a large amount If a node n on the access path is at depth d before the splay, it is at about depth d/2 after the splay Exceptions are the root, the child of the root, and the node splayed (Splay Trees) Data Structures and Programming Spring / 27
8 Splaying Example (Splay Trees) Data Structures and Programming Spring / 27
9 Splaying Example (Cont d) (Splay Trees) Data Structures and Programming Spring / 27
10 Splaying Example (Cont d) (Splay Trees) Data Structures and Programming Spring / 27
11 Splaying Example (Cont d) (Splay Trees) Data Structures and Programming Spring / 27
12 Splaying Example (Cont d) (Splay Trees) Data Structures and Programming Spring / 27
13 Locality Locality if an item is accessed, it is likely to be accessed again soon Why? Assume m n access in a tree of size n Total worst case time is O(m log n) O(log n) per access amortized time Suppose only k distinct items are accessed in the m accesses. Time is O(n log n + m log k) O(n log n) getting those k items near root m log k those k items are all at the top of the tree Compare with O(m log n) for AVL tree (Splay Trees) Data Structures and Programming Spring / 27
14 Splay Operations: Insert To insert, could do an ordinary BST insert but would not fix up tree A BST insert followed by a find (splay)? Better idea: do the splay before the insert! How? Split(T, x) creates two BSTs L and R All elements of T are in either L or R All elements in L are x All elements in R are > x L and R share no elements Insert as root, with children L and R (Splay Trees) Data Structures and Programming Spring / 27
15 Splitting in Splay Trees How can we split? We have the splay operation We can find x or the parent of where x would be if we were to insert it as an ordinary BST We can splay x or the parent to the root Then break one of the links from the root to a child (Splay Trees) Data Structures and Programming Spring / 27
16 Split (Splay Trees) Data Structures and Programming Spring / 27
17 Back to Insert (Splay Trees) Data Structures and Programming Spring / 27
18 Insert Example (Splay Trees) Data Structures and Programming Spring / 27
19 Delete (Splay Trees) Data Structures and Programming Spring / 27
20 Join Join(L, R): given two trees such that L < R, merge them Splay on the maximum element in L then attach R (Splay Trees) Data Structures and Programming Spring / 27
21 Delete Completed (Splay Trees) Data Structures and Programming Spring / 27
22 Delete Example (Splay Trees) Data Structures and Programming Spring / 27
23 Top-down Splay Case 1: X is the node we are splaying Case 2: (zig-zig) The node we are splaying is in the subtree rooted at X (Splay Trees) Data Structures and Programming Spring / 27
24 Top-down Splay (Cont d) Case 3: (zig-zag) The node we are splaying is in the subtree rooted at X Case 4: (the last step) X is the node we wish to splay (Splay Trees) Data Structures and Programming Spring / 27
25 Example Splay at B (Splay Trees) Data Structures and Programming Spring / 27
26 Example Splay at B (Cont d) (Splay Trees) Data Structures and Programming Spring / 27
27 Summary Splay trees are arguably the most practical kind of self-balancing trees If number of finds is much larger than n, then locality is crucial! Also supports efficient Split and Join operations useful for other tasks (Splay Trees) Data Structures and Programming Spring / 27
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