Lecture 14 splay tree
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This document discusses splay trees, a type of self-balancing binary search tree where frequently accessed elements are moved closer to the root through rotations. Splay trees provide O(log n) time for search, insert, and delete operations while being simpler to implement than other self-balancing trees. When an element is accessed in a splay tree, it is "splayed" or rotated to the root through zig, zig-zig, or zig-zag steps depending on its position. Splay trees prioritize frequently accessed elements but may become unbalanced if all elements are accessed sequentially. The document provides examples of splaying tree rotations and constructs splay trees for two example sequences.
Lecture 14 splay tree
- 1. Lecture 14 Splay Tree Abirami Sivaprasad
- 2. Splay Tree A splay tree is a self-balancing binary search tree with an additional property that recently accessed elements can be re-accessed fast. It is said to be an efficient binary tree because it performs basic operations such as insertion, search and deletion operations in O(log(n)) time. When a node in a splay tree is accessed, it is rotated or "splayed" to the root thereby changing the structure of the tree. Since the most frequently accessed node is always moved closer to the starting point of the search (or the root node), those nodes are therefore located faster. A simple idea behind it is that if an element is accessed, it is likely that it will be accessed again.
- 3. Advantages A splay tree gives good performance for search, insert and delete operations. Splay trees are considerably simpler to implement than other self-balancing binary search trees, such as red-black trees or AVL trees, while their average-case performance is just as efficient. Splay tree minimizes memory requirements . Unlike other types of self balancing trees, splay trees gives good performance (O(log n)) with nodes containing identical keys.
- 4. Disadvantages While sequentially accessing all the nodes of the tree in a sorted order, the resultant tree becomes completely unbalanced. This takes n accesses of the tree in which each access takes O(log n) time. For uniform access, the performance of a splay tree will be considerably worse than a somewhat balanced simple binary search tree.
- 5. Splaying Factors to consider: Whether N is the left or right child of its parent P Whether P is the root or not, and if not Whether P is the left or right child of its parent, G (N’s grandparent).
- 6. Zig Step P N T 1 T 2 T 3 N T 1 P T 2 T 3 The zig operation is done when P (the parent of N) is the root of the splay tree. In the zig step, the tree is rotated on the edge between N and P. Zig step is usually performed as the last step in a splay operation and only when N has odd depth at the beginning of the operation.
- 7. Zig-zig Step: The zig- zig operation is performed when P is not the root. In addition to this, N and P are either both right children or are both left children of their parent’s. Figure shows the case where N and P are the left children. During the zig- zig step, first the tree is rotated on the edge joining P and its parent G, and then again rotated on the edge joining N and P. G P T 4 N T 1 T 2 T 3 P G T 4 N T 1 T 2 N T 1 P G T 4 T 2 T 3
- 8. Zig-zag Step: The zig-zag operation is performed when P is not the root. In addition to this, N is a right child of P and P is a left child of G or vice versa. In zig-zag step, the tree is first rotated on the edge between N and P, and then rotated on the edge between P and G. G P T 4 T 1 N T 2 T 3 G N T 4 P T 1 T 2 T 3 N G P T 1 T 2 T 1 T 2
- 9. Exercise 1. Construct the Splay Tree for the following sequence 10, 30, 50, 40, 60,70 2. Construct the Splay Tree for the following sequence 34, 67, 45, 89, 12, 37
- 10. Thank U