Disjoint Set Neil Tang 02/26/2008 CS223 Advanced Data Structures and Algorithms

CS223 Advanced Data Structures and Algorithms Class Overview Disjoint Set and An Application Basic Operations Linked-list Implementation Array Implementation Union-by-Size and Union-by-Height(Rank) Find with Path Compression Worst-Case Time Complexity CS223 Advanced Data Structures and Algorithms

CS223 Advanced Data Structures and Algorithms Disjoint Set Given a set of elements, we can have a collection S = {S1, S2, ... Sk} of disjoint dynamic (sub) sets. Representative of a set: We choose one element of a set to identify the set, e.g., we use the root of a tree to identify a tree, or the head element of a linked list to access the linked list. Usually, we want to find out if two elements belong to the same set. CS223 Advanced Data Structures and Algorithms

CS223 Advanced Data Structures and Algorithms An Application Given an undirected graph G = (V, E) We may want to find all connected components, whether the graph is connected or whether two given nodes belong to the same connected component. a b c d g e f h i CS223 Advanced Data Structures and Algorithms

CS223 Advanced Data Structures and Algorithms Basic Operations find(x): find which disjoint set x belongs to Union(x,y): Union set x and set y. CS223 Advanced Data Structures and Algorithms

Linked-list Implementation f nil head tail a b c find(b) a b c f nil tail union(f, b) CS223 Advanced Data Structures and Algorithms

CS223 Advanced Data Structures and Algorithms Array Implementation Assume that all the elements are numbered sequentially from 0 to N-1. CS223 Advanced Data Structures and Algorithms

CS223 Advanced Data Structures and Algorithms Array Implementation CS223 Advanced Data Structures and Algorithms

CS223 Advanced Data Structures and Algorithms Array Implementation CS223 Advanced Data Structures and Algorithms

CS223 Advanced Data Structures and Algorithms Union Operation Time complexity: O(1) CS223 Advanced Data Structures and Algorithms

CS223 Advanced Data Structures and Algorithms Find Operation Time complexity: O(N) CS223 Advanced Data Structures and Algorithms

CS223 Advanced Data Structures and Algorithms Union-by-Size Make the smaller tree a subtree of the larger and break ties arbitrarily. CS223 Advanced Data Structures and Algorithms

Union-by-Height (Rank) Make the shallow tree a subtree of the deeper and break ties arbitrarily. CS223 Advanced Data Structures and Algorithms

CS223 Advanced Data Structures and Algorithms Size and Height -1 4 -5 6 1 2 3 4 5 6 7 -1 4 -3 6 CS223 Advanced Data Structures and Algorithms

Union-by-Height (Rank) Time complexity: O(logN) CS223 Advanced Data Structures and Algorithms

CS223 Advanced Data Structures and Algorithms Worst-Case Tree CS223 Advanced Data Structures and Algorithms

Find with Path Compression CS223 Advanced Data Structures and Algorithms

Find with Path Compression CS223 Advanced Data Structures and Algorithms

Find with Path Compression Fully compatible with union-by-size. Not compatible with union-by-height. Union-by-size is usually as efficient as union-by-height. CS223 Advanced Data Structures and Algorithms

Worst-Case Time Complexity If both union-by-rank and path compression heuristics are used, the worst-case running time for any sequence of M union/find operations is O(M * (M,N)), where (M, N) is the inverse Ackermann function which grows even slower than logN. For any practical purposes, (M, N) < 4. CS223 Advanced Data Structures and Algorithms