Union-Find with Constant Time Deletions

Abstract

A union-find data structure maintains a collection of disjoint sets under the operations makeset, union, and find. Kaplan, Shafrir, and Tarjan [SODA 2002] designed data structures for an extension of the union-find problem in which items of the sets maintained may be deleted. The cost of a delete operation in their implementations is essentially the same as the cost of a find operation; namely, O (log n ) worst-case and O (α ⌈ M / N ⌉ ( n )) amortized, where n is the number of items in the set returned by the find operation, N is the total number of makeset operations performed, M is the total number of find operations performed, and α ⌈ M / N ⌉ ( n ) is a functional inverse of Ackermann’s function. They left open the question whether delete operations can be implemented more efficiently than find operations, for example, in o (log n ) worst-case time. We resolve this open problem by presenting a relatively simple modification of the classical union-find data structure that supports delete, as well as makeset and union operations, in constant worst-case time, while still supporting find operations in O (log n ) worst-case time and O (α ⌈ M/N⌉ ( n )) amortized time. Our analysis supplies, in particular, a very concise potential-based amortized analysis of the standard union-find data structure that yields an O (α ⌈ M / N ⌉ ( n )) amortized bound on the cost of find operations. All previous potential-based analyses yielded the weaker amortized bound of O (α ⌈ M / N ⌉ ( N )). Furthermore, our tighter analysis extends to one-path variants of the path compression technique such as path splitting .