Work-efficient parallel union-find with applications to incremental graph connectivity

Abstract

On an undirected graph, how can one quickly answer whether two vertices are connected while allowing more edges to be added incrementally? This is the well-studied incremental graph connectivity (IGC) problem, a fundamental problem that can be efficiently solved using solutions to the classical union-find problem. Motivated by the need to handle larger and rapidly-changing graphs, this paper presents the first shared-memory parallel algorithm for IGC and equivalently, Union-Find that is provably work-efficient (i.e., does no more work than the sequential optimal) and has polylogarithmic parallel depth. It performs path compression in parallel without a lock or speculative execution. We also present a simpler algorithm with slightly worse theoretical properties, but which is easier to implement, and has good practical performance.