A linear time algorithm for maximal clique enumeration in large sparse graphs

Abstract

A maximal clique is one of the most fundamental dense substructures in an undirected graph, and maximal clique enumeration (MCE) plays an essential role in densely connected subgraphs discovering. Existing algorithms of maximal clique enumeration employ recursive iteration of adjacent nodes as guiding thought, which incurs high time complexity. In this paper, we propose a linear time algorithm, CM-Constructor (Candidate Map Constructor), for maximal clique enumeration in large sparse graphs which generates a novel data structure called candidate map as result. A candidate map holds not only all maximal cliques of an undirected graph but also some non-maximal cliques that can be easily discarded via an inverted clique tree. To the best of our knowledge, CM-Constructor is the first algorithm to tackle maximal clique enumeration problem utilizing linear procedure. It generates all maximal cliques without duplications for an undirected graph G=(V,E) within O(|E|) time.