New Algorithms for Enumerating All Maximal Cliques

Abstract

Abstract. In this paper, we consider the problems of generating all maximal (bipartite) cliques in a given (bipartite) graph G = (V, E) with n vertices and m edges. We propose two algorithms for enumerating all maximal cliques. One runs with O(M (n)) time delay and in O(n 2 ) space and the other runs with O(∆ 4 ) time delay and in O(n + m) space, where ∆ denotes the maximum degree of G, M (n) denotes the time needed to multiply two n × n matrices, and the latter one requires O(nm) time as a preprocessing. For a given bipartite graph G, we propose three algorithms for enumerating all maximal bipartite cliques. The first algorithm runs with O(M (n)) time delay and in O(n 2 ) space, which immediately follows from the algorithm for the nonbipartite case. The second one runs with O(∆ 3 ) time delay and in O(n + m) space, and the last one runs with O(∆ 2 ) time delay and in O(n + m + N ∆) space, where N denotes the number of all maximal bipartite cliques in G and both algorithms require O(nm) time as a preprocessing. Our algorithms improve upon all the existing algorithms, when G is either dense or sparse. Furthermore, computational experiments show that our algorithms for sparse graphs have significantly good performance for graphs which are generated randomly and appear in real-world problems.

A Correspondence Between Maximal Complete Bipartite Subgraphs and Closed Patterns

“…The problem of enumerating all maximal complete bipartite subgraphs (called maximal bipartite cliques there) from a bipartite graph has been investigated by (Makino & Uno, 2004). The difference is that our work is to enumerate all the subgraphs from any graphs (without self loops and undirected), but Makino and Uno's work is limited to enumerating from only bipartite graphs.…”

“…Interest in graphs and their applications has grown exponentially in the past two decades (Gross & Yellen, 2004;Makino & Uno, 2004), largely due to the usefulness of graphs as models in many areas such as mathematical research, electrical engineering, computer programming, business administration, sociology, economics, marketing, biology, and networking and communications. In particular, many problems can be modelled with maximal complete bipartite subgraphs (see the definition below) formed by grouping two non-overlapping subsets of vertices of a certain graph that show a kind of full connectivity between them.…”

Mining maximal cliques from a large graph using MapReduce: Tackling highly uneven subproblem sizes

“…These algorithms stem from the Tsukiyama et al [41] algorithm, which has a running time of O(|V | |E|µ), where µ is the number of maximal cliques. Other output sensitive algorithms include [8,25,21,31], with [31] providing one of the best theoretical guarantees. However, these output sensitive algorithms tend not to perform as well as the worst case optimal algorithms in practice [40,13].…”

“…While our algorithm maybe more broadly applicable, in this work we focus our implementation on the widely used MapReduce [10,11,17] framework for cluster computing. While MCE is widely studied in the sequential setting [4,5,8,25,13,23,21,31,40,41], there is relatively less work on parallel methods [45,12,38,43,30].…”

Detection of Novel-Type Brute Force Attacks Used Ephemeral Springboard IPs as CamouflageDetection of Novel-Type Brute Force Attacks Used Ephemeral Springboard IPs as Camouflage

“…Threshold of correlation varies 0.8, 0.6 and 0.4. Note that we calculate correlation based on Pearson product-moment correlation coefficient and we use MACE in order to apply enumeration maximal cliques [25]. Fig.…”