## New Algorithms for Enumerating All Maximal Cliques (2004)

Citations: | 33 - 1 self |

### BibTeX

@INPROCEEDINGS{Makino04newalgorithms,

author = {Kazuhisa Makino and Takeaki Uno},

title = {New Algorithms for Enumerating All Maximal Cliques},

booktitle = {},

year = {2004},

pages = {260--272},

publisher = {Springer-Verlag}

}

### Years of Citing Articles

### OpenURL

### 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. 1

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Citation Context ...ated randomly and appear in real-world problems. We show that our algorithm is much faster than the algorithm of Tsukiyama et al.sListing all maximal bipartite cliques is also well-studied (see e.g., =-=[5, 11, 17, 18]-=-). Let us first note that the generation of all maximal bipartite cliques in a bipartite graph G = (V1 ∪V2, E) can be seen as the one of all maximal cliques in the graph ˆ G obtained from G by adding ... |

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Citation Context ...sets of T and maximal bipartite cliques in GT . Hence our algorithms can enumerate all closed item sets in polynomial time delay. Since GT constructed from a database T is usually sparse, it is shown =-=[24]-=- that our algorithms for sparse graphs work pretty well. 3 Definitions and Notations This section introduces some notions and notations of graphs used in the subsequent sections. Let G = (V, E) be a g... |

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Two general methods to reduce delay and change of enumeration algorithms
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Citation Context ...tputs K before all its recursive calls, if the depth of the current recursion is odd; output K after all its recursive calls, otherwise. Although we skip the details, due to the space limitation (see =-=[23]-=- for more details), this reduces the delay to O(∆ 4 ). Theorem 2. For a given graph G = (V, E), all maximal cliques of G can be generated with O(∆ 4 ) time delay and in O(n + m) space, where O(nm) tim... |

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