Topic
K-tree
About: K-tree is a research topic. Over the lifetime, 427 publications have been published within this topic receiving 12096 citations.
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TL;DR: Algorithms are provided to track all maximal cliques in a fully dynamic graph to solve fuzzy clustering problems in models with non-disjunct clusters.
Abstract: Clustering applications dealing with perception based or biased data lead to models with non-disjunct clusters. There, objects to be clustered are allowed to belong to several clusters at the same time which results in a fuzzy clustering. It can be shown that this is equivalent to searching all maximal cliques in dynamic graphs like Gt e (V,Et), where Et − 1 ⊂ Et, t e 1,…,T; E0 e p. In this article algorithms are provided to track all maximal cliques in a fully dynamic graph.
90 citations
89 citations
TL;DR: It is shown that a perfect elimination scheme of a chordal graph can be computed in O(log n ) time with O( n 3 ) processors on CRCW PRAM once a clique tree of the graph is given.
88 citations
01 Apr 2016
TL;DR: This paper designs an algorithm to maintain k candidates in the process of maximal clique enumeration, which has limited memory footprint and can achieve a guaranteed approximation ratio, and introduces a novel light-weight $$\mathsf {PNP}$$PNP-$$\ mathsf {Index}$$Index, based on which an optimal maximal cliques maintenance algorithm is designed.
Abstract: Maximal clique enumeration is a fundamental problem in graph theory and has been extensively studied. However, maximal clique enumeration is time-consuming in large graphs and always returns enormous cliques with large overlaps. Motivated by this, in this paper, we study the diversified top-k clique search problem which is to find top-k cliques that can cover most number of nodes in the graph. Diversified top-k clique search can be widely used in a lot of applications including community search, motif discovery, and anomaly detection in large graphs. A naive solution for diversified top-k clique search is to keep all maximal cliques in memory and then find k of them that cover most nodes in the graph by using the approximate greedy max k-cover algorithm. However, such a solution is impractical when the graph is large. In this paper, instead of keeping all maximal cliques in memory, we devise an algorithm to maintain k candidates in the process of maximal clique enumeration. Our algorithm has limited memory footprint and can achieve a guaranteed approximation ratio. We also introduce a novel light-weight $$\mathsf {PNP}$$PNP-$$\mathsf {Index}$$Index, based on which we design an optimal maximal clique maintenance algorithm. We further explore three optimization strategies to avoid enumerating all maximal cliques and thus largely reduce the computational cost. Besides, for the massive input graph, we develop an I/O efficient algorithm to tackle the problem when the input graph cannot fit in main memory. We conduct extensive performance studies on real graphs and synthetic graphs. One of the real graphs contains 1.02 billion edges. The results demonstrate the high efficiency and effectiveness of our approach.
82 citations
TL;DR: The well-known partial κ-tree (resp. treewidth) approach belongs to this kind of research and bases on a tree structure of constant-size bounded maximal cliques that characterizes chordal graphs.
81 citations