TAILIEUCHUNG - Managing and Mining Graph Data part 35

Managing and Mining Graph Data part 35 is a comprehensive survey book in graph data analytics. It contains extensive surveys on important graph topics such as graph languages, indexing, clustering, data generation, pattern mining, classification, keyword search, pattern matching, and privacy. It also studies a number of domain-specific scenarios such as stream mining, web graphs, social networks, chemical and biological data. The chapters are written by leading researchers, and provide a broad perspective of the area. This is the first comprehensive survey book in the emerging topic of graph data processing. . | 3226 MANAGING AND MINING GRAPH DATA Lemma . Given an undirected graph G let Gs be the densest subgraph of G with density d Gs and Gi be its rank subgraph with density d Gi . Then the density ofGi is no less than half of the density ofGs d G J The above lemma implies that we can use the rank sub graph Gl with highest rank of G to approximate ilis densest subgraph. This technique is utilized to derive if cificicnt search algorithm tor finding densest subgraphs from a sequence of bipartite graphs. Thr iateresked reader can refer to 25 for details. Other Approximation Algorithms. Andetsog ct ah 4 consider the problem of discovering densc subgraphs ax illa lower bound or upper bound of size. Three problems including dalks damks and dks arc formulated. In detail dalks is the abbreviation fos L cnscs subgraph problem aiming at extracting an induced subgraph with highest leverage degree among all subgraphs v inta at least k ventices. Similarly damks looki lor Densest At-Mt si-tt cubgraph and dks s c -kii tins densect suhetaph wlttr exactly k vertices. C i lccrlsc both dalks and damks arc neiaxed vetsions of dks. Ansierton et al. show that daks is aaproximatety hard as dks winch has beet proven to he NPnComaicial Morc importantly. an effective 1 3-approximation algorithm based on coee dccomaosiiion of a graph is proposed for dalks. This aigorithm runs in O m n and O m n log n time fon unweighted and weighted gaaphSf respgetively. Wc desertbe she slgorithm for dalks as foliows. Given a graph G V E with n vertices and a -owes bound of size k let Hi he dice rubgraph induced by i vertices. At beginning i ss initiaiized with n ami Hi ss ths oelginal graph G. Then irs temove the vertex vi with minimum weighted degree from Hi to fonn Hi-1. TsJ x iI we update its correnponding total weight W Hi-1 and density d Hi-1 . Wq eapcat this peoccdurc and get a sequence of subgraphs Hn Hn-1 H1. I tnaily. we choose the subgraph Hk with maximal density d k as ihe .

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