A Near-optimal Algorithm for Edge Connectivity-based Hierarchical Graph Decomposition
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Driven by many applications in graph analytics, the problem of computing k-edge connected components (k-ECCs) of a graph G for a user-given k has been extensively studied recently. In this paper, we investigate the problem of constructing the hierarchy of edge connectivity-based graph decomposition, which compactly represents the k-ECCs of a graph for all possible k values. This is based on the fact that each k-ECC is entirely contained in a (k-1)-ECC. In contrast to the existing approaches that conduct the computation either in a bottom-up or a top-down manner, we propose a binary search-based framework which invokes a k-ECC computation algorithm as a black box. Let T_kecc(G) be the time complexity of computing all k-ECCs of G for a specific k value. We prove that the time complexity of our framework is O( (δ(G))× T_kecc(G)), where δ(G) is the degeneracy of G and equals the maximum value among the minimum vertex degrees of all subgraphs of G. As δ(G) is typically small for real-world graphs, this time complexity is optimal up to a logarithmic factor.
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