The problem of computing k-edge connected components (k-ECC\documentclass[12pt]{minimal}
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\begin{document}$$\mathsf {ECC}$$\end{document}s) of a graph G for a specific k is a fundamental graph problem and has been investigated recently. In this paper, we study the problem of ECC\documentclass[12pt]{minimal}
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\begin{document}$$\mathsf {ECC}$$\end{document} decomposition, which computes the k-ECC\documentclass[12pt]{minimal}
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\begin{document}$$\mathsf {ECC}$$\end{document}s of a graph G for all possible k values. ECC\documentclass[12pt]{minimal}
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\begin{document}$$\mathsf {ECC}$$\end{document} decomposition can be widely applied in a variety of applications such as graph-topology analysis, community detection, Steiner Component Search, and graph visualization. A straightforward solution for ECC\documentclass[12pt]{minimal}
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\begin{document}$$\mathsf {ECC}$$\end{document} decomposition is to apply the existing k-ECC\documentclass[12pt]{minimal}
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\begin{document}$$\mathsf {ECC}$$\end{document} computation algorithm to compute the k-ECC\documentclass[12pt]{minimal}
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\begin{document}$$\mathsf {ECC}$$\end{document}s for all k values. However, this solution is not applicable to large graphs for two challenging reasons. First, all existing k-ECC\documentclass[12pt]{minimal}
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\begin{document}$$\mathsf {ECC}$$\end{document} computation algorithms are highly memory intensive due to the complex data structures used in the algorithms. Second, the number of possible k values can be very large, resulting in a high computational cost when each k value is independently considered. In this paper, we address the above challenges, and study I/O efficient ECC\documentclass[12pt]{minimal}
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\begin{document}$$\mathsf {ECC}$$\end{document} decomposition via graph reduction. We introduce two elegant graph reduction operators which aim to reduce the size of the graph loaded in memory while preserving the connectivity information of a certain set of edges to be computed for a specific k. We also propose three novel I/O efficient algorithms, Bottom\documentclass[12pt]{minimal}
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\begin{document}$$\mathsf {Bottom}$$\end{document}-Up\documentclass[12pt]{minimal}
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\begin{document}$$\mathsf {Up}$$\end{document}, Top\documentclass[12pt]{minimal}
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\begin{document}$$\mathsf {Down}$$\end{document}, and Hybrid\documentclass[12pt]{minimal}
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\begin{document}$$\mathsf {Hybrid}$$\end{document}, that explore the k values in different orders to reduce the redundant computations between different k values. We analyze the I/O and memory costs for all proposed algorithms. In addition, we extend our algorithm to build an efficient index for Steiner Component Search. We show that our index can be used to perform Steiner Component Search in optimal I/Os when only the node information of the graph is allowed to be loaded in memory. In our experiments, we evaluate our algorithms using seven real large datasets with various graph properties, one of which contains 1.95 billion edges. The experimental results show that our proposed algorithms are scalable and efficient.
