Fast Algorithms for Minimum Cycle Basis and Minimum Homology Basis
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We study the problem of finding a minimum homology basis, that is, a shortest set of cycles that generates the 1-dimensional homology classes with ℤ_2 coefficients in a given simplicial complex K. This problem has been extensively studied in the last few years. For general complexes, the current best deterministic algorithm, by Dey et al., runs in O(N^ω + N^2 g) time, where N denotes the number of simplices in K, g denotes the rank of the 1-homology group of K, and ω denotes the exponent of matrix multiplication. In this paper, we present two conceptually simple randomized algorithms that compute a minimum homology basis of a general simplicial complex K. The first algorithm runs in Õ(m^ω) time, where m denotes the number of edges in K, whereas the second algorithm runs in O(m^ω + N m^ω-1) time. We also study the problem of finding a minimum cycle basis in an undirected graph G with n vertices and m edges. The best known algorithm for this problem runs in O(m^ω) time. Our algorithm, which has a simpler high-level description, but is slightly more expensive, runs in Õ(m^ω) time.
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