Abstract
The problem of finding the minimum cut of an undirected unweighted graph is studied on the external memory model. First, a lower bound of Ω((E/V)Sort(V)) on the number of I/Os is shown for the problem, where V is the number of vertices and E is the number of edges. Then the following are presented, (1) a deterministic minimum cut algorithm that usesO(min{c7(log3+ϵE)(MST(V,E)),c(logE)(MST(V,E))+(V/M)Sort(V)}) I/Os; here ϵ>0 is a small constant, MST(V,E) is the number of I/Os needed to compute a minimum spanning tree of the graph, and c is the value of the minimum cut. The algorithm performs better on dense graphs than the algorithm of [1], which requires O(E+c2Vlog(V/c)) I/Os, when executed on the external memory model. For a δ-fat graph (for δ>0, the maximum tree packing of the graph is at least (1+δ)c/2), our algorithm computes a minimum cut in O(c(logE)MST(V,E)) I/Os. (2) A randomized algorithm that computes minimum cut with high probability in O(c(logE)⋅MST(V,E)+Sort(E)log2V+VBSort(V)logV) I/Os. (3) A (2+ϵ)-minimum cut algorithm that requires O((E/V)MST(V,E)) I/Os.
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