Almost-tight distributed minimum cut algorithms

Abstract

We study the problem of computing the minimum cut in a weighted distributed message-passing networks (the CONGEST model). Let λ be the minimum cut, n be the number of nodes (processors) in the network, and D be the network diameter. Our algorithm can compute λ exactly in time. To the best of our knowledge, this is the first paper that explicitly studies computing the exact minimum cut in the distributed setting. Previously, non-trivial sublinear time algorithms for this problem are known only for unweighted graphs when λ ≤ 3 due to Pritchard and Thurimella’s O(D)-time and O(D+n1/2log∗ n)-time algorithms for computing 2-edge-connected and 3-edge-connected components [ACM Transactions on Algorithms 2011]. By using the edge sampling technique of Karger [STOC 1994], we can convert this algorithm into a (1 + ∈)-approximation -time algorithm for any ∈ >0. This improves over the previous (2 + ∈)-approximation time algorithm and O(∈−1)-approximation -time algorithm of Ghaffari and Kuhn [DISC 2013]. Due to the lower bound of Ω(D + n1/2/ log n) by Das Sarma et al. [SICOMP 2013] which holds for any approximation algorithm, this running time is tight up to a poly log n factor. To get the stated running time, we developed an approximation algorithm which combines the ideas of Thorup’s algorithm [Combinatorica 2007] and Matula’s contraction algorithm [SODA 1993]. It saves an ∈−9log7n factor as compared to applying Thorup’s tree packing theorem directly. Then, we combine Kutten and Peleg’s tree partitioning algorithm [J. Algorithms 1998] and Karger’s dynamic programming [JACM 2000] to achieve an efficient distributed algorithm that finds the minimum cut when we are given a spanning tree that crosses the minimum cut exactly once.