Parallel Minimum Cuts in O ( m log 2 n ) Work and Low Depth

Abstract

We present a randomized O ( m log 2 n ) work, O (polylog n ) depth parallel algorithm for minimum cut. This algorithm matches the work bounds of a recent sequential algorithm by Gawrychowski, Mozes, and Weimann [ICALP’20], and improves on the previously best parallel algorithm by Geissmann and Gianinazzi [SPAA’18], which performs O ( m log 4 n ) work in O (polylog n ) depth. Our algorithm makes use of three components that might be of independent interest. First, we design a parallel data structure that efficiently supports batched mixed queries and updates on trees. It generalizes and improves the work bounds of a previous data structure of Geissmann and Gianinazzi and is work efficient with respect to the best sequential algorithm. Second, we design a parallel algorithm for approximate minimum cut that improves on previous results by Karger and Motwani. We use this algorithm to give a work-efficient procedure to produce a tree packing, as in Karger’s sequential algorithm for minimum cuts. Last, we design an efficient parallel algorithm for solving the minimum 2-respecting cut problem.