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

TL;DR

This paper introduces a new parallel algorithm for minimum cut that achieves optimal work bounds and low depth, utilizing innovative data structures and algorithms for tree queries, approximate minimum cuts, and 2-respecting cuts.

Contribution

It presents a parallel minimum cut algorithm with optimal work and depth, along with new data structures and algorithms for tree queries and cut approximations.

Findings

Achieves $O(m \log^2 n)$ work and polylogarithmic depth for minimum cut.

Introduces a parallel data structure for batched tree queries with improved work bounds.

Develops a parallel algorithm for approximate minimum cut and 2-respecting cut problems.

Abstract

We present a randomized $O(m \log^2 n)$ work, $O(\text{polylog } n)$ 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 $O(\text{polylog } n)$ depth. Our algorithm makes use of three components that might be of independent interest. Firstly, 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. Secondly, 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. Lastly, we design an efficient parallel algorithm for solving the minimum $2$-respecting cut problem.