Work-Optimal Parallel Minimum Cuts for Non-Sparse Graphs

TL;DR

This paper introduces the first work-optimal parallel algorithm for minimum cut in non-sparse graphs, achieving optimal work and depth, and significantly improving previous methods.

Contribution

It presents a work-efficient, parallel, approximation algorithm for minimum cut that matches the best sequential runtime and improves upon prior parallel algorithms.

Findings

Achieves $O(m \,\log n)$ work and $O(\log^3 n)$ depth for non-sparse graphs.

Matches the best known sequential algorithm's runtime.

Improves previous parallel algorithms by a factor of $O(\log^3 n)$.

Abstract

We present the first work-optimal polylogarithmic-depth parallel algorithm for the minimum cut problem on non-sparse graphs. For $m\geq n^{1+\epsilon}$ for any constant $\epsilon>0$, our algorithm requires $O(m \log n)$ work and $O(\log^3 n)$ depth and succeeds with high probability. Its work matches the best $O(m \log n)$ runtime for sequential algorithms [MN STOC 2020, GMW SOSA 2021]. This improves the previous best work by Geissmann and Gianinazzi [SPAA 2018] by $O(\log^3 n)$ factor, while matching the depth of their algorithm. To do this, we design a work-efficient approximation algorithm and parallelize the recent sequential algorithms [MN STOC 2020; GMW SOSA 2021] that exploit a connection between 2-respecting minimum cuts and 2-dimensional orthogonal range searching.