Work-Optimal Parallel Minimum Cuts for Non-Sparse Graphs

We present the first work-optimal polylogarithmic-depth parallel algorithm for the minimum cut problem on non-sparse graphs. For $m\ge n^{1+ϵ}$ for any constant $ϵ>0$, 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’20; GMW SOSA’21]. This improves the previous best work by Geissmann and Gianinazzi [SPAA’18] 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’21; GMW SOSA’21] that exploit a connection between 2-respecting minimum cuts and 2-dimensional orthogonal range searching.