Strongly Polynomial Parallel Work-Depth Tradeoffs for Directed SSSP

TL;DR

This paper introduces new strongly polynomial parallel algorithms for directed SSSP, achieving improved work-depth tradeoffs and sublinear depth for dense graphs, with implications for related problems like min-cost flow and minimum mean cycle.

Contribution

It presents the first nearly work-efficient strongly polynomial parallel algorithm for directed SSSP with sublinear depth and extends these techniques to related problems and large edge weights.

Findings

Achieved $ ilde{O}(m+n^{2- ext{epsilon}})$ work and $ ilde{O}(n^{1- ext{epsilon}})$ depth for directed SSSP.

Provided the first strongly polynomial parallel algorithms for min-cost flow and assignment problem.

Developed efficient parallel algorithms for SSSP variants with exponentially large edge weights.

Abstract

In this paper, we show new strongly polynomial work-depth tradeoffs for computing single-source shortest paths (SSSP) in non-negatively weighted directed graphs in parallel. Most importantly, we prove that directed SSSP can be solved within $\tilde{O}(m+n^{2-\epsilon})$ work and $\tilde{O}(n^{1-\epsilon})$ depth for some positive $\epsilon>0$. In particular, for dense graphs with non-negative real weights, we provide the first nearly work-efficient strongly polynomial algorithm with sublinear depth. Our result immediately yields improved strongly polynomial parallel algorithms for min-cost flow and the assignment problem. It also leads to the first non-trivial strongly polynomial dynamic algorithm for minimum mean cycle. Moreover, we develop efficient parallel algorithms in the Word RAM model for several variants of SSSP in graphs with exponentially large edge weights.