A Deterministic Parallel APSP Algorithm and its Applications
Adam Karczmarz, Piotr Sankowski

TL;DR
This paper introduces a deterministic parallel algorithm for all-pairs shortest paths in real-weighted directed graphs, achieving a novel trade-off between work and depth, and improving upon previous randomized methods.
Contribution
It presents the first deterministic parallel APSP algorithm with a specific work-depth trade-off and introduces a new method for computing hitting sets of shortest paths.
Findings
Achieves $ ilde{O}(nm+(n/d)^3)$ work and $ ilde{O}(d)$ depth for APSP
Improves parallelism over previous randomized algorithms for transitive closure
Provides a deterministic $ ilde{O}(nm)$-time algorithm for detecting shortest negative cycles
Abstract
In this paper we show a deterministic parallel all-pairs shortest paths algorithm for real-weighted directed graphs. The algorithm has work and depth for any depth parameter . To the best of our knowledge, such a trade-off has only been previously described for the real-weighted single-source shortest paths problem using randomization [Bringmann et al., ICALP'17]. Moreover, our result improves upon the parallelism of the state-of-the-art randomized parallel algorithm for computing transitive closure, which has work and depth [Ullman and Yannakakis, SIAM J. Comput. '91]. Our APSP algorithm turns out to be a powerful tool for designing efficient planar graph algorithms in both parallel and sequential regimes. One notable ingredient of our parallel APSP algorithm is a simple deterministic…
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