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In this paper we show a deterministic parallel all-pairs shortest paths algorithm for realweighted directed graphs. The algorithm has Õ(nm + (n/d)) work and Õ(d) depth for any depth parameter d ∈ [1, n]. 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 Õ(nm+ n/d) work and Õ(d) 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. By suitably adjusting the depth parameter d and applying known techniques, we obtain: (1) nearly work-efficient Õ(n)-depth parallel algorithms for the real-weighted single-source shortest paths problem and finding a bipartite perfect matching in a planar graph, (2) an Õ(n)-time sequential strongly polynomial algorithm for computing a minimum mean cycle or a minimum cost-to-time-ratio cycle of a planar graph, (3) a slightly faster algorithm for computing so-called external dense distance graphs of all pieces of a recursive decomposition of a planar graph. One notable ingredient of our parallel APSP algorithm is a simple deterministic Õ(nm)-work Õ(d)-depth procedure for computing Õ(n/d)-size hitting sets of shortest d-hop paths between all pairs of vertices of a real-weighted digraph. Such hitting sets have also been called d-hub sets. Hub sets have previously proved especially useful in designing parallel or dynamic shortest paths algorithms and are typically obtained via random sampling. Our procedure implies, for example, an Õ(nm)-time deterministic algorithm for finding a shortest negative cycle of a real-weighted digraph. Such a near-optimal bound for this problem has been so far only achieved using a randomized algorithm [Orlin et al., Discret. Appl. Math. ’18]. ∗ a.karczmarz@mimuw.edu.pl. Supported by ERC Consolidator Grant 772346 TUgbOAT, the Polish National Science Centre 2018/29/N/ST6/00757 grant, and by the Foundation for Polish Science (FNP) via the START programme. † sank@mimuw.edu.pl. Supported by ERC Consolidator Grant 772346 TUgbOAT.

A Deterministic Parallel APSP Algorithm and its Applications