In this paper we show a deterministic parallel all-pairs shortest paths algorithm for real-weighted directed graphs. The algorithm has $tilde{O}(nm+(n/d)^3)$ work and $tilde{O}(d)$ depth for any depth parameter $din [1,n]$. 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., ICALP17]. Moreover, our result improves upon the parallelism of the state-of-the-art randomized parallel algorithm for computing transitive closure, which has $tilde{O}(nm+n^3/d^2)$ work and $tilde{O}(d)$ 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. One notable ingredient of our parallel APSP algorithm is a simple deterministic $tilde{O}(nm)$-work $tilde{O}(d)$-depth procedure for computing $tilde{O}(n/d)$-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 $tilde{O}(nm)$-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].
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