A Deterministic Parallel APSP Algorithm and its Applications

In this paper we show a deterministic parallel all-pairs shortest paths algorithm for real-weighted directed graphs. The algorithm has tildeO(nm+(n/d)3) work and tildeO(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 tildeO(nm+n3/d2) work and tildeO(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 tildeO(nm)-work tildeO(d)-depth procedure for computing tildeO(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 tildeO(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].