Deterministic PRAM Approximate Shortest Paths in Polylogarithmic Time and Slightly Super-Linear Work
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We study a (1+ϵ)-approximate single-source shortest paths (henceforth, (1+ϵ)-SSSP) in n-vertex undirected, weighted graphs in the parallel (PRAM) model of computation. A randomized algorithm with polylogarithmic time and slightly super-linear work Õ(|E|· n^ρ), for an arbitrarily small ρ>0, was given by Cohen [Coh94] more than 25 years ago. Exciting progress on this problem was achieved in recent years [ElkinN17,ElkinN19,Li19,AndoniSZ19], culminating in randomized polylogarithmic time and Õ(|E|) work. However, the question of whether there exists a deterministic counterpart of Cohen's algorithm remained wide open. In the current paper we devise the first deterministic polylogarithmic-time algorithm for this fundamental problem, with work Õ(|E|· n^ρ), for an arbitrarily small ρ>0. This result is based on the first efficient deterministic parallel algorithm for building hopsets, which we devise in this paper.
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