Deterministic PRAM Approximate Shortest Paths in Polylogarithmic Time and Slightly Super-Linear Work

TL;DR

This paper presents the first deterministic parallel algorithm for approximate shortest paths in undirected weighted graphs, achieving polylogarithmic time with slightly super-linear work, based on a new deterministic hopset construction.

Contribution

It introduces the first deterministic polylogarithmic-time algorithm for approximate shortest paths, utilizing a novel efficient deterministic hopset construction.

Findings

Achieves polylogarithmic time with slightly super-linear work for approximate shortest paths.

Provides the first deterministic parallel algorithm for hopset construction.

Bridges the gap between randomized and deterministic approaches in parallel shortest path algorithms.

Abstract

We study a $(1+\epsilon)$-approximate single-source shortest paths (henceforth, $(1+\epsilon)$-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 $\tilde{O}(|E|\cdot n^\rho)$, for an arbitrarily small $\rho>0$, 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 $\tilde{O}(|E|)$ 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 $\tilde{O}(|E|\cdot n^\rho)$, for an arbitrarily small $\rho>0$. This result is based on the first efficient deterministic parallel algorithm for building hopsets, which we devise in this paper.