Faster parallel algorithm for approximate shortest path

We present the first m polylog(n) work, polylog(n) time algorithm in the PRAM model that computes (1+є)-approximate single-source shortest paths on weighted, undirected graphs. This improves upon the breakthrough result of Cohen [JACM’00] that achieves O(m 1+є0 ) work and polylog(n) time. While most previous approaches, including Cohen’s, leveraged the power of hopsets, our algorithm builds upon the recent developments in continuous optimization, studying the shortest path problem from the lens of the closely-related minimum transshipment problem. To obtain our algorithm, we demonstrate a series of near-linear work, polylogarithmic-time reductions between the problems of approximate shortest path, approximate transshipment, and ℓ1-embeddings, and establish a recursive algorithm that cycles through the three problems and reduces the graph size on each cycle. As a consequence, we also obtain faster parallel algorithms for approximate transshipment and ℓ1-embeddings with polylogarithmic distortion. The minimum transshipment algorithm in particular improves upon the previous best m 1+o(1) work sequential algorithm of Sherman [SODA’17]. To improve readability, the paper is almost entirely self-contained, save for several staple theorems in algorithms and combinatorics.