Faster Parallel Algorithm for Approximate Shortest Path
release_huswgevzijdipnd3jvtdxifrhu
by
Jason Li
2020
Abstract
We present the first m polylog(n) work, polylog(n) time
algorithm in the PRAM model that computes (1+ϵ)-approximate
single-source shortest paths on weighted, undirected graphs. This improves upon
the breakthrough result of Cohen [JACM'00] that achieves O(m^1+ϵ_0)
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 ℓ_1-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 ℓ_1-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.
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