Massively parallel algorithms for approximate shortest paths

We present fast algorithms for approximate shortest paths in the massively parallel computation (MPC) model. We provide randomized algorithms that take poly(łogłogn ) rounds in the near-linear memory MPC model. Our results are for unweighted undirected graphs with n vertices and m edges. Our first contribution is a (1+ϵ)-approximation algorithm for Single-Source Shortest Paths (SSSP) that takes poly(łogłogn ) rounds in the near-linear MPC model, where the memory per machine is Õ(n) and the total memory is Õ (mnρ ), where ρ is a small constant. Our second contribution is a distance oracle that allows to approximate the distance between any pair of vertices. The distance oracle is constructed in poly(łogłogn ) rounds and allows to query a (1+ϵ)(2k-1)-approximate distance between any pair of vertices u and v in O(1) additional rounds. The algorithm is for the near-linear memory MPC model with total memory of size Õ((m+n1+ρ )n1/k ), where ρ is a small constant. While our algorithms are for the near-linear MPC model, in fact they only use one machine with Õ(n) memory, where the rest of machines can have sublinear memory of size O(nγ) for a small constant γ< 1. All previous algorithms for approximate shortest paths in the near-linear MPC model either required ω(łogn ) rounds or had an ω(łogn ) approximation. Our approach is based on fast construction of near-additive emulators, limited-scale hopsets and limited-scale distance sketches that are tailored for the MPC model. While our end-results are for the near-linear MPC model, many of the tools we construct such as hopsets and emulators are constructed in the more restricted sublinear MPC model.