Massively parallel algorithms for approximate shortest paths - Dory, Michal; Matar, Shaked (Distributed computing, 2025)
We present fast algorithms for approximate shortest paths in the massively parallel computation (MPC) model. We provide randomized algorithms that take poly ( log log n ) 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 ( log log n ) rounds in the near-linear MPC model, where the memory per machine is O ~ ( n ) and the total memory is O ~ ( m n ρ ) , 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 ( log log n ) rounds and allows to query a ( 1 + ϵ ) ( 2 k - 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 O ~ ( ( m + n 1 + ρ ) n 1 / 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 O ~ ( n ) memory, where the rest of machines can have sublinear memory of size O ( n γ ) for a small constant γ < 1 . All previous algorithms for approximate shortest paths in the near-linear MPC model either required Ω ( log n ) rounds or had an Ω ( log n ) 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.