Improved Massively Parallel Shortest Path Computation

Abstract

We present new algorithms for approximate shortest path computation in weighted undirected graphs within the Massively Parallel Computation (MPC) model. Specifically, we construct an $a$-approximate Single-Source Shortest Path (SSSP) oracle: Given a query consisting of a source vertex $u \in V$, the oracle returns $a$-approximate distances $δ(u,v)$ for all $v \in V$. Fix an $n$-vertex weighted undirected graph. We consider two regimes of the MPC model: the near-linear regime, where each machine has $\tildeΘ(n)$ memory, and the heterogeneous regime, where all machines have $Θ(n^γ)$ memory for some constant $γ\in (0,1]$, except for a single machine with $\tildeΘ(n)$ memory. We develop randomised algorithms that construct $O(1)$-approximate SSSP oracles in $O(\log\log n)$ rounds in the near-linear regime and in $O(\log^2\log n)$ rounds in the heterogeneous regime. These oracles answer distance queries in $O(1)$ rounds. Our results improve upon the recent work of Dory and Matar (SPAA 2024) which, for the same total memory, requires $\left(\log\log n\right)^{2+O(1/γ)}$ rounds and applies only to unweighted graphs in the heterogeneous setting. We say that $\tilde N_k(u)$ is the approximate $k$-nearest neighbourhood of a vertex $u$ if there exists an approximate distance estimate $δ$ such that $\tilde N_k(u)$ consists of the $k$ vertices closest to $u$ according to $δ$, breaking ties by vertex ID. The main technical ingredient is a new method for efficiently computing approximate $n^{O(γ)}$-nearest neighbourhoods in the heterogeneous MPC model in $O(\log^2\log n)$ rounds.