Improved Massively Parallel Shortest Path Computation
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.