# Massively Parallel Computation in a Heterogeneous Regime

## TL;DR

This paper demonstrates that adding a single large machine to the sublinear MPC model significantly improves the efficiency of various graph algorithms, overcoming previous hardness results.

## Contribution

It introduces a heterogeneous MPC model with a large machine, enabling faster algorithms for MST, spanners, and matching, and extends existing algorithms to this setting.

## Key findings

- MST algorithm runs in O(log log(m/n)) rounds.
- O(k)-spanner construction in O(1) rounds.
- Maximal matching in O(√log(m/n)) log log(m/n) rounds.

## Abstract

Massively-parallel graph algorithms have received extensive attention over the past decade, with research focusing on three memory regimes: the superlinear regime, the near-linear regime, and the sublinear regime. The sublinear regime is the most desirable in practice, but conditional hardness results point towards its limitations. In this work we study a \emph{heterogeneous} model, where the memory of the machines varies in size. We focus mostly on the heterogeneous setting created by adding a single near-linear machine to the sublinear MPC regime, and show that even a single large machine suffices to circumvent most of the conditional hardness results for the sublinear regime: for graphs with $n$ vertices and $m$ edges, we give (a) an MST algorithm that runs in $O(\log\log(m/n))$ rounds; (b) an algorithm that constructs an $O(k)$-spanner of size $O(n^{1+1/k})$ in $O(1)$ rounds; and (c) a maximal-matching algorithm that runs in $O(\sqrt{\log(m/n)}\log\log(m/n))$ rounds. We also observe that the best known near-linear MPC algorithms for several other graph problems which are conjectured to be hard in the sublinear regime (minimum cut, maximal independent set, and vertex coloring) can easily be transformed to work in the heterogeneous MPC model with a single near-linear machine, while retaining their original round complexity in the near-linear regime. If the large machine is allowed to have \emph{superlinear} memory, all of the problems above can be solved in $O(1)$ rounds.