Simulations between Strongly Sublinear MPC and Node-Capacitated Clique

TL;DR

This paper explores the relationship between the strongly sublinear MPC model and the Node-Capacitated Clique, providing conditions for when simulations between these models are possible or impossible, with implications for distributed graph algorithms.

Contribution

It introduces techniques for round-preserving simulations between the models and establishes fundamental impossibility results, clarifying the limits of such simulations in distributed computing.

Findings

Simulation techniques with constant overhead

Conditions for successful round-preserving simulations

Impossibility results for certain problem classes

Abstract

We study how the strongly sublinear MPC model relates to the classic, graph-centric distributed models, focusing on the Node-Capacitated Clique (NCC), a bandwidth-parametrized generalization of the Congested Clique. In MPC, $M$ machines with per-machine memory $S$ hold a partition of the input graph, in NCC, each node knows its full neighborhood but can send/receive only a bounded number of $C$ words per round. We are particularly interested in the strongly sublinear regime where $S=C=n^\delta$ for some constant $0 < \delta <1$. Our goal is determine when round-preserving simulations between these models are possible and when they are not, when total memory and total bandwidth $SM=nC$ in both models are matched, for different problem families and graph classes. On the positive side, we provide techniques that allow us to replicate the specific behavior regarding input representation, number of machines and local memory from one model to the other to obtain simulations with only constant overhead. On the negative side, we prove simulation impossibility results, which show that the limitations of our simulations are necessary.