Non-Local Probes Do Not Help with Graph Problems

TL;DR

This paper demonstrates that for many graph problems, the power of centralized probing does not surpass distributed message-passing, allowing lower bounds from distributed algorithms to be applied to centralized local algorithms.

Contribution

It shows that centralized local algorithms can be simulated by distributed algorithms, bridging the gap between models and enabling transfer of lower bounds.

Findings

Centralized probes do not improve solving certain graph problems.

Distributed algorithms can simulate centralized local algorithms efficiently.

Lower bounds from distributed models apply to centralized local algorithms.

Abstract

This work bridges the gap between distributed and centralised models of computing in the context of sublinear-time graph algorithms. A priori, typical centralised models of computing (e.g., parallel decision trees or centralised local algorithms) seem to be much more powerful than distributed message-passing algorithms: centralised algorithms can directly probe any part of the input, while in distributed algorithms nodes can only communicate with their immediate neighbours. We show that for a large class of graph problems, this extra freedom does not help centralised algorithms at all: for example, efficient stateless deterministic centralised local algorithms can be simulated with efficient distributed message-passing algorithms. In particular, this enables us to transfer existing lower bound results from distributed algorithms to centralised local algorithms.