Stochastic Distance in Property Testing

TL;DR

This paper introduces stochastic distance, a probabilistic measure for property testing, which is particularly effective in distributed settings, enabling ultra-fast algorithms for properties like k-edge-connectivity.

Contribution

The paper proposes stochastic distance as a new concept for property testing and demonstrates its effectiveness in designing fast distributed algorithms for graph properties.

Findings

Stochastic distance aligns with probabilistic edge addition.

Efficient distributed testers for k-edge-connectivity.

New framework for property testing in distributed models.

Abstract

We introduce a novel concept termed "stochastic distance" for property testing. Diverging from the traditional definition of distance, where a distance $t$ implies that there exist $t$ edges that can be added to ensure a graph possesses a certain property (such as $k$-edge-connectivity), our new notion implies that there is a high probability that adding $t$ random edges will endow the graph with the desired property. While formulating testers based on this new distance proves challenging in a sequential environment, it is much easier in a distributed setting. Taking $k$-edge-connectivity as a case study, we design ultra-fast testing algorithms in the CONGEST model. Our introduction of stochastic distance offers a more natural fit for the distributed setting, providing a promising avenue for future research in emerging models of computation.