Stochastic Distance in Property Testing

TL;DR

This work introduces stochastic distance, a probabilistic augmentation-based distance for property testing, motivated by distributed dynamic networks. In the CONGEST model, it yields two testers: a deterministic $O(s)$-round tester for connectivity vs $\Omega((n \log n)/s)$-stochastically-far and a randomized $\tilde{O}(s^4)$-round tester for $k$-connectivity vs $\Omega\bigl((k n \log n)/s\bigr)$-stochastically-far. The authors connect stochastic distance to minimal-cut structure via the parameter $s_k(G)$, proving that a graph is $O\big( (k n \log n)/s_k \big)$-stochastically-close to being $k$-connected, with matching lower bounds up to constants. This framework aligns with distributed computation by modeling resilience against random edge failures and enabling ultra-fast distributed testing, suggesting rich directions for future research in new computation models and augmentation-based robustness.

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.

Theorems & Definitions (13)

• Theorem 1.1
• Theorem 1.2
• Definition 1: Random addition of edges
• Lemma 3.1
• proof
• Theorem 3.1
• proof
• Theorem 4.1
• Proposition 4.2
• proof