Distributed Detection of Cycles

Distributed property testing in networks has been introduced by Brakerski and Patt-Shamir (2011), with the objective of detecting the presence of large dense sub-networks in a distributed manner. Recently, Censor-Hillel et al. (2016) have shown how to detect 3-cycles in a constant number of rounds by a distributed algorithm. In a follow up work, Fraigniaud et al. (2016) have shown how to detect 4-cycles in a constant number of rounds as well. However, the techniques in these latter works were shown not to generalize to larger cycles $C_k$ with $k\geq 5$. In this paper, we completely settle the problem of cycle detection, by establishing the following result. For every $k\geq 3$, there exists a distributed property testing algorithm for $C_k$-freeness, performing in a constant number of rounds. All these results hold in the classical CONGEST model for distributed network computing. Our algorithm is 1-sided error. Its round-complexity is $O(1/\epsilon)$ where $\epsilon\in(0,1)$ is the property testing parameter measuring the gap between legal and illegal instances.

💡 Research Summary

The paper addresses the long‑standing open problem of testing Cₖ‑freeness (absence of a k‑node cycle) in the CONGEST model for arbitrary k ≥ 3. While prior work succeeded in constant‑round distributed testers for triangles (k = 3) and 4‑cycles (k = 4), those techniques break down for larger cycles because of combinatorial constructions (Behrend graphs) that cause message congestion.

The authors present a unified algorithm that works for every constant‑size cycle length, achieving a one‑sided error tester that runs in O(1/ε) rounds, where ε is the distance parameter in the sparse‑model definition of “ε‑far”. The algorithm consists of two phases:

Phase 1 – Random edge selection.
Each edge e is assigned to the endpoint with the smaller identifier. The owning node draws a random rank r(e) uniformly from