Distributed Detection of Cycles

Abstract

Distributed property testing in networks has been introduced by Brakerski and Patt-Shamir [6], with the objective of detecting the presence of large dense sub-networks in a distributed manner. Recently, Censor-Hillel et al. [7] have revisited this notion and formalized it in a broader context. In particular, they 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. [21] 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 ≥ 5. In this article, we completely settle the problem of cycle detection by establishing the following result: For every k ≥ 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 (1ϵ) where ϵ ∈ (0,1) is the property-testing parameter measuring the gap between legal and illegal instances.