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 Ck with k≥5. In this paper, 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 Ck-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/ϵ) where ϵ∈(0,1) is the property testing parameter measuring the gap between legal and illegal instances.
Cite
@article{arxiv.1706.03992,
title = {Distributed Detection of Cycles},
author = {Pierre Fraigniaud and Dennis Olivetti},
journal= {arXiv preprint arXiv:1706.03992},
year = {2017}
}