# Distributed Detection of Cycles

## TL;DR

This paper presents a unified distributed property testing algorithm that detects cycles of any length in constant rounds within the CONGEST model, advancing cycle detection in network graphs.

## Contribution

It introduces a general constant-round distributed algorithm for detecting any cycle length $C_k$, extending previous results limited to small cycles.

## Key findings

- Works for all cycle lengths $k \\geq 3$
- Operates in constant rounds in the CONGEST model
- Has a round complexity of $O(1/\\epsilon)$

## Abstract

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.