Deterministic Even-Cycle Detection in Broadcast CONGEST

TL;DR

This paper presents a deterministic distributed algorithm for detecting even cycles in graphs within optimal round complexity in the Broadcast CONGEST model, matching randomized algorithms and establishing new combinatorial bounds.

Contribution

It introduces a deterministic algorithm for even-cycle detection with optimal round complexity and a novel combinatorial bound on local density in graphs without 2k-cycles.

Findings

Deterministic algorithm matches randomized complexity for certain cycle lengths.

Optimal round complexity achieved for detecting even cycles in broadcast networks.

New combinatorial bound on local density of graphs without 2k-cycles.

Abstract

We show that, for every $k\geq 2$, $C_{2k}$-freeness can be decided in $O(n^{1-1/k})$ rounds in the Broadcast CONGEST model, by a deterministic algorithm. This (deterministic) round-complexity is optimal for $k=2$ up to logarithmic factors thanks to the lower bound for $C_4$-freeness by Drucker et al. [PODC 2014], which holds even for randomized algorithms. Moreover it matches the round-complexity of the best known randomized algorithms by Censor-Hillel et al. [DISC 2020] for $k\in\{3,4,5\}$, and by Fraigniaud et al. [PODC 2024] for $k\geq 6$. Our algorithm uses parallel BFS-explorations with deterministic selections of the set of paths that are forwarded at each round, in a way similar to what was done for the detection of odd-length cycles, by Korhonen and Rybicki [OPODIS 2017]. However, the key element in the design and analysis of our algorithm is a new combinatorial result bounding the "local density" of graphs without $2k$-cycles, which we believe is interesting on its own.