The Impact of Locality on the Detection of Cycles in the Broadcast Congested Clique Model

Abstract

The broadcast congested clique model is a message-passing model of distributed computation where n nodes communicate with each other in synchronous rounds. The joint input to the n nodes is an undirected graph G on the same set of nodes, with each node receiving the list of its immediate neighbors in G. In each round each node sends the same message to all other nodes, depending on its own input, on the messages it has received so far, and on a public sequence of random bits. One parameter of this model is an upper bound b on the size of the messages, also known as bandwidth. In this paper we introduce another parameter to the model. We study the situation where the nodes, initially, instead of knowing their immediate neighbors, know their neighborhood up to a fixed radius r.

In this new framework we study one of the hardest problems in distributed graph algorithms, this is, the problem of detecting, in G, an induced cycle of length at most k ($\textsc {Cycle}_{\le k}$) and the problem of detecting in G an induced cycle of length at least k $+$ 1 ($\textsc {Cycle}_{>k}$). For $r=1$, we exhibit a deterministic, one-round algorithm for solving $\textsc {Cycle}_{\le k}$ with $b = \mathcal {O}(n^{2/k} \log n)$ if k is even, and $b = \mathcal {O}(n^{2/(k-1)}\log n)$ if k is odd. We also prove, assuming the Erdős Girth Conjecture, that this result is tight for $k \ge 4$, up to logarithmic factors. More precisely, if each node, instead of being able to see only its immediate neighbors, is allowed to see up to distance $r={\lfloor k/4 \rfloor }$, and if we also allowed randomization and multiple rounds, then the number of rounds R needed to solve $\textsc {Cycle}_{\le k}$ must be such that $R\cdot b = \varOmega ( n^{2/k})$ if k is even, and $R\cdot b = \varOmega ( n^{2/(k-1)})$ if k is odd.

On the other hand, we show that, if each node is allowed to see up to distance $r={\lfloor k/2 \rfloor + 1}$, then a polylogarithmic bandwidth is sufficient for solving $\textsc {Cycle}_{>k}$ with only two rounds. Nevertheless, if nodes were allowed to see up to distance $r=\lfloor k/3 \rfloor$, then any one-round algorithm that solves $\textsc {Cycle}_{>k}$ needs the bandwidth b to be at least $\varOmega (n/\log n)$.

Additional support from CONICYT via Basal in Applied Mathematics (P.M. and I.R.) and Fondecyt 1170021 (I.R.).