Two Rounds Are Enough for Reconstructing Any Graph (Class) in the Congested Clique Model

Abstract

In this paper we study the reconstruction problem in the congested clique model. In the reconstruction problem nodes are asked to recover all the edges of the input graph G. Formally, given a class of graphs $\mathcal G$, the problem is defined as follows: if $G \notin {\mathcal G}$, then every node must reject; on the other hand, if $G \in {\mathcal G}$, then every node must end up knowing all the edges of G. It is not difficult to see that the cost Rb of any algorithm that solves this problem (even with public coins) is at least $\varOmega (\log |{\mathcal {G}}_n|/n)$, where ${\mathcal {G}}_n$ is the subclass of all n-node labeled graphs in $\mathcal G$, R is the number of rounds and b is the bandwidth.

We prove here that the lower bound above is in fact tight and that it is possible to achieve it with only $R=2$ rounds and private coins. More precisely, we exhibit (i) a one-round algorithm that achieves this bound for hereditary graph classes; and (ii) a two-round algorithm that achieves this bound for arbitrary graph classes. Later, we show that the bound $\varOmega (\log |{\mathcal {G}}_n|/n)$ cannot be achieved in one round for arbitrary graph classes, and we give tight algorithms for that case.

From (i) we recover all known results concerning the reconstruction of graph classes in one round and bandwidth $\mathcal {O}(\log n)$: forests, planar graphs, cographs, etc. But we also get new one-round algorithms for other hereditary graph classes such as unit-disc graphs, interval graphs, etc. From (ii), we can conclude that any problem restricted to a class of graphs of size $2^{\mathcal {O}(n\log n)}$ can be solved in the congested clique model in two rounds, with bandwidth $\mathcal {O}(\log n)$. Moreover, our general two-round algorithm is valid for any set of labeled graphs, not only for graph classes.

Partially supported by CONICYT PIA/Apoyo a Centros Científicos y Tecnológicoss de Excelencia AFB 170001 (P.M. and I.R.), Fondecyt 1170021 (I.R.) and CONICYT + PAI + CONVOCATORIA NACIONAL SUBVENCIÓN A INSTALACIÓN EN LA ACADEMIA CONVOCATORIA AÑO 2017 + PAI77170068 (P.M.).