Deterministic graph connectivity in the broadcast congested clique

TL;DR

This paper introduces deterministic protocols for graph connectivity in the broadcast congested clique model, achieving constant-round solutions with sublinear communication per node, and presents a novel deterministic sparse linear sketch technique.

Contribution

The paper develops deterministic constant-round protocols for graph connectivity in the broadcast congested clique, including a new sparse linear sketch method for deterministic recovery.

Findings

Deterministic protocols for connectivity in constant rounds.

Protocols operate with sublinear communication per node.

Introduction of deterministic sparse linear sketches for recovery.

Abstract

We present deterministic constant-round protocols for the graph connectivity problem in the model where each of the $n$ nodes of a graph receives a row of the adjacency matrix, and broadcasts a single sublinear size message to all other nodes. Communication rounds are synchronous. This model is sometimes called the broadcast congested clique. Specifically, we exhibit a deterministic protocol that computes the connected components of the input graph in $\lceil 1/\epsilon \rceil$ rounds, each player communicating $\mathcal{O}(n^{\epsilon} \cdot \log n)$ bits per round, with $0 < \epsilon \leq 1$. We also provide a deterministic one-round protocol for connectivity, in the model when each node receives as input the graph induced by the nodes at distance at most $r>0$, and communicates $\mathcal{O}(n^{1/r} \cdot \log n)$ bits. This result is based on a $d$-pruning protocol, which consists in successively removing nodes of degree at most $d$ until obtaining a graph with minimum degree larger than $d$. Our technical novelty is the introduction of deterministic sparse linear sketches: a linear compression function that permits to recover sparse Boolean vectors deterministically.