Tight Distributed Sketching Lower Bound for Connectivity

TL;DR

This paper establishes a tight lower bound of Omega(log^3 n) bits on the message length for distributed sketching of graph connectivity, matching existing upper bounds and showing the problem's inherent complexity.

Contribution

It proves a new lower bound for distributed connectivity sketching, demonstrating the problem's difficulty matches known upper bounds, and extends previous bounds from spanning forest computation.

Findings

Expected message length must be at least Omega(log^3 n) bits.

Lower bound matches the AGM sketch upper bound.

Connectivity is as hard as its search version in this model.

Abstract

In this paper, we study the distributed sketching complexity of connectivity. In distributed graph sketching, an $n$-node graph $G$ is distributed to $n$ players such that each player sees the neighborhood of one vertex. The players then simultaneously send one message to the referee, who must compute some function of $G$ with high probability. For connectivity, the referee must output whether $G$ is connected. The goal is to minimize the message lengths. Such sketching schemes are equivalent to one-round protocols in the broadcast congested clique model. We prove that the expected average message length must be at least $\Omega(\log^3 n)$ bits, if the error probability is at most $1/4$. It matches the upper bound obtained by the AGM sketch [AGM12], which even allows the referee to output a spanning forest of $G$ with probability $1-1/\mathrm{poly}\, n$. Our lower bound strengthens the previous $\Omega(\log^3 n)$ lower bound for spanning forest computation [NY19]. Hence, it implies that connectivity, a decision problem, is as hard as its "search" version in this model.