Fooling Views: A New Lower Bound Technique for Distributed Computations under Congestion

TL;DR

This paper introduces a new lower bound technique called fooling views for distributed graph algorithms under bandwidth constraints, proving fundamental limits for triangle detection in the CONGEST model.

Contribution

It presents the first lower bounds for triangle membership in the CONGEST model using fooling views, surpassing previous techniques.

Findings

Any 1-round algorithm requires bandwidth proportional to Δ log n.

In the CONGEST(1) model, triangle detection requires at least Ω(log* n) rounds.

First separation between LOCAL and CONGEST models for deterministic triangle membership.

Abstract

We introduce a novel lower bound technique for distributed graph algorithms under bandwidth limitations. We define the notion of \emph{fooling views} and exemplify its strength by proving two new lower bounds for triangle membership in the CONGEST(B) model: (i) Any $1$-round algorithm requires $B\geq c\Delta \log n$ for a constant $c>0$. (ii) If $B=1$, even in constant-degree graphs any algorithm must take $\Omega(\log^* n)$ rounds. The implication of the former is the first proven separation between the LOCAL and the CONGEST models for deterministic triangle membership. The latter result is the first non-trivial lower bound on the number of rounds required, even for \emph{triangle detection}, under limited bandwidth. All previous known techniques are provably incapable of giving these bounds. We hope that our approach may pave the way for proving lower bounds for additional problems in various settings of distributed computing for which previous techniques do not suffice.