Near-Linear Lower Bounds for Distributed Distance Computations, Even in Sparse Networks

TL;DR

This paper introduces a new technique for constructing sparse graphs to establish near-linear lower bounds on the round complexity of distributed distance computations in the CONGEST model, extending previous results to sparse and constant-degree graphs.

Contribution

The authors develop a novel degree-reduction technique and a graph construction method to prove near-linear lower bounds for various distance-related problems in sparse networks.

Findings

Proves an  lower bound for diameter in sparse networks.

Establishes near-linear lower bounds for approximations of diameter, radius, and eccentricities.

Shows an almost-linear lower bound for verifying -spanners.

Abstract

We develop a new technique for constructing sparse graphs that allow us to prove near-linear lower bounds on the round complexity of computing distances in the CONGEST model. Specifically, we show an $\widetilde{\Omega}(n)$ lower bound for computing the diameter in sparse networks, which was previously known only for dense networks [Frishknecht et al., SODA 2012]. In fact, we can even modify our construction to obtain graphs with constant degree, using a simple but powerful degree-reduction technique which we define. Moreover, our technique allows us to show $\widetilde{\Omega}(n)$ lower bounds for computing $(\frac{3}{2}-\varepsilon)$-approximations of the diameter or the radius, and for computing a $(\frac{5}{3}-\varepsilon)$-approximation of all eccentricities. For radius, we are unaware of any previous lower bounds. For diameter, these greatly improve upon previous lower bounds and are tight up to polylogarithmic factors [Frishknecht et al., SODA 2012], and for eccentricities the improvement is both in the lower bound and in the approximation factor [Holzer and Wattenhofer, PODC 2012]. Interestingly, our technique also allows showing an almost-linear lower bound for the verification of $(\alpha,\beta)$-spanners, for $\alpha < \beta+1$.