Distributed Distance Approximation

TL;DR

This paper characterizes the trade-offs between approximation ratios and round complexity for distributed algorithms approximating graph parameters like diameter, radius, and eccentricities, including bi-chromatic variants, with new bounds and algorithms.

Contribution

It provides the first distributed bounds for bi-chromatic parameters and introduces the concept of approximate pseudo-centers with efficient algorithms.

Findings

Established near-complete trade-offs for approximation and round complexity.

Provided the first distributed bounds for bi-chromatic diameter, radius, and eccentricities.

Developed an efficient distributed algorithm for approximate pseudo-centers.

Abstract

Diameter, radius and eccentricities are fundamental graph parameters, which are extensively studied in various computational settings. Typically, computing approximate answers can be much more efficient compared with computing exact solutions. In this paper, we give a near complete characterization of the trade-offs between approximation ratios and round complexity of distributed algorithms for approximating these parameters, with a focus on the weighted and directed variants. Furthermore, we study \emph{bi-chromatic} variants of these parameters defined on a graph whose vertices are colored either red or blue, and one focuses only on distances for pairs of vertices that are colored differently. Motivated by applications in computational geometry, bi-chromatic diameter, radius and eccentricities have been recently studied in the sequential setting [Backurs et al. STOC'18, Dalirrooyfard et al. ICALP'19]. We provide the first distributed upper and lower bounds for such problems. Our technical contributions include introducing the notion of \emph{approximate pseudo-center}, which extends the \emph{pseudo-centers} of [Choudhary and Gold SODA'20], and presenting an efficient distributed algorithm for computing approximate pseudo-centers. On the lower bound side, our constructions introduce the usage of new functions into the framework of reductions from 2-party communication complexity to distributed algorithms.