Communication Costs in a Geometric Communication Network

TL;DR

This paper investigates the communication costs for solving fundamental geometric problems in a spatially-embedded network model, extending previous planar results to arbitrary topologies.

Contribution

It extends the analysis of communication costs for geometric problems from planar networks to networks of arbitrary topology in the asynchronous CONGEST model.

Findings

Established bounds for communication costs of convex hull, diameter, and closest pair problems.

Extended previous planar network results to general topologies.

Provided insights into the complexity of geometric computations in distributed settings.

Abstract

A communication network is a graph in which each node has only local information about the graph and nodes communicate by passing messages along its edges. Here, we consider the {\it geometric communication network} where the nodes also occupy points in space and the distance between points is the Euclidean distance. Our goal is to understand the communication cost needed to solve several fundamental geometry problems, including Convex Hull, Diameter, Closest Pair, and approximations of these problems, in the asynchronous CONGEST KT1 model. This extends the 2011 result of Rajsbaum and Urrutia for finding a convex hull of a planar geometric communication network to networks of arbitrary topology.