Communication costs in a geometric communication network

Abstract

We represent a communication network as a graph in which each node has only local information about the graph, except for an upper bound on the number of nodes, and nodes communicate by passing messages along its edges. Here, we consider a 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 Farthest Pair, Convex Hull, Closest Pair, and approximations of these problems, in the asynchronous CONGEST KT1 model, where each node knows its ID and those of its neighbors. 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.We define a new model where each node has a position on the plane and nodes can communicate to each other if and only if there is an edge between them. We motivate the model and study a number of geometric problems in this model. We prove lower bounds on the communication complexity of the problems in this new model and present approximation algorithms for them. We prove lower bounds on the number of expected bits required for any randomized algorithm to solve the problems.