Communication-Efficient Construction of the Plane Localized Delaunay Graph

TL;DR

This paper introduces a highly communication-efficient local algorithm for constructing a plane spanner of the unit-disk graph in the plane, reducing message complexity significantly while maintaining geometric properties.

Contribution

It presents a 2-local, one-round algorithm that constructs a plane spanner with improved message bounds compared to previous methods.

Findings

Constructs a plane $rac{4 \pi \sqrt{3}}{9}$-spanner of the UDG

Uses only one round of communication with at most 5 messages per point

Improves previous message complexity bounds from 11 to 5

Abstract

Let $V$ be a finite set of points in the plane. We present a 2-local algorithm that constructs a plane $\frac{4 \pi \sqrt{3}}{9}$-spanner of the unit-disk graph $\UDG(V)$. This algorithm makes only one round of communication and each point of $V$ broadcasts at most 5 messages. This improves the previously best message-bound of 11 by Ara\'{u}jo and Rodrigues (Fast localized Delaunay triangulation, Lecture Notes in Computer Science, volume 3544, 2004).