On Geometric Spanners of Euclidean and Unit Disk Graphs

TL;DR

This paper presents simple, efficient algorithms for constructing planar geometric spanners with bounded degree and stretch factor for Euclidean and unit disk graphs, improving upon previous methods and ensuring the inclusion of a Euclidean Minimum Spanning Tree.

Contribution

It introduces a linear-time algorithm for Euclidean graphs and a distributed localized algorithm for unit disk graphs to create spanners with bounded degree, stretch factor, and EMST inclusion.

Findings

Algorithms achieve bounded degree and stretch factor guarantees.

Constructed spanners include Euclidean Minimum Spanning Trees.

Results significantly improve previous bounds and efficiency.

Abstract

We consider the problem of constructing bounded-degree planar geometric spanners of Euclidean and unit-disk graphs. It is well known that the Delaunay subgraph is a planar geometric spanner with stretch factor $C_{del\approx 2.42$; however, its degree may not be bounded. Our first result is a very simple linear time algorithm for constructing a subgraph of the Delaunay graph with stretch factor $\rho =1+2\pi(k\cos{\frac{\pi{k)^{-1$ and degree bounded by $k$, for any integer parameter $k\geq 14$. This result immediately implies an algorithm for constructing a planar geometric spanner of a Euclidean graph with stretch factor $\rho \cdot C_{del$ and degree bounded by $k$, for any integer parameter $k\geq 14$. Moreover, the resulting spanner contains a Euclidean Minimum Spanning Tree (EMST) as a subgraph. Our second contribution lies in developing the structural results necessary to transfer our analysis and algorithm from Euclidean graphs to unit disk graphs, the usual model for wireless ad-hoc networks. We obtain a very simple distributed, {\em strictly-localized algorithm that, given a unit disk graph embedded in the plane, constructs a geometric spanner with the above stretch factor and degree bound, and also containing an EMST as a subgraph. The obtained results dramatically improve the previous results in all aspects, as shown in the paper.