Local Routing in Sparse and Lightweight Geometric Graphs

TL;DR

This paper presents a simple planar geometric graph construction with low weight and degree that supports efficient local routing, improving routing strategies in sparse networks.

Contribution

It introduces a new planar graph with constant degree and near-minimal weight that enables $O(1)$-competitive local routing strategies, applicable broadly to planar graphs.

Findings

Constructed a planar graph with constant degree and low weight.

Established an $O(1)$-competitive local routing strategy for the graph.

Technique applies generally to planar geometric graphs.

Abstract

Online routing in a planar embedded graph is central to a number of fields and has been studied extensively in the literature. For most planar graphs no $O(1)$-competitive online routing algorithm exists. A notable exception is the Delaunay triangulation for which Bose and Morin [Online routing in triangulations. SIAM Journal on Computing, 33(4):937-951, 2004] showed that there exists an online routing algorithm that is $O(1)$-competitive. However, a Delaunay triangulation can have $\Omega(n)$ vertex degree and a total weight that is a linear factor greater than the weight of a minimum spanning tree. We show a simple construction, given a set $V$ of $n$ points in the Euclidean plane, of a planar geometric graph on $V$ that has small weight (within a constant factor of the weight of a minimum spanning tree on $V$), constant degree, and that admits a local routing strategy that is $O(1)$-competitive. Moreover, the technique used to bound the weight works generally for any planar geometric graph whilst preserving the admission of an $O(1)$-competitive routing strategy.