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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 (SIAM J Comput 33(4):937–951, 2004) showed that there exists an online routing algorithm that is O(1)-competitive. However, a Delaunay triangulation can have varOmega (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.
Highlights
We show: Given a set V of n points in the plane, together with two parameters 0 < θ < π/2 and r > 0, we show how to construct in O(n log n) time a planar ((1+1/r )·τ )-spanner with degree at most 5 2π/θ, and weight at most ((2r + 1) · τ ) times the weight of a minimum spanning tree of V, where τ = 1.998 · max(π/2, π sin(θ/2) + 1)
While we focus on our construction, we note that the techniques used to bound the weight of the graph apply generally to any planar geometric graph
(I.I) If Ai−1 is not contained in the cone with apex vi sweeping clockwise from vi u1 to vi um, simulating a step of the relaxation of Bonichon et al.’s Routing Algorithm [8] is analogous to the method used for the first step: determine if vi u1 or vi um is a middle edge and use a Guided or Unguided Face Walk to reach the proper vertex of Ai
Summary
The aim of this paper is to design a graph on V (a finite set of points in the Euclidean plane) that is cheap to build and easy to route on. We show: Given a set V of n points in the plane, together with two parameters 0 < θ < π/2 and r > 0, we show how to construct in O(n log n) time a planar ((1+1/r )·τ )-spanner with degree at most 5 2π/θ , and weight at most ((2r + 1) · τ ) times the weight of a minimum spanning tree of V , where τ = 1.998 · max(π/2, π sin(θ/2) + 1) This construction admits an O(1)-memory deterministic 1-local routing algorithm with a routing ratio of no more than 5.90 · (1 + 1/r ) · max(π/2, π sin(θ/2) + 1). Using techniques similar to the ones we use, it may be possible to extend the results by Bose et al [11] to obtain other routing algorithms for bounded-degree light spanners
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