Optimal Local Routing on Delaunay Triangulations Defined by Empty Equilateral Triangles

Abstract

We present a deterministic local routing algorithm that is guaranteed to find a path between any pair of vertices in a half-θ 6 -graph 1 (the half-θ 6 -graph is equivalent to the Delaunay triangulation where the empty region is an equilateral triangle). The length of the path is at most 5/ √ 3 ≈ 2.887 times the Euclidean distance between the pair of vertices. Moreover, we show that no local routing algorithm can achieve a better routing ratio, thereby proving that our routing algorithm is optimal. This is somewhat surprising because the spanning ratio of the half-θ 6 -graph is 2, meaning that even though there always exists a path whose lengths is at most twice the Euclidean distance, we cannot always find such a path when routing locally.Since every triangulation can be embedded in the plane as a half-θ 6 -graph using O(log n) bits per vertex coordinate via Schnyder's embedding scheme (SODA 1990), our result provides a competitive local routing algorithm for every such embedded triangulation. Finally, we show how our routing algorithm can be adapted to provide a routing ratio of 15/ √ 3 ≈ 8.660 on two bounded degree subgraphs of the half-θ 6 -graph.

On Plane Constrained Bounded-Degree Spanners

“…Bose and Keil [7] showed that the Constrained Delaunay Triangulation is a 2.42-spanner of Vis(P, S). In this article, we show that the constrained half-θ 6 -graph (which is identical to the constrained Delaunay graph whose empty visible region is an equilateral triangle) is a plane 2-spanner of Vis(P, S) by generalizing the approach used by Bose et al [6]. This improves the upper bound on the spanning ratio of 36 implied by Bose et al [4].…”

Local Routing in Spanners Based on WSPDs

“…They can achieve spanning ratios arbitrarily close to 1 by choosing arbitrarily small values for this parameter. Significant work has gone into finding competitive local and low-memory routing algorithms for graphs in the first category, including Delaunay graphs (classical-, L 1 -, L ∞ -, TD-, and generalized convex Delaunay triangulations) [6,7,10,11,15]. In most cases, proving tight spanning ratios and routing ratios for graphs in this category is difficult.…”

“…Still, for certain cone-based spanners, there have been some refined results on competitive routing algorithms that produce exceptionally low competitive ratios. For example, Bose et al [10] present a routing algorithm for the TD-Delaunay triangulation (which is equivalent to the Half-θ 6 -graph) with a competitive ratio of 2.887. They prove that this is optimal, thereby proving a separation between the routing ratio and the spanning ratio of a graph since the spanning ratio of the TD-Delaunay triangulation is 2 [16].…”

Routing in polygonal domains

“…Another approach is called geometric routing. Here, the graph is embedded in a geometric space, and the routing algorithm has to determine the next vertex for the data packet based on the location of the source and the target vertex, the current vertex, and its neighbourhood, see for instance [9,10] and the references therein. The most notable difference between geometric routing and our setting is that in geometric routing, vertices are generally not allowed to store routing tables, so that routing decisions are based solely on the geometric information available at the current vertex (and possibly information stored in the message).…”