Expected Complexity of Routing in $\Theta$ 6 and Half-$\Theta$ 6 Graphs

TL;DR

This paper introduces new online routing algorithms for $ heta$6 and half-$ heta$6 graphs, achieving significantly better average-case routing ratios than worst-case bounds, especially on random point sets.

Contribution

It presents simpler, memory-efficient online routing algorithms with proven average-case performance improvements for $ heta$6 and half-$ heta$6 graphs.

Findings

Expected routing ratio of 1.161 for $ heta$6-graph on random points

Expected routing ratio of 1.43 for half-$ heta$6-graph with memoryless routing

Algorithms outperform worst-case routing ratios of 2 and 2.89

Abstract

We study online routing algorithms on the $\Theta$6-graph and the half-$\Theta$6-graph (which is equivalent to a variant of the Delaunay triangulation). Given a source vertex s and a target vertex t in the $\Theta$6-graph (resp. half-$\Theta$6-graph), there exists a deterministic online routing algorithm that finds a path from s to t whose length is at most 2 st (resp. 2.89 st) which is optimal in the worst case [Bose et al., siam J. on Computing, 44(6)]. We propose alternative, slightly simpler routing algorithms that are optimal in the worst case and for which we provide an analysis of the average routing ratio for the $\Theta$6-graph and half-$\Theta$6-graph defined on a Poisson point process. For the $\Theta$6-graph, our online routing algorithm has an expected routing ratio of 1.161 (when s and t random) and a maximum expected routing ratio of 1.22 (maximum for fixed s and t where all other points are random), much better than the worst-case routing ratio of 2. For the half-$\Theta$6-graph, our memoryless online routing algorithm has an expected routing ratio of 1.43 and a maximum expected routing ratio of 1.58. Our online routing algorithm that uses a constant amount of additional memory has an expected routing ratio of 1.34 and a maximum expected routing ratio of 1.40. The additional memory is only used to remember the coordinates of the starting point of the route. Both of these algorithms have an expected routing ratio that is much better than their worst-case routing ratio of 2.89.