Competitive Local Routing with Constraints

TL;DR

This paper studies local routing algorithms on constrained $ heta$-graphs, showing limitations of deterministic algorithms and providing new algorithms with competitive path guarantees in constrained geometric graphs.

Contribution

It introduces the first local routing algorithms with path length guarantees for constrained $ heta$-graphs, including a 2-competitive algorithm for the full graph and an 18-competitive one for a subgraph.

Findings

No deterministic 1-local routing algorithm is o(√n)-competitive.

A 1-local routing algorithm guarantees a 2-competitive path.

An 18-competitive routing algorithm exists for the constrained half-$ heta_6$-graph.

Abstract

Let $P$ be a set of $n$ vertices in the plane and $S$ a set of non-crossing line segments between vertices in $P$, called constraints. Two vertices are visible if the straight line segment connecting them does not properly intersect any constraints. The constrained $\Theta_m$-graph is constructed by partitioning the plane around each vertex into $m$ disjoint cones, each with aperture $\theta = 2 \pi/m$, and adding an edge to the `closest' visible vertex in each cone. We consider how to route on the constrained $\Theta_6$-graph. We first show that no deterministic 1-local routing algorithm is $o(\sqrt{n})$-competitive on all pairs of vertices of the constrained $\Theta_6$-graph. After that, we show how to route between any two visible vertices of the constrained $\Theta_6$-graph using only 1-local information. Our routing algorithm guarantees that the returned path is 2-competitive. Additionally, we provide a 1-local 18-competitive routing algorithm for visible vertices in the constrained half-$\Theta_6$-graph, a subgraph of the constrained $\Theta_6$-graph that is equivalent to the Delaunay graph where the empty region is an equilateral triangle. To the best of our knowledge, these are the first local routing algorithms in the constrained setting with guarantees on the length of the returned path.