Routing on the Visibility Graph

TL;DR

This paper introduces two deterministic local routing algorithms for visibility graphs with obstacles, guaranteeing linear path length and minimal memory use, and establishes lower bounds on routing efficiency.

Contribution

The paper presents the first deterministic local routing algorithms on visibility graphs that do not rely on plane subgraphs, with proven efficiency bounds.

Findings

Algorithms guarantee paths of linear size between vertices.

Routing algorithms operate with constant memory and local information.

Lower bounds on routing ratios for deterministic algorithms are established.

Abstract

We consider the problem of routing on a network in the presence of line segment constraints (i.e., obstacles that edges in our network are not allowed to cross). Let $P$ be a set of $n$ points in the plane and let $S$ be a set of non-crossing line segments whose endpoints are in $P$. We present two deterministic 1-local $O(1)$-memory routing algorithms that are guaranteed to find a path of at most linear size between any pair of vertices of the \emph{visibility graph} of $P$ with respect to a set of constraints $S$ (i.e., the algorithms never look beyond the direct neighbours of the current location and store only a constant amount of additional information). Contrary to {\em all} existing deterministic local routing algorithms, our routing algorithms do not route on a plane subgraph of the visibility graph. Additionally, we provide lower bounds on the routing ratio of any deterministic local routing algorithm on the visibility graph.