# 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.

## Key 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.