A Combinatorial Bound for Beacon-based Routing in Orthogonal Polygons

TL;DR

This paper establishes a tight combinatorial bound on the size of beacon sets needed for routing in orthogonal polygons, showing the maximum size is roughly one-third of the polygon's vertices minus a constant.

Contribution

It provides the first tight bound on the minimum number of beacons required for complete routing in orthogonal polygons, improving understanding of beacon-based navigation.

Findings

Maximum beacon set size is approximately (n-4)/3.

Bound is tight; cannot be improved further.

Provides constructive methods for routing with optimal beacon counts.

Abstract

Beacon attraction is a movement system whereby a robot (modeled as a point in 2D) moves in a free space so as to always locally minimize its Euclidean distance to an activated beacon (which is also a point). This results in the robot moving directly towards the beacon when it can, and otherwise sliding along the edge of an obstacle. When a robot can reach the activated beacon by this method, we say that the beacon attracts the robot. A beacon routing from $p$ to $q$ is a sequence $b_1, b_2,$ ..., $b_{k}$ of beacons such that activating the beacons in order will attract a robot from $p$ to $b_1$ to $b_2$ ... to $b_{k}$ to $q$, where $q$ is considered to be a beacon. A routing set of beacons is a set $B$ of beacons such that any two points $p, q$ in the free space have a beacon routing with the intermediate beacons $b_1, b_2,$ ..., $b_{k}$ all chosen from $B$. Here we address the question of "how large must such a $B$ be?" in orthogonal polygons, and show that the answer is "sometimes as large as $[(n-4)/3]$, but never larger."