Combinatorics of Beacon-based Routing in Three Dimensions

TL;DR

This paper extends the concept of beacon-based routing from 2D polygons to 3D polyhedra, establishing bounds on the number of beacons needed for routing based on tetrahedral decompositions.

Contribution

It introduces the first results on beacon routing in three dimensions, providing bounds on the number of beacons required in polyhedra.

Findings

eacons suffice for routing in 3D polyhedra

Bounds depend on the number of tetrahedra in the decomposition

First extension of beacon routing theory to three dimensions

Abstract

A beacon is a point-like object which can be enabled to exert a magnetic pull on other point-like objects in space. Those objects then move towards the beacon in a greedy fashion until they are either stuck at an obstacle or reach the beacon's location. Beacons placed inside polyhedra can be used to route point-like objects from one location to another. A second use case is to cover a polyhedron such that every point-like object at an arbitrary location in the polyhedron can reach at least one of the beacons once the latter is activated. The notion of beacon-based routing and guarding was introduced by Biro et al. [FWCG'11] in 2011 and covered in detail by Biro in his PhD thesis [SUNY-SB'13], which focuses on the two-dimensional case. We extend Biro's result to three dimensions by considering beacon routing in polyhedra. We show that $\lfloor\frac{m+1}{3}\rfloor$ beacons are always sufficient and sometimes necessary to route between any pair of points in a given polyhedron $P$, where $m$ is the number of tetrahedra in a tetrahedral decomposition of $P$. This is one of the first results that show that beacon routing is also possible in three dimensions.