LATIN 2018: LATIN 2018: Theoretical Informatics pp 346-360

# Combinatorics of Beacon-Based Routing in Three Dimensions

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10807)

## 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 Ph.D. 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.

## Notes

### Acknowledgments

We thank the anonymous reviewers for their thorough reading of the paper and helpful suggestions.

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## Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

## Authors and Affiliations

1. 1.Institut für InformatikFreie Universität BerlinBerlinGermany