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Combinatorics of Beacon-Based Routing in Three Dimensions

  • Jonas Cleve
  • Wolfgang Mulzer
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

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