Combinatorics of Beacon-based Routing in Three Dimensions

A beacon is a point-like object that 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. The notion of beacon-based routing was introduced by Biro et al. [FWCG’11] in 2011 for two dimensions and covered in detail by Biro in his PhD thesis [SUNY-SB’13]. We extend Biro’s results to polyhedra in three dimensions. We show that ⌊ m+1 3 ⌋ 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.