A beacon $b\in {\mathbb{R}}^d$ is a point-shaped object in d-dimensional space that can exert a magnetic pull on any other point-shaped object $p\in {\mathbb{R}}^d$. This object p then moves greedily towards b. The motion stops when p gets stuck at an obstacle or when p reaches b. By placing beacons inside a d-dimensional polyhedron P, we can implement a scheme to route point-shaped objects between any two locations in P. We can also place beacons to guard P, which means that any point-shaped object in P can reach at least one activated beacon.

The notion of beacon-based routing and guarding was introduced in 2011 by Biro et al. [FWCG'11]. The two-dimensional setting is discussed in great detail in Biro's 2013 PhD thesis [SUNY-SB'13].

Here, we consider combinatorial aspects of beacon routing in three dimensions. We show that $\lfloor (m+1)/3\rfloor$ beacons are always sufficient and sometimes necessary to route between any two points in a given polyhedron P, where m is the smallest size of a tetrahedral decomposition of P. This is one of the first results to show that beacon routing is also possible in higher dimensions.

信标 $b\in {\mathbb{[R}}^d$是d维空间中的点形对象，可以对任何其他点形对象施加磁拉$p\in {\mathbb{[R}}^d$。然后，该对象p贪婪地向b移动。当p卡在障碍物上或p到达b时，运动停止。通过将信标放置在d维多面体P中，我们可以实现一种在P中任意两个位置之间路由点形对象的方案。我们还可以将信标来保护P，这意味着在任何点状对象P可达到至少一个激活的信标。

Biro等人于2011年引入了基于信标的路由和防护的概念。[FWCG'11]。Biro的2013年博士学位论文[SUNY-SB'13]中对二维设置进行了详细讨论。

在这里，我们从三个维度考虑信标路由的组合方面。我们证明$\lfloor （米+\mathrm{1个}）/3\rfloor$在给定多面体P中的任何两个点之间路由时，信标总是足够的，有时是必要的，其中m是P的四面体分解的最小大小。这是表明信标路由在更高维度上也是可行的第一个结果。