On Interference Among Moving Sensors and Related Problems

We study geometric hypergrahs in a kinetic setting. That is, the set of vertices of the hypergraph is a set of moving points in R with coordinates that are polynomials in time. The hyperedges are all subsets that can be realized by intersecting the set of points at some fixed time with some “simple” geometric shape, such as, say, a halfspace. We show that for many of the static cases where the VC-dimension of the hypergraph is bounded, the kinetic counterpart also has bounded VC-dimension. This allows us to prove our main result: for any set of n moving points in R and any parameter 1 < k < n, one can select a non-empty subset of the points of size O(k log k) such that the Voronoi diagram of this subset is “balanced” at any given time. By that, we mean that at any time, each Voronoi cell contains at most O(n/k) of the points. We also show that the bound O(k log k) is near optimal already for the one dimensional case (i.e., d = 1) and points moving linearly (i.e., with constant speed). As an application, we show that we can assign a communication radius to a collection of n moving sensors so that at any given time, their interference is O( √ n log n). This is optimal up to an O( √ log n) factor.