On interference among moving sensors and related problems

TL;DR

This paper demonstrates how to select a small, fixed subset of moving sensors to maintain balanced Voronoi diagrams over time, with applications in reducing interference and extending epsilon-net theory to dynamic settings.

Contribution

It introduces a method for selecting a small subset of moving points to keep Voronoi diagrams balanced at all times, extending epsilon-net theory to kinetic environments.

Findings

A subset of size O(k log k) maintains balanced Voronoi diagrams for moving points.

Interference among sensors can be bounded by O(√n log n) with appropriate radius assignment.

Extended epsilon-net results to kinetic (moving) point sets.

Abstract

We show that for any set of $n$ points moving along "simple" trajectories (i.e., each coordinate is described with a polynomial of bounded degree) in $\Re^d$ and any parameter $2 \le k \le n$, one can select a fixed 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 (i.e., it contains $O(n/k)$ points per cell). We also show that the bound $O(k \log k)$ is near optimal even for the one dimensional case in which points move linearly in time. As applications, we show that one can assign communication radii to the sensors of a network of $n$ moving sensors so that at any given time their interference is $O(\sqrt{n\log n})$. We also show some results in kinetic approximate range counting and kinetic discrepancy. In order to obtain these results, we extend well-known results from $\varepsilon$-net theory to kinetic environments.