On interference among moving sensors and related problems

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.

💡 Research Summary

The paper tackles a fundamental problem in kinetic computational geometry: given n points moving in ℝ^d along “simple” trajectories (each coordinate is a univariate polynomial of bounded degree), select a small fixed subset of these points that can serve as facilities (or communication hubs) such that at any instant the load on each facility is balanced. The authors prove that a subset of size O(k log k) suffices to guarantee that the Voronoi diagram of the selected points is “balanced”: every Voronoi cell contains at most O(n/k) of the moving points, for any integer 2 ≤ k ≤ n. They also show that this bound is essentially optimal even in one dimension with linear motion.

The technical core lies in extending ε‑net theory and VC‑dimension arguments from static range spaces to kinetic ones. A kinetic hypergraph (P,R) is defined where P is the moving point set and R a family of geometric ranges (e.g., halfspaces, cones). The authors prove that for many classic families (halfspaces, bounded cones, any semi‑algebraic set of constant description complexity) the kinetic hypergraph still has bounded VC‑dimension, specifically O(d log d + log s log log s) where s is the maximum polynomial degree of the trajectories. The proof hinges on the observation that a combinatorial change in the hypergraph occurs only when d + 1 points become affinely dependent, which translates into a polynomial equation of degree at most d·s. Consequently, the number of such events over all time is polynomially bounded, leading to a polynomial shatter function and, via a standard lemma, a constant‑order VC‑dimension.

With bounded VC‑dimension in hand, the classical ε‑net theorem guarantees an ε‑net of size O((1/ε)·log(1/ε)). Setting ε = 1/k yields a net N of size O(k log k). The authors then argue that for any time t and any auxiliary point set S, the Voronoi diagram of N(t) ∪ S has the desired balance property: each cell corresponds to a bounded cone (or a similar range) that contains at least n/k points, and by the ε‑net property such a cone must intersect N, preventing any cell from swallowing more than O(n/k) points.

The balanced Voronoi structure directly leads to an application in wireless sensor networks. By assigning each sensor a communication radius proportional to the size of its Voronoi cell, the number of sensors that can directly communicate with a given sensor (its interference) becomes O(n/k). Choosing k ≈ √(n/ log n) yields an interference bound of O(√(n log n)), which matches the known lower bound Ω(√n) up to a logarithmic factor and simultaneously guarantees a hop‑diameter of three for the communication graph.

Additional applications are discussed: kinetic ε‑approximations for approximate range counting and kinetic discrepancy bounds. Because the kinetic hypergraphs retain low VC‑dimension, existing static algorithms for these problems can be lifted to the kinetic setting with only modest overhead.

In summary, the paper establishes that many static geometric tools—VC‑dimension, ε‑nets, balanced partitions—remain powerful in dynamic environments where points follow polynomial trajectories. It provides near‑optimal bounds for facility placement and sensor interference, and opens the door to a suite of kinetic algorithms for range searching, counting, and discrepancy minimization. The results are both theoretically elegant and practically relevant for mobile ad‑hoc networks and other systems with continuously moving geometric data.