Nonuniform SINR+Voroni diagrams are effectively uniform

Abstract

This paper concerns the behavior of an SINR diagram of wireless systems, composed of a set S of n stations embedded in ℝd, when restricted to the corresponding Voronoi diagram imposed on S. The diagram obtained by restricting the SINR zones to their corresponding Voronoi cells is referred to hereafter as an SINR+Voronoi diagram. While uniform SINR diagrams (where all stations transmit with the same power) are simple and nicely structured (e.g., the station reception zones are convex and “fat”) [3], nonuniform SINR diagrams might be complex (e.g., the reception zones might be fractured and their boundaries might contain many singular points) [9]. In this paper, we establish the (perhaps surprising) fact that a nonuniform SINR+Voronoi diagram is topologically almost as nice as a uniform SINR diagram. In particular, it is convex and effectively (In the sense that its fatness measure does not depend on the number of stations n but only on parameters typically bounded by a constant.) fat. This holds for every power assignment, every path-loss parameter α and every dimension d ≥ 1. The convexity property also holds for every SINR threshold β > 0, and the effective fatness holds for any β > 1. These fundamental properties provide a theoretical justification to engineering practices basing zonal tessellations on the Voronoi diagram, and helps to explain the soundness and efficacy of such practices. We also consider two algorithmic applications. The first concerns the Power Control with Voronoi Diagram (PCVD) problem, where given n stations embedded in some polygon P, it is required to find the power assignment that optimizes the SINR threshold of the transmission station si for any given reception point p ∈ P in its Voronoi cell Vor(si). The second application is approximate point location; we show that for SINR+Voronoi zones, this task can be solved considerably more efficiently than in the general non-uniform case.