Nonuniform SINR+Voroni Diagrams Are Effectively Uniform

This paper concerns the behavior of an SINR diagram of wireless systems, composed of a set S of n stations embedded in R, 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. The study of SINR+Voronoi diagrams is motivated by the following two facts. (1) Uniform SINR diagrams (where all stations transmit with the same power) are simple and nicely structured. In particular, the reception zone of each station is convex and “fat”; this can be used to devise an efficient algorithm for the fundamental problem of point location [3]. (2) In contrast, nonuniform SINR diagrams (where transmission energies are arbitrary) might be complex; the reception zone of each station might be fractured and its boundary might contain many singular points [9]. This makes it harder to understand the geometry of nonuniform SINR diagrams, as well as to design efficient point location algorithms for this setting. 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 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 affective fatness holds for any β > 1. These fundamental properties provide a theoretical justification to engineering practices basing zoneal tessellations on the Voronoi diagram, and helps to explain the soundness and efficacy of such practices. We then 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. ∗MIT, CSAIL. E-mail: [email protected]. †Department of Communication Systems Engineering, Ben Gurion University, Beer-Sheva, Israel. E-mail: [email protected]. ‡Supported in part by the Israel Science Foundation (grant 894/09). §Department of Computer Science and Applied Mathematics, The Weizmann Institute of Science, Rehovot, Israel. E-mail: {merav.parter,david.peleg}@ weizmann.ac.il. 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.