The topology of wireless communication

In this paper we study the topological properties of wireless communication maps and their usability in algorithmic design. We consider the SINR model, which compares the received power of a signal at a receiver against the sum of strengths of other interfering signals plus background noise. To describe the behavior of a multi-station network, we use the convenient representation of a reception map. In the SINR model, the resulting SINR diagram partitions the plane into reception zones, one per station, and the complementary region of the plane where no station can be heard. SINR diagrams have been studied in [3] for the specific case where all stations use the same power. It is shown that the reception zones are convex (hence connected) and fat, and this is used to devise an efficient algorithm for the fundamental problem of point location. Here we consider the more general (and common) case where transmission energies are arbitrary (or non-uniform). Under that setting, the reception zones are not necessarily convex or even connected. This poses the algorithmic challenge of designing efficient point location techniques for the non-uniform setting, as well as the theoretical challenge of understanding the geometry of SINR diagrams (e.g., the maximal number of connected components they might have). We achieve several results in both directions. We establish a form of weaker convexity in the case where stations are aligned on a line and use this to derive a tight bound on the number of connected components in this case. In addition, one of our key results concerns the behavior of a (d+1)-dimensional map, i.e., a map in one dimension higher than the dimension in which stations are embedded. Specifically, although the d-dimensional map might be highly fractured, drawing the map in one dimension higher "heals" the zones, which become connected (in fact hyperbolically connected). In addition, as a step toward establishing a weaker form of convexity for the d-dimensional map, we study the interference function and show that it satisfies the maximum principle. This is done through an analysis technique based on looking at the behavior of systems composed on lines of densely placed weak stations, as the number of stations tends to infinity, keeping their total transmission energy fixed. Finally, we turn to consider algorithmic applications, and propose a new variant of approximate point location.