Batched Point Location in SINR Diagrams via Algebraic Tools

Abstract

The SINR (Signal to Interference plus Noise Ratio) model for the quality of wireless connections has been the subject of extensive recent study. It attempts to predict whether a particular transmitter is heard at a specific location, in a setting consisting of n simultaneous transmitters and background noise. The SINR model gives rise to a natural geometric object, the SINR diagram , which partitions the space into n regions where each of the transmitters can be heard and the remaining space where no transmitter can be heard. Efficient point location in the SINR diagram, i.e., being able to build a data structure that facilitates determining, for a query point, whether any transmitter is heard there, and if so, which one, has been recently investigated in several articles. These planar data structures are constructed in time at least quadratic in n and support logarithmic-time approximate queries. Moreover, the performance of some of the proposed structures depends strongly not only on the number n of transmitters and on the approximation parameter ε , but also on some geometric parameters that cannot be bounded a priori as a function of n or ε . In this article, we address the question of batched point location queries, i.e., answering many queries simultaneously. Specifically, in one dimension, we can answer n queries exactly in amortized polylogarithmic time per query, while in the plane we can do it approximately. In another result, we show how to answer n 2 queries exactly in amortized polylogarithmic time per query, assuming the queries are located on a possibly non-uniform n × n grid. All these results can handle arbitrary power assignments to the transmitters. Moreover, the amortized query time in these results depends only on n and ε . We also show how to speed up the preprocessing in a previously proposed point-location structure in SINR diagram for uniform-power sites, by almost a full order of magnitude. For this, we obtain results on the sensitivity of the reception regions to slight changes in the reception threshold, which are of independent interest. Finally, these results demonstrate the (so far underutilized) power of combining algebraic tools with those of computational geometry and other fields.