Batched Point Location in SINR Diagrams via Algebraic Tools
  The SINR model for wireless networks has been extensively studied recently. It tries to model whether a particular transmitter is heard at a specific location, with $n$ transmitting simultaneously. The SINR diagram consists of $n$ regions where each transmitter can be heard and the remaining space where no one can be heard.   Efficient point location in the SINR diagram, i.e., building a data structure to determine, for a query point, whether any transmitter is heard there, and if so, which one, has been recently investigated. Previous such planar data structures are constructed in time at least quadratic in $n$ and support logarithmic-time approximate queries. Moreover, the performance of these depends not only on the number $n$ of transmitters and on the approximation parameter $\varepsilon$, but also on some geometric parameters that cannot be bounded a priori as a function of $n$ or $\varepsilon$.   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.   All these results can handle arbitrary power assignments to the transmitters. Moreover, the amortized query time in these results depends only on $n$ and $\varepsilon$.   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. 
Boris Aronov and Matthew J. Katz
cs.CG