Improved algorithms in geometric optimization via expanders

We incorporate into our expander-based technique for solving problems in geometric optimization, as developed in Katz and Sharir (1993), a technique which is in some sense equivalent to (though much more flexible than) Cole's technique for accelerating parametric searching (1987). This enables us to obtain, in some cases, deterministic algorithms that are asymptotically faster by a logarithmic factor than the best previously known algorithms (which are mostly based on parametric searching). We exemplify the enhanced technique on two problems, the planar distance selection problem and the planar two-line center problem. To obtain our more efficient solutions, we also develop some auxiliary results concerning batched range searching where the ranges are congruent discs or annuli. For example, we show that it is possible to compute deterministically a compact representation of the set of all point-disc containments among a set of n congruent discs and a set of m points in the plane, in time O((m^2/3n^2/3+m+n) log n), slightly better than what was previously known. We believe these results are of independent interest.