An expander-based approach to geometric optimization

We present a new approach to problems in geometric optimization that are traditionally solved using the parametric-searching technique of Megiddo [J. ACM, 30 (1983), pp. 852-865]. Our new approach is based on expander graphs and range-searching techniques. It is conceptually simpler, has more explicit geometric flavor, and does not require parallelization or randomization. In certain cases, our approach yields algorithms that are asymptotically faster than those currently known (e.g., the second and third problems below) by incorporating into our (basic) technique a subtechnique that is equivalent to (though much more flexible than) Cole's technique for speeding up parametric searching [J. ACM, 34 (1987), pp. 200-208]. We exemplify the technique on three main problems - the slope selection problem, the planar distance selection problem, and the planar two-line center problem. For the first problem we develop an O(n log³ n) solution, which, although suboptimal, is very simple. The other two problems are more typical examples of our approach. Our solutions have running time O(n^4/3 log² n) and O(n² log⁴ n), respectively, slightly better than the previous respective solutions of [Agarwal et al., Algorithmica, 9 (1993), pp. 495-514], [Agarwal and Sharir, Algorithmica, 11 (1994), pp. 185-195]. We also briefly mention two other problems that can be solved efficiently by our technique. In solving these problems, we also obtain 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 incidences 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), again slightly better than what was previously known.