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. Our new approach is based on expander graphs and is conceptually much simpler and has more explicit geometric flavor. It does not require parallelization or randomization, and it exploits recent range-searching techniques of Matousˇek and others. We exemplify the technique on three problems, the <italic>slope selection</italic> problem, the planar <italic>distance selection</italic> problem, and the planar <italic>two-center</italic> problem. For the first problem we develop an <italic>O(n log<supscrpt>3</supscrpt>n)</italic>) solution, which, although suboptimal, is very simple. The second and third problems are more typical examples of our approach. Our solutions have, respectively, running time <italic>O(n<supscrpt>4/3</supscrpt> log<supscrpt>3+δ</supscrpt> n)</italic>, for any δ > 0, and <italic>O(n<supscrpt>2</supscrpt> log<supscrpt>3</supscrpt> n)</italic>, comparable with the respective solutions of [2, 5].