Optimal Slope Selection Via Expanders

Given n points in the plane and an integer k, the slope selection problem is to nd the pair of points whose connecting line has the k-th smallest slope. (In dual setting, given n lines in the plane, we want to nd the vertex of their arrangement with the k-th smallest x-coordinate.) Cole et al. 6] have given an O(n log n) solution (which is optimal), using the parametric searching technique of Megiddo. We obtain another optimal (deterministic) solution that does not depend on parametric searching and uses expander graphs instead. Our solution is somewhat simpler than that of 6] and has a more explicit geometric interpretation.