On some geometric selection and optimization problems via sorted matrices

In this paper we apply the selection and optimization technique of Frederickson and Johnson to a number of geometric selection and optimization problems, some of which have previously been solved by parametric search, and provide efficient and simple algorithms. Our technique improves the solutions obtained by parametric search by a log n factor. For example, we apply the technique to the two-line center problem, where we want to find two strips that cover a given set S of n points in the plane, so as to minimize the width of the largest of the two strips.