Geometric p-Center Problems with Centers Constrained to Two Lines

We first consider the weighted p-center problem, in which the centers are constrained to lie on two axis-parallel lines. Given a set of n points in the plane, which are sorted according to their x-coordinates, we show how to test in O(n log n) time if p piercing points placed on two lines, parallel to the x-axis, can pierce all the disks of different radii centered at the n given points. This leads to an O(n log(2) n) time algorithm for the weighted p-center problem. We then consider the unweighted case, where the centers are constrained to be on two perpendicular lines. Our algorithm runs in O(n log(2) n) time in this case as well.