Parallel line centers with guaranteed separation

초록

Given a set P of n points in the plane and an integer k >= 1, the k-line-center problem asks k slabs whose union covers P that minimizes the maximum width of the k slabs. In this paper, we introduce a new variant of the k-line-center problem for k >= 2, in which the resulting k lines are parallel and a prescribed separation between two line centers is guaranteed. More precisely, we define a measure of separation, namely the gap-ratio of k parallel slabs, to be the minimum distance between any two slabs, divided by the width of the smallest slab enclosing the k slabs. We present efficient algorithms for the following problems: (1) Given a real 0<rho <= 1, compute k parallel slabs of minimum width that cover P with gap-ratio at least rho. (2) Compute k parallel slabs that cover P with maximum possible gap-ratio. Our algorithms run in O(rho(-k).(nlogn+kn)) and O(rho(-k)(max).(nlogn+kn)) time, respectively, using O(knlogk) space, where rho(max) denotes the maximum possible gap-ratio of any k parallel slabs that cover P. Using linear space, the running times only slightly increase to O((rho-k).knlogn) and O(rho(-k)(max).knlogn).