Parallel line centers with guaranteed separation

Abstract

Given a set P of n points in the plane and an integer $k\ge 1$, 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 $k\ge 2$, 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 $0<\rho \le 1$, compute k parallel slabs of minimum width that cover P with gap-ratio at least ρ. (2) Compute k parallel slabs that cover P with maximum possible gap-ratio. Our algorithms run in $O({\rho }^{-k}\cdot (n\log n+kn\left)\right)$ and $O({\rho }_{\max }^{-k}\cdot (n\log n+kn\left)\right)$ time, respectively, using $O(kn\log k)$ 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 $O({\rho }^{-k}\cdot kn\log n)$ and $O({\rho }_{\max }^{-k}\cdot kn\log n)$.