Linear Time Algorithm for 1-Center in $\mathfrak {R}^d$ Under Convex Polyhedral Distance Function

Abstract

In this paper we present algorithms for computing 1-center of a set of points for convex polyhedral distance function in $\mathfrak {R}^d$ for any d. Given polyhedral P of size m, the running time of our algorithm for computing 1-center of n points in $\mathfrak {R}^2$ for convex polygonal distance function $d_P$ is $O(nm\log ^2 m)$. For $d>2$, we present an $O(3^{3d^2} nm^2\log ^d m)$ algorithm to compute 1-center of n points in $\mathfrak {R}^d$ for convex polyhedral distance function $d_P$, $|P|=m$. Both the algorithms are linear time for fixed d and fixed polyhedron P.