Approximate Minimum Diameter

Abstract

We study the minimum diameter problem for a set of inexact points. By inexact, we mean that the precise location of the points is not known. Instead, the location of each point is restricted to a contineus region (\impre model) or a finite set of points (\indec model). Given a set of inexact points in one of \impre or \indec models, we wish to provide a lower-bound on the diameter of the real points. In the first part of the paper, we focus on \indec model. We present an $O(2^{\frac{1}{\epsilon^d}} \cdot \epsilon^{-2d} \cdot n^3 )$ time approximation algorithm of factor $(1+\epsilon)$ for finding minimum diameter of a set of points in $d$ dimensions. This improves the previously proposed algorithms for this problem substantially. Next, we consider the problem in \impre model. In $d$-dimensional space, we propose a polynomial time $\sqrt{d}$-approximation algorithm. In addition, for $d=2$, we define the notion of $\alpha$-separability and use our algorithm for \indec model to obtain $(1+\epsilon)$-approximation algorithm for a set of $\alpha$-separable regions in time $O(2^{\frac{1}{\epsilon^2}}\allowbreak . \frac{n^3}{\epsilon^{10} .\sin(\alpha/2)^3} )$