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 $O(2^{\frac{1}{{\epsilon }^2}}.\frac{n^3}{{\epsilon }^{10}.\sin (\alpha /2{)}^3})$.