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 continuous region (${Imprecise}$ model) or a finite set of points (${Indecisive}$ model). Given a set of inexact points in one of ${Imprecise}$ or ${Indecisive}$ 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 ${Indecisive}$ 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 ${Imprecise}$ 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 ${Indecisive}$ model to obtain $(1+\epsilon )$-approximation algorithm for a set of $\alpha$-separable regions in time $O(2^{\frac{1}{\epsilon ^2}}. \frac{n^3}{\epsilon ^{10} . \sin (\alpha /2)^3} )$.