Approximate Minimum Diameter
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 $O(2^{\frac{1}{ε^d}} \cdot ε^{-2d} \cdot n^3 )$ time approximation algorithm of factor $(1+ε)$ 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 $α$-separability and use our algorithm for $\indec$ model to obtain $(1+ε)$-approximation algorithm for a set of $α$-separable regions in time $O(2^{\frac{1}{ε^2}}\allowbreak . \frac{n^3}{ε^{10} .\sin(α/2)^3} )$.