# Approximate Minimum Diameter

## TL;DR

This paper introduces improved approximation algorithms for estimating the minimum diameter of inexact point sets under different models, with significant complexity and approximation guarantees in various dimensions.

## Contribution

It presents a new $O(2^{1/ ext{epsilon}^d} ext{epsilon}^{-2d} n^3)$ time approximation algorithm for the indec model and a polynomial-time $ ext{sqrt}(d)$-approximation for the imprecise model, also addressing $ ext{alpha}$-separable regions.

## Key findings

- Improved approximation algorithms for minimum diameter in inexact point sets.
- Complexity bounds for algorithms in different models and dimensions.
- Specialized algorithms for $ ext{alpha}$-separable regions in 2D.

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