Efficiently Approximating the Minimum-Volume Bounding Box of a Point Set in Three Dimensions

TL;DR

This paper introduces two efficient algorithms for approximating the minimum-volume bounding box of a 3D point set, with theoretical guarantees and experimental validation, improving computational efficiency in geometric data analysis.

Contribution

It proposes a novel approximation algorithm with improved runtime and a simpler alternative, both for computing near-optimal bounding boxes in three dimensions.

Findings

The first algorithm runs in $O(n + 1/ ext{ε}^{4.5})$ time with approximation guarantees.

The second, simpler algorithm runs in $O(n ext{log} n + n / ext{ε}^3)$ time.

Experimental results demonstrate practical effectiveness of the algorithms.

Abstract

$\renewcommand{\Re}{\mathbb{R}}$We present an efficient $O (n + 1/\varepsilon^{4.5})$-time algorithm for computing a $(1+\varepsilon$)-approximation of the minimum-volume bounding box of $n$ points in $\Re^3$. We also present a simpler algorithm (for the same purpose) whose running time is $O (n \log{n} + n / \varepsilon^3)$. We give some experimental results with implementations of various variants of the second algorithm. The implementation of the algorithm described in this paper is available online https://github.com/sarielhp/MVBB.