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

Gill Barequet, Sariel Har-Peled

TL;DR

This work delivers two near-linear-time algorithms for computing a $(1+\varepsilon)$-approximation of the minimum-volume bounding box for 3D point sets: a main method with runtime $O(n + 1/\varepsilon^{4.5})$ and a simpler variant with $O(n \log n + n/\varepsilon^{3})$. The approach replaces the convex hull with a low-complexity polyhedron and solves the bounding-box problem exactly on that surrogate, ensuring provable accuracy. A more practical grid-search variant with $O(n \log n + n/\varepsilon^{3})$ time is also proposed and extensively evaluated, demonstrating favorable trade-offs in practice. The paper also discusses extensions, potential improvements, and open problems, including dynamic MVBB maintenance and extensions to higher dimensions or bounding ellipsoids.

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.

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

TL;DR

This work delivers two near-linear-time algorithms for computing a -approximation of the minimum-volume bounding box for 3D point sets: a main method with runtime and a simpler variant with . The approach replaces the convex hull with a low-complexity polyhedron and solves the bounding-box problem exactly on that surrogate, ensuring provable accuracy. A more practical grid-search variant with time is also proposed and extensively evaluated, demonstrating favorable trade-offs in practice. The paper also discusses extensions, potential improvements, and open problems, including dynamic MVBB maintenance and extensions to higher dimensions or bounding ellipsoids.

Abstract

We present an efficient -time algorithm for computing a )-approximation of the minimum-volume bounding box of points in . We also present a simpler algorithm (for the same purpose) whose running time is . 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.

Paper Structure

This paper contains 8 sections, 9 theorems, 15 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Notations and Definitions
  3. An Efficient Approximation Algorithm
  4. Approximating the Diameter
  5. Computing an Approximating Box
  6. An Alternative Practical Algorithm
  7. Experimental Results
  8. Conclusion

Key Result

Lemma 3.2

Given a point set $S$ in $\mathbb{R}^d$ (for a fixed $d$), one can compute in $O (n)$ time a pair of points $s,t \in S$, such that $|st| \leq {\cal D} (S) \leq \sqrt{d} |st|$.

Figures (4)

  • Figure 3.1: The convex hull of the projection of $S$ contains a large quadrangle
  • Figure 4.1: An exhaustive grid-based search algorithm for approximating $B_{\hbox{\scriptsize \rm opt}} (S)$
  • Figure 4.2: The construction used in \ref{['theo:aprox:ext']}
  • Figure 5.1: Bounding boxes of three spatial point sets

Theorems & Definitions (10)

  • Definition 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5
  • Lemma 3.6
  • Theorem 3.7
  • Lemma 4.1
  • Lemma 4.2
  • Theorem 4.3