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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
The paper seeks efficient approximation algorithms for the minimum-volume bounding box enclosing a point set in R^3, a problem used in hierarchical point containers. It constructs approximating boxes through grid-based geometric procedures and presents two algorithms with runtimes O(n + 1/ε^4.5) and O(n log n + n/ε^3), alongside experiments with variants of the simpler algorithm.
Problem
The problem is to enclose n points in R^3 with a box whose volume approximates the minimum-volume bounding box by factor 1 + ε.
Method
The algorithms approximate the point set on a grid, form a low-complexity convex-hull polyhedron or search favorable grid-point directions, and compute constrained minimum-volume boxes.
Results
The paper presents two nearly linear-in-n approximation algorithms running in O(n + 1/ε^4.5) and O(n log n + n/ε^3) time.
Takeaways & Limitations
Implemented variants of the simpler algorithm produce good approximating boxes quickly and perform considerably better than uniform-grid heuristics on long and skinny point sets.
Takeaways & Limitations
The more laborious alternative algorithm may be too slow in practice because its hidden big-Oh constant can be large.
Abstract
from arXiv · showhide
$\renewcommand{\Re}{\mathbb{R}}$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.
1. Introduction
The paper addresses efficiently approximating the minimum-volume arbitrarily oriented bounding box for points in R^3, motivated by applications requiring nearly optimal enclosing boxes. It improves substantially on the known exact algorithm and reports two nearly linear-time approximation algorithms.
- Problem: Given n points in R^3 and 0 < ε ≤ 1, the task is to find an enclosing box within factor 1 + ε of minimum volume.
- Motivation: Three-dimensional enclosing boxes support hierarchical point-set partitioning applications including rendering, collision detection, and range-search queries.
- Prior Work: Existing heuristics generally compute bounding boxes in O(n) time and space but provide no guaranteed approximation factor.
- Prior Work: O(n^3) time is required by the known exact algorithm for arbitrarily oriented minimum-volume bounding boxes.
- Contributions: The paper presents the first two (1 + ε)-approximation algorithms for this problem, both nearly linear in n.
- Contributions: The two algorithms run in O(n + 1/ε^4.5) and O(n log n + n/ε^3) time.
2. Notations and Definitions
This section establishes notation for point sets, convex hulls, boxes, rectangles, volumes, constrained optima, and grids used throughout the algorithms.
- Point Sets and Hulls: Q denotes a two-dimensional point set, S a three-dimensional point set, and CH(Q) or CH(S) their corresponding convex hulls.
- Boxes and Rectangles: R and B denote rectangles and boxes, while B(S) denotes any bounding box of S.
- Boxes and Rectangles: B = (b1, b2, b3) represents a box through three orthogonal edge vectors, and Area(R) and Vol(B) denote rectangle area and box volume.
- Optimal Bounding Objects: Ropt(Q) and Bopt(S) denote the minimum-area bounding rectangle and minimum-volume bounding box, respectively.
- Constrained Optima: Bopt(S, V) denotes the minimum-volume bounding box of S whose directions contain the orthogonal-vector set V.
- Grid Notation: Grid(B) is the set of all integer linear combinations of the box-edge vectors b1, b2, and b3.
3. An Efficient Approximation Algorithm
The paper develops approximation algorithms for the minimum-volume bounding box in three dimensions, combining diameter estimates, constant-factor boxes, grid constructions, and convex-polyhedral approximations. The main algorithm achieves a (1 + ε)-approximation in O(n + 1/ε^4.5) time.
- Approximating the diameter: O(n + 1/ε^2(d−1)) time computes two points whose distance is at least (1 − ε)D(S) for fixed-dimensional point sets.In three dimensions, a more specialized method computes a (1 − ε)-diameter approximation in O(n + (1/ε^3/2) log(1/ε)) time.
- Constant-factor initialization: A linear-time procedure computes a bounding box whose volume is between Vol(Bopt(S)) and 6Vol(Bopt(S)).This constant-factor box provides an initial approximation for subsequent constructions.
- Polyhedral approximation: The main algorithm expands CH(S) to a low-complexity convex polyhedron P and computes Bopt(P) exactly.The polyhedron satisfies CH(S) ⊆ P and is constructed from grid vertices; its complexity is O(1/ε^3/2), leading to O(1/ε^4.5) exhaustive search time.
- Constant-factor initialization: A stronger linear-time box B*(S) has volume at most 15Vol(Bopt(S)) and contains, after translation, a scaled copy of itself inside CH(S).Specifically, a translated copy of (1/107)B*(S) lies within the convex hull.
- Degenerate inputs: The method remains valid for planar inputs, producing a degenerate bounding box of volume 0.For a planar set, the intermediate minimum-area bounding rectangle degenerates to a segment.
- Main result: O(n + 1/ε^4.5) time yields a bounding box with volume at most (1 + ε)Vol(Bopt(S)).This is the stated main theorem for point sets S in R3 and 0 < ε ≤ 1.
4. An Alternative Practical Algorithm
The alternative algorithm searches grid-induced directions near the origin to find a direction whose constrained minimum-volume box approximates the optimum. Its guarantee is O(n log n + n/ε^3), although a potentially large hidden constant may limit practical speed.
- Correctness: A favorable grid direction exists such that its constrained box is a (1 + ε)-approximation of the unconstrained optimum.The proof identifies a grid point v* near the origin whose induced direction provides the required approximation.
- Algorithm: The algorithm computes a constant-approximation box, builds a grid, and evaluates constrained minimum-volume boxes for nearby grid directions.The grid is defined from the constant-approximation box, while an auxiliary grid is used only in the correctness proof.
- Correctness: The proof uses an auxiliary grid and projection geometry to show that the optimal box lies inside a prism and a constructed comparison box.The projection preserves or reduces rectangle area, supporting the volume comparison.
- Guarantee: O(n log n + n/ε^3) time suffices to compute a bounding box B(S) with volume at most (1 + ε) times optimal.The theorem provides the algorithm’s approximation guarantee and running time.
- Practicality: The algorithm may be too slow in practice because the constant hidden in its asymptotic running time can be large.This motivates a heuristic that tests directions induced by grid points close to the origin.
5. Experimental Results
The experiments compare exact and approximate bounding-box heuristics on three spatial point sets, measuring box volumes and software running times. Faster variants can perform reasonably well, with B∗(S)-G(5) plus improvement especially effective for long, skinny point sets.
- Implementation: The implementation computes exact planar minimum-area rectangles and exact three-dimensional minimum-volume boxes with one prescribed direction.These tools support several approximation heuristics implemented in plain C.
- Heuristic behavior: Local projection-and-rectangle improvement can converge to a local rather than global minimum-volume bound.The procedure projects onto a plane perpendicular to a box direction, computes a minimum-area rectangle, and uses it to improve the box.
- Point-set examples: Figure 5.1 illustrates B∗(S), improved B∗(S)-G(20), and boxes aligned with hull diagonals for rotated, arbitrary, and spherical point sets.The examples include four points, 48 arbitrary points, and 100 points uniformly sampled on the unit sphere.
- Performance comparison: Table 5.1 compares bounding-box volumes and software running times across several approximation heuristics.It includes all-pairs directions, grid-based variants, and the number of Bopt(S, {v}) calls.
- Observed performance: B∗(S)-G(5) with the improvement step produces good approximating boxes quickly and outperforms the uniform-grid heuristic on long and skinny point sets.The paper contrasts this faster variant with more laborious heuristics that require much time to run.
6. Conclusion
The paper presents efficient and simpler algorithms for approximating the minimum-volume bounding box of points in R3, alongside implementations and experiments. It also identifies a faster variant using Dudley’s method and poses dynamic and minimum-volume bounding-ellipsoid extensions as open problems.
- Conclusion: The paper presents an efficient approximation algorithm and a simpler implemented algorithm for the minimum-volume bounding box of a point set in R3.The simpler algorithm was experimentally evaluated on numerous three-dimensional point sets.
- Conclusion: Dudley’s method reduces the first algorithm’s O(1/ε^4.5) term to O(1/ε^3) while adding an O(n log n) term.The resulting running time is O(n log n + 1/ε^3).
- Conclusion: The implementation of the algorithm is available online.The paper concludes by identifying two open problems concerning dynamic point sets and minimum-volume bounding ellipsoids.