## Efficiently Approximating the Minimum-Volume Bounding Box of a Point Set in Three Dimensions (2001)

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Venue: | In Proc. 10th ACM-SIAM Sympos. Discrete Algorithms |

Citations: | 79 - 12 self |

### BibTeX

@INPROCEEDINGS{Barequet01efficientlyapproximating,

author = {Gill Barequet and Sariel Har-peled},

title = {Efficiently Approximating the Minimum-Volume Bounding Box of a Point Set in Three Dimensions},

booktitle = {In Proc. 10th ACM-SIAM Sympos. Discrete Algorithms},

year = {2001},

pages = {38--91}

}

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### Abstract

We present an efficient O(n + 1/ε^4.5)-time algorithm for computing a (1 + 1/ε)-approximation of the minimum-volume bounding box of n points in R³. We also present a simpler algorithm (for the same purpose) whose running time is O(n log n+n/ε³). 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 [Har00].

### Citations

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Citation Context ...wn R-tree, the packed R-tree [RL85], the R + -tree [SRF87], the R ∗ -tree [BKSS90], etc. [HKM95] use a minimum-volume AABB trimmed in a fixed number of directions for speeding up collision detection. =-=[GLM96]-=- implement in their RAPID system the OBB-tree (a tree of arbitrarily-oriented bounding boxes), where each box which encloses a set of polygons is aligned with the principal components of the distribut... |

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Citation Context ...compute the axis-aligned bounding box (AABB) of the point set. Two-dimensional variants of this heuristic include the well-known R-tree, the packed R-tree [RL85], the R + -tree [SRF87], the R ∗ -tree =-=[BKSS90]-=-, etc. [HKM95] use a minimum-volume AABB trimmed in a fixed number of directions for speeding up collision detection. [GLM96] implement in their RAPID system the OBB-tree (a tree of arbitrarily-orient... |

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Citation Context ...and v. The volume of the computed box is at most d! times the volume of the optimal (minimum-volume) bounding box of S. 3 Moreover, the side lengths of the bounding box are found in decreasing order. =-=[FL95]-=- use a similar method for visualizing a 3 Here we can trade time for approximation quality. By investing less time we can compute a (1/ √ 2)- (resp., (1/ √ 3)-) approximation of the diameter in two (r... |

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Citation Context ...Lemma 3.3 can be improved to O We can do even better in three dimensions if we are willing to sacrifice simplicity. In this case we compute the exact diameter of CH(SG) in O((1/ε3/2 ) log (1/ε)) time =-=[CS89]-=-. Overall, we compute a (1 − ε)-approximation of the diameter of S ∈ R3 in O(n + (1/ε3/2 ) log (1/ε)) time. 3.2 Computing an Approximating Box � (1/ε) 2d(d−1) d+1 Let Q be a set of n points in R 2 . C... |

304 | The R+ tree: A dynamic index for multi-dimensional objects
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Citation Context ...istic was naturally to compute the axis-aligned bounding box (AABB) of the point set. Two-dimensional variants of this heuristic include the well-known R-tree, the packed R-tree [RL85], the R + -tree =-=[SRF87]-=-, the R ∗ -tree [BKSS90], etc. [HKM95] use a minimum-volume AABB trimmed in a fixed number of directions for speeding up collision detection. [GLM96] implement in their RAPID system the OBB-tree (a tr... |

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Citation Context ...her method (e.g., by computing the exact minimum-area bounding rectangle of the projection of the points into a plane orthogonal to the first chosen direction). Other generic shapes, such as a sphere =-=[Hub95]-=-, a cone [Sam89], or a prism [FP87, BCG + 96] were also used for maintaining a hierarchical data structure of point containers. Most of these heuristics require O(n) time and space for computing the b... |

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Citation Context ...nts. The simplest heuristic was naturally to compute the axis-aligned bounding box (AABB) of the point set. Two-dimensional variants of this heuristic include the well-known R-tree, the packed R-tree =-=[RL85]-=-, the R + -tree [SRF87], the R ∗ -tree [BKSS90], etc. [HKM95] use a minimum-volume AABB trimmed in a fixed number of directions for speeding up collision detection. [GLM96] implement in their RAPID sy... |

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Citation Context ... the diameter of S ∈ R3 in O(n + (1/ε3/2 ) log (1/ε)) time. 3.2 Computing an Approximating Box � (1/ε) 2d(d−1) d+1 Let Q be a set of n points in R 2 . Computing Ropt(Q) can be done in O(n log n) time =-=[Tou83]-=-. (Hence, given a set S of n points and a direction v in R 3 , one can compute Bopt(S, {v}) in O(n log n) time.) The bottleneck of the cited algorithm is the computation of CH(Q); when the latter is g... |

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Citation Context ...nd63] showed that h = O(1/ε (d−1)d/(d+1) ). For d ≤ 3 we compute SCH(SG) by computing the entire convex-hull of SG in O(|SG| log |SG|) time. For higher dimensions we use an output-sensitive algorithm =-=[Cha96]-=-. Let m denote the number of points in SG. Clearly, m = O(1/ε (d−1) ). The time required for computing SG is � O m log d+2 h + (mh) 1− 1 ⌊d/2⌋+1 log O(1) � m � = O m 2d � d+1 = O � �1 ε � 2d(d−1) � d+... |

69 | Evaluation of collision detection methods for virtual reality fly-throughs
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Citation Context ...s-aligned bounding box (AABB) of the point set. Two-dimensional variants of this heuristic include the well-known R-tree, the packed R-tree [RL85], the R + -tree [SRF87], the R ∗ -tree [BKSS90], etc. =-=[HKM95]-=- use a minimum-volume AABB trimmed in a fixed number of directions for speeding up collision detection. [GLM96] implement in their RAPID system the OBB-tree (a tree of arbitrarily-oriented bounding bo... |

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38 |
Spatial Data Structures: Quadtree, Octrees and Other Hierarchical Methods
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Citation Context ..., by computing the exact minimum-area bounding rectangle of the projection of the points into a plane orthogonal to the first chosen direction). Other generic shapes, such as a sphere [Hub95], a cone =-=[Sam89]-=-, or a prism [FP87, BCG + 96] were also used for maintaining a hierarchical data structure of point containers. Most of these heuristics require O(n) time and space for computing the bounding box (or ... |

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Inner and outer j-radii of convex bodies in finite-dimensional normed spaces
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Citation Context ..., Width(P ) ≥ Vol(P )/ � 3π 4 � ≥ 4 45π , where Width(P ) is the minimum distance between two parallel planes supporting P . Let r(P ) be the radius of the largest ball K inscribed in P . It is known =-=[GK92]-=- that r(P ) ≥ Width(P )/(2 √ 3). This implies r(P ) ≥ 2/(45 √ 3π). Finally, K inscribes an axis-parallel cube C ′ whose side is of length (2/ √ 3)r(P ) ≥ 1/107, and C ′ ⊆ K ⊆ P , as asserted. Let A an... |

30 | Finding minimal enclosing boxes - O’Rourke - 1985 |

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Citation Context ...in any dimension; see [EK89].) Actually, we can find any arbitrarily-good approximation of the diameter. Since we were not able to find any reference with a proof of the following folklore lemma (see =-=[Har99]-=- for a similar result), we include an easy proof of it here. Lemma 3.3 Given a set S of n points in R d (for a fixed d) and ε > 0, one can compute in O(n + 1/ε 2(d−1) ) time a pair of points s, t ∈ S ... |

14 | Eciently approximating the minimum-volume bounding box of a point set in three dimensions - Barequet, Har-Peled - 2001 |

11 | BOXTREE: A hierarchical representation of surfaces - Barequet, Chazelle, et al. - 1996 |

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7 | Covering a set of points by two axis-parallel boxes
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Citation Context ...ese heuristics require O(n) time and space for computing the bounding box (or another shape) but do not provide a guaranteed value (approximation factor of the optimum) of the output. An algorithm of =-=[BS97]-=- solves a similar problem, in which the n points are to be contained in two axis-aligned boxes, and the goal is to minimize the volume (or any other monotone measure) of the larger box. Their algorith... |

6 | Approximating the diameter of a set of points in the Euclidean space
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Citation Context ... approximate in linear time the diameter of a point set in three dimensions by a factor of 1/ √ 3. (In fact, one can find in linear time a (1/ √ 3)-approximation of the diameter in any dimension; see =-=[EK89]-=-.) Actually, we can find any arbitrarily-good approximation of the diameter. Since we were not able to find any reference with a proof of the following folklore lemma (see [Har99] for a similar result... |

4 | Inner and outer j-radii of convex bodies in normed spaces - Gritzmann, Klee - 1992 |

2 |
Source code of program for computing and approximating the diameter of a point-set in 3d, 2000. http://www.uiuc.edu/ ~ sariel/papers/00/diameter /diam prog.html. [LGM] Large geometric models archive. Georgia Tech. http://www.cc.gatech.edu/projects/large m
- Har-Peled
(Show Context)
Citation Context ... O(n log n+n/ε 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 =-=[Har00]-=-. 1 Introduction In this paper we give efficient algorithms for solving the following problem: Given a set S of n points in R 3 and a parameter 0 < ε ≤ 1, find a box that encloses S and approximates t... |

1 | An object-centered hierarchical representation for 3-d objects: The prism-tree - Faugeras, Ponce - 1987 |

1 | BOXTREE: A hierarchical representation for surfaces - Mitchell, TaI - 1996 |

1 | The r--tree: An efficient and robust [sm8Ql access method for points aud rectangles - Beckmann, Kriegel, et al. - 1990 |

1 | Output-Sensitive Results on Convex Hulls, Extreme Points, and Related Prob lems. Discrete - Gwm - 1996 |

1 | Metric entropy of some classes of sets with differentiable boundaries - Dud - 1974 |

1 | Approximate Shortest paths and Geodesic diameters on convex polytopes in three dimensions - Gwm - 1995 |

1 | Direct spatial search on pictorial databases using packed rtrees - Sci - 1985 |

1 | The R+-tree: A dynamic index for multidimensional o bjects - Sellis, Roussopoulos, et al. - 1987 |