SIGGRAPH '86 Proceedings of the 13th annual conference on Computer graphics and interactive techniques
An object centered hierarchical representation for 3D objects: the prism tree
Computer Vision, Graphics, and Image Processing
Automatic Creation of Object Hierarchies for Ray Tracing
IEEE Computer Graphics and Applications
Fast ray tracing by ray classification
SIGGRAPH '87 Proceedings of the 14th annual conference on Computer graphics and interactive techniques
The R*-tree: an efficient and robust access method for points and rectangles
SIGMOD '90 Proceedings of the 1990 ACM SIGMOD international conference on Management of data
Minimizing the sum of diameters efficiently
Computational Geometry: Theory and Applications
Approximating polyhedra with spheres for time-critical collision detection
ACM Transactions on Graphics (TOG)
OBBTree: a hierarchical structure for rapid interference detection
SIGGRAPH '96 Proceedings of the 23rd annual conference on Computer graphics and interactive techniques
Rectilinear and polygonal p-piercing and p-center problems
Proceedings of the twelfth annual symposium on Computational geometry
Real-time collision detection for motion simulation within complex environments
SIGGRAPH '96 ACM SIGGRAPH 96 Visual Proceedings: The art and interdisciplinary programs of SIGGRAPH '96
Approximation schemes for Euclidean k-medians and related problems
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
Efficient algorithms for geometric optimization
ACM Computing Surveys (CSUR)
Fast collision detection using QuOSPO trees
I3D '99 Proceedings of the 1999 symposium on Interactive 3D graphics
A fast and robust GJK implementation for collision detection of convex objects
Journal of Graphics Tools
Covering a set of points by two axis-parallel boxes
Information Processing Letters
Improved Computational Methods for Ray Tracing
ACM Transactions on Graphics (TOG)
Box-trees and R-trees with near-optimal query time
SCG '01 Proceedings of the seventeenth annual symposium on Computational geometry
Approximate clustering via core-sets
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
Minimal hierarchical collision detection
VRST '02 Proceedings of the ACM symposium on Virtual reality software and technology
R-trees: a dynamic index structure for spatial searching
SIGMOD '84 Proceedings of the 1984 ACM SIGMOD international conference on Management of data
Collision Detection for Interactive Graphics Applications
IEEE Transactions on Visualization and Computer Graphics
Efficient Collision Detection Using Bounding Volume Hierarchies of k-DOPs
IEEE Transactions on Visualization and Computer Graphics
Efficient collision detection of complex deformable models using AABB trees
Journal of Graphics Tools
Rapid Collision Detection by Dynamically Aligned DOP-Trees
VRAIS '98 Proceedings of the Virtual Reality Annual International Symposium
Efficient collision detection for interactive three-dimensional graphics and virtual environments
Efficient collision detection for interactive three-dimensional graphics and virtual environments
Clustering to minimize the sum of cluster diameters
Journal of Computer and System Sciences - STOC 2001
Approximate minimum enclosing balls in high dimensions using core-sets
Journal of Experimental Algorithmics (JEA)
Polynomial time approximation schemes for base station coverage with minimum total radii
Computer Networks and ISDN Systems
Minimum-cost coverage of point sets by disks
Proceedings of the twenty-second annual symposium on Computational geometry
Geometric clustering to minimize the sum of cluster sizes
ESA'05 Proceedings of the 13th annual European conference on Algorithms
Subspace Discovery for Promotion: A Cell Clustering Approach
DS '09 Proceedings of the 12th International Conference on Discovery Science
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We study the problem of covering a set of points or polyhedra in R3 with two axis-aligned boxes in order to minimize a function of the measures of the two boxes, such as the sum or the maximum of their volumes. This 2-box cover problem arises naturally in the construction of bounding volume hierarchies, as well as in shape approximation and clustering. Existing algorithms solve the min-max version of the exact problem in quadratic time. Our results are more general, addressing min-max, min-sum and other versions. Our results give the first approximation schemes for the problem, which run in nearly linear time, as well as some new exact algorithms. We give (1+ε)-approximation algorithms for minimizing the maximum or sum of volumes (or surface areas, diameters, widths, or girths) of the two boxes in R3. We investigate also the problem of computing balanced coverings, in which each box covers at least a fraction of the input objects, and we discuss the application to constructing provably-good bounding volume hierarchies of polyhedra. We also generalize our results to higher dimension.