Algorithms for two-box covering

We study the problem of covering a set of points or polyhedra in 3 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 3 . 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 prov...

• Type: Conference
• Year: 2006
• Where: COMPGEOM
• Authors: Esther M. Arkin, Gill Barequet, Joseph S. B. Mitchell
• Keywords: 2-box Cover Problem, Algorithms, Bounding Volume Hierarchies, COMPGEOM 2006