An improved algorithm for Kleeʼs measure problem on fat boxes

The measure problem of Klee asks for the volume of the union of n axis-parallel boxes in a fixed dimension d. We give an O(n(d+2)/3) time algorithm for the special case of all boxes being cubes or, more generally, fat boxes. Previously, the fastest run-time was nd/22O(log⁎n), achieved by the general...

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Bibliographic Details
Published inComputational geometry : theory and applications Vol. 45; no. 5-6; pp. 225 - 233
Main Author Bringmann, Karl
Format Journal Article
LanguageEnglish
Published Elsevier B.V 01.07.2012
Subjects
Algorithms
Computational geometry
Cubes
Dimensional measurements
Geometric data structures
Run time (computers)
Union of cubes
Unions
Union of cubes
Geometric data structures
Online AccessGet full text
ISSN0925-7721
DOI10.1016/j.comgeo.2011.12.001

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Summary:The measure problem of Klee asks for the volume of the union of n axis-parallel boxes in a fixed dimension d. We give an O(n(d+2)/3) time algorithm for the special case of all boxes being cubes or, more generally, fat boxes. Previously, the fastest run-time was nd/22O(log⁎n), achieved by the general case algorithm of Chan [SoCG 2008]. For the general problem our run-time would imply a breakthrough for the k-clique problem.
Bibliography:ObjectType-Article-2
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ISSN:0925-7721
DOI:10.1016/j.comgeo.2011.12.001