Article

Computing the volume of the union of cubes

Authors:

Pankaj K. Agarwal,

Haim Kaplan,

Micha SharirAuthors Info & Claims

SCG '07: Proceedings of the twenty-third annual symposium on Computational geometry

Pages 294 - 301

https://doi.org/10.1145/1247069.1247121

Published: 06 June 2007 Publication History

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Abstract

Let C be a set of n axis-aligned cubes in R³, and let U(C) denote the union of C. We present an algorithmthat can compute the volume of U(C) in time O(n^4/3 log n). The previously best known algorithm, by Overmars and Yap, computes the volume of the union ofany n axis-aligned boxes in R³ in O(n^3/2log n) time.

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Cited By

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Kunnemann M(2022)A tight (non-combinatorial) conditional lower bound for Klee’s Measure Problem in 3D2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS54457.2022.00059(555-566)Online publication date: Oct-2022
https://doi.org/10.1109/FOCS54457.2022.00059
Yıldız HSuri S(2015)Computing Klee's Measure of Grounded BoxesAlgorithmica10.1007/s00453-013-9797-971:2(307-329)Online publication date: 1-Feb-2015
https://dl.acm.org/doi/10.1007/s00453-013-9797-9
Chan T(2013)Klee's Measure Problem Made EasyProceedings of the 2013 IEEE 54th Annual Symposium on Foundations of Computer Science10.1109/FOCS.2013.51(410-419)Online publication date: 26-Oct-2013
https://dl.acm.org/doi/10.1109/FOCS.2013.51
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Index Terms

Computing the volume of the union of cubes
1. Theory of computation
  1. Design and analysis of algorithms

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Published In

SCG '07: Proceedings of the twenty-third annual symposium on Computational geometry

June 2007

404 pages

ISBN:9781595937056

DOI:10.1145/1247069

Program Chair:
Jeff Erickson
University of Illinois, Urbana-Champaign, USA

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 06 June 2007

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Cited By

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Kunnemann M(2022)A tight (non-combinatorial) conditional lower bound for Klee’s Measure Problem in 3D2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS54457.2022.00059(555-566)Online publication date: Oct-2022
https://doi.org/10.1109/FOCS54457.2022.00059
Yıldız HSuri S(2015)Computing Klee's Measure of Grounded BoxesAlgorithmica10.1007/s00453-013-9797-971:2(307-329)Online publication date: 1-Feb-2015
https://dl.acm.org/doi/10.1007/s00453-013-9797-9
Chan T(2013)Klee's Measure Problem Made EasyProceedings of the 2013 IEEE 54th Annual Symposium on Foundations of Computer Science10.1109/FOCS.2013.51(410-419)Online publication date: 26-Oct-2013
https://dl.acm.org/doi/10.1109/FOCS.2013.51
Yildiz HSuri SDey TWhitesides S(2012)On Klee's measure problem for grounded boxesProceedings of the twenty-eighth annual symposium on Computational geometry10.1145/2261250.2261267(111-120)Online publication date: 17-Jun-2012
https://dl.acm.org/doi/10.1145/2261250.2261267
Chen DMisioek E(2012)Computing feasible toolpaths for 5-axis machinesTheoretical Computer Science10.1016/j.tcs.2011.11.032447(13-25)Online publication date: 1-Aug-2012
https://dl.acm.org/doi/10.1016/j.tcs.2011.11.032
Bringmann K(2012)An improved algorithm for Klee's measure problem on fat boxesComputational Geometry: Theory and Applications10.1016/j.comgeo.2011.12.00145:5-6(225-233)Online publication date: 1-Jul-2012
https://dl.acm.org/doi/10.1016/j.comgeo.2011.12.001
Yíldíz HFoschini LHershberger JSuri S(2011)The union of probabilistic boxesProceedings of the 19th European conference on Algorithms10.5555/2040572.2040637(591-602)Online publication date: 5-Sep-2011
https://dl.acm.org/doi/10.5555/2040572.2040637
Yıldız HFoschini LHershberger JSuri S(2011)The Union of Probabilistic Boxes: Maintaining the VolumeAlgorithms – ESA 201110.1007/978-3-642-23719-5_50(591-602)Online publication date: 2011
https://doi.org/10.1007/978-3-642-23719-5_50
Chen DMisiołek E(2010)Computing toolpaths for 5-axis NC machinesProceedings of the 4th international conference on Combinatorial optimization and applications - Volume Part I10.5555/1940390.1940413(270-284)Online publication date: 18-Dec-2010
https://dl.acm.org/doi/10.5555/1940390.1940413
Agarwal PKirkpatrick DMitchell J(2010)An improved algorithm for computing the volume of the union of cubesProceedings of the twenty-sixth annual symposium on Computational geometry10.1145/1810959.1811000(230-239)Online publication date: 13-Jun-2010
https://dl.acm.org/doi/10.1145/1810959.1811000
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