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An improved algorithm for computing the volume of the union of cubes

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Pankaj K. AgarwalAuthors Info & Claims

SoCG '10: Proceedings of the twenty-sixth annual symposium on Computational geometry

Pages 230 - 239

https://doi.org/10.1145/1810959.1811000

Published: 13 June 2010 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 algorithm that computes the volume of U(C) in time O(n polylog(n)). The previously best known algorithm takes O(n^4/3 log² 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
Barbay JPérez-Lantero PRojas-Ledesma J(2021)Computing the depth distribution of a set of boxesTheoretical Computer Science10.1016/j.tcs.2021.06.007883(69-82)Online publication date: Sep-2021
https://doi.org/10.1016/j.tcs.2021.06.007
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
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Index Terms

An improved algorithm for computing the volume of the union of cubes
1. Theory of computation
  1. Randomness, geometry and discrete structures
    1. Computational geometry

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SoCG '10: Proceedings of the twenty-sixth annual symposium on Computational geometry

June 2010

452 pages

ISBN:9781450300162

DOI:10.1145/1810959

Program Chairs:
David Kirkpatrick
University of British Columbia, Canada
,
Joseph Mitchell
SUNY Stony Brook, 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

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Published: 13 June 2010

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SoCG '10: Symposium on Computational Geometry

June 13 - 16, 2010

Utah, Snowbird, USA

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Overall Acceptance Rate 625 of 1,685 submissions, 37%

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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
Barbay JPérez-Lantero PRojas-Ledesma J(2021)Computing the depth distribution of a set of boxesTheoretical Computer Science10.1016/j.tcs.2021.06.007883(69-82)Online publication date: Sep-2021
https://doi.org/10.1016/j.tcs.2021.06.007
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
Bringmann K(2013)Bringing Order to Special Cases of Klee’s Measure ProblemMathematical Foundations of Computer Science 201310.1007/978-3-642-40313-2_20(207-218)Online publication date: 2013
https://doi.org/10.1007/978-3-642-40313-2_20
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
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
Bringmann KKirkpatrick DMitchell J(2010)Klee's measure problem on fat boxes in time ∂(n(+2)/3)Proceedings of the twenty-sixth annual symposium on Computational geometry10.1145/1810959.1810999(222-229)Online publication date: 13-Jun-2010
https://dl.acm.org/doi/10.1145/1810959.1810999

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Computing the volume of the union of cubes

Klee's measure problem on fat boxes in time ∂(n^(d+2)/3)

Orthogonal range searching on the RAM, revisited

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Computing the volume of the union of cubes

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Orthogonal range searching on the RAM, revisited

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