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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| Published in | Computational geometry : theory and applications Vol. 45; no. 5-6; pp. 225 - 233 |
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| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
Elsevier B.V
01.07.2012
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| Online Access | Get full text |
| ISSN | 0925-7721 |
| DOI | 10.1016/j.comgeo.2011.12.001 |
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| Abstract | 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. |
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| AbstractList | 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. 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 n d / 2 2 O ( 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. |
| Author | Bringmann, Karl |
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| Cites_doi | 10.1145/1810959.1811000 10.1145/359545.359553 10.1137/S0097539702404389 10.1145/1247069.1247121 10.2307/2318871 10.1007/978-3-540-92182-0_40 10.1016/j.comgeo.2009.01.007 10.1007/s00454-003-0729-3 10.1137/0220065 10.1145/220279.220288 |
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| References | K. Bringmann, T. Friedrich, Approximating the volume of unions and intersections of high-dimensional geometric objects, in: Proc. 19th International Symposium on Algorithms and Computation (ISAACʼ08), in: LNCS vol. 5369, 2008, pp. 436–447. Chan (br0060) 2003; 32 P.K. Agarwal, H. Kaplan, M. Sharir, Computing the volume of the union of cubes, in: Proc. 23rd Annual Symposium on Computational Geometry (SoCGʼ07), 2007, pp. 294–301. H. Kaplan, N. Rubin, M. Sharir, E. Verbin, Counting colors in boxes, in: Proc. 18th Annual ACM-SIAM Symposium on Discrete Algorithms (SODAʼ07), 2007, pp. 785–794. Nešetřil, Poljak (br0130) 1985; 26 Fredman, Weide (br0100) 1978; 21 Overmars, Yap (br0140) 1991; 20 S. Suzuki, T. Ibaraki, An average running time analysis of a backtracking algorithm to calculate the measure of the union of hyperrectangles in J.-D. Boissonnat, M. Sharir, B. Tagansky, M. Yvinec, Voronoi diagrams in higher dimensions under certain polyhedral distance functions, in: Proc. 11th Annual Symposium on Computational Geometry (SoCGʼ95), 1995, pp. 79–88. Dumitrescu, Mitchell, Sharir (br0080) 2004; 31 dimensions, in: Proc. 16th Canadian Conference on Computational Geometry (CCCGʼ04), 2004, pp. 196–199. P.K. Agarwal, An improved algorithm for computing the volume of the union of cubes, in: Proc. 26th Annual Symposium on Computational Geometry (SoCGʼ10), 2010, pp. 230–239. J.L. Bentley, Algorithms for Kleeʼs rectangle problems, Department of Computer Science, Carnegie Mellon University, 1977, unpublished notes. Erickson (br0090) 1998 Klee (br0120) 1977; 84 Chan (br0070) 2010; 43 Chan (10.1016/j.comgeo.2011.12.001_br0060) 2003; 32 Chan (10.1016/j.comgeo.2011.12.001_br0070) 2010; 43 10.1016/j.comgeo.2011.12.001_br0010 10.1016/j.comgeo.2011.12.001_br0110 10.1016/j.comgeo.2011.12.001_br0050 Nešetřil (10.1016/j.comgeo.2011.12.001_br0130) 1985; 26 10.1016/j.comgeo.2011.12.001_br0040 10.1016/j.comgeo.2011.12.001_br0150 10.1016/j.comgeo.2011.12.001_br0030 10.1016/j.comgeo.2011.12.001_br0020 Erickson (10.1016/j.comgeo.2011.12.001_br0090) Dumitrescu (10.1016/j.comgeo.2011.12.001_br0080) 2004; 31 Klee (10.1016/j.comgeo.2011.12.001_br0120) 1977; 84 Overmars (10.1016/j.comgeo.2011.12.001_br0140) 1991; 20 Fredman (10.1016/j.comgeo.2011.12.001_br0100) 1978; 21 |
| References_xml | – volume: 21 start-page: 540 year: 1978 end-page: 544 ident: br0100 article-title: On the complexity of computing the measure of publication-title: Commun. ACM – reference: P.K. Agarwal, An improved algorithm for computing the volume of the union of cubes, in: Proc. 26th Annual Symposium on Computational Geometry (SoCGʼ10), 2010, pp. 230–239. – volume: 84 start-page: 284 year: 1977 end-page: 285 ident: br0120 article-title: Can the measure of publication-title: American Mathematical Monthly – reference: K. Bringmann, T. Friedrich, Approximating the volume of unions and intersections of high-dimensional geometric objects, in: Proc. 19th International Symposium on Algorithms and Computation (ISAACʼ08), in: LNCS vol. 5369, 2008, pp. 436–447. – year: 1998 ident: br0090 article-title: Kleeʼs measure problem – reference: P.K. Agarwal, H. Kaplan, M. Sharir, Computing the volume of the union of cubes, in: Proc. 23rd Annual Symposium on Computational Geometry (SoCGʼ07), 2007, pp. 294–301. – volume: 20 start-page: 1034 year: 1991 end-page: 1045 ident: br0140 article-title: New upper bounds in Kleeʼs measure problem publication-title: SIAM J. Comput. – reference: J.-D. Boissonnat, M. Sharir, B. Tagansky, M. Yvinec, Voronoi diagrams in higher dimensions under certain polyhedral distance functions, in: Proc. 11th Annual Symposium on Computational Geometry (SoCGʼ95), 1995, pp. 79–88. – volume: 31 start-page: 207 year: 2004 end-page: 227 ident: br0080 article-title: Binary space partitions for axis-parallel segments, rectangles, and hyperrectangles publication-title: Discrete & Computational Geometry – volume: 32 start-page: 700 year: 2003 end-page: 716 ident: br0060 article-title: Semi-online maintenance of geometric optima and measures publication-title: SIAM J. Comput. – reference: H. Kaplan, N. Rubin, M. Sharir, E. Verbin, Counting colors in boxes, in: Proc. 18th Annual ACM-SIAM Symposium on Discrete Algorithms (SODAʼ07), 2007, pp. 785–794. – volume: 26 start-page: 415 year: 1985 end-page: 419 ident: br0130 article-title: On the complexity of the subgraph problem publication-title: Commentationes Mathematicae Universitatis Carolinae – volume: 43 start-page: 243 year: 2010 end-page: 250 ident: br0070 article-title: A (slightly) faster algorithm for Kleeʼs measure problem publication-title: Computational Geometry: Theory and Applications – reference: J.L. Bentley, Algorithms for Kleeʼs rectangle problems, Department of Computer Science, Carnegie Mellon University, 1977, unpublished notes. – reference: dimensions, in: Proc. 16th Canadian Conference on Computational Geometry (CCCGʼ04), 2004, pp. 196–199. – reference: S. Suzuki, T. Ibaraki, An average running time analysis of a backtracking algorithm to calculate the measure of the union of hyperrectangles in – ident: 10.1016/j.comgeo.2011.12.001_br0010 doi: 10.1145/1810959.1811000 – ident: 10.1016/j.comgeo.2011.12.001_br0110 – ident: 10.1016/j.comgeo.2011.12.001_br0090 – volume: 21 start-page: 540 year: 1978 ident: 10.1016/j.comgeo.2011.12.001_br0100 article-title: On the complexity of computing the measure of ⋃[ai,bi] publication-title: Commun. ACM doi: 10.1145/359545.359553 – volume: 32 start-page: 700 year: 2003 ident: 10.1016/j.comgeo.2011.12.001_br0060 article-title: Semi-online maintenance of geometric optima and measures publication-title: SIAM J. Comput. doi: 10.1137/S0097539702404389 – ident: 10.1016/j.comgeo.2011.12.001_br0020 doi: 10.1145/1247069.1247121 – volume: 26 start-page: 415 year: 1985 ident: 10.1016/j.comgeo.2011.12.001_br0130 article-title: On the complexity of the subgraph problem publication-title: Commentationes Mathematicae Universitatis Carolinae – volume: 84 start-page: 284 year: 1977 ident: 10.1016/j.comgeo.2011.12.001_br0120 article-title: Can the measure of ⋃[ai,bi] be computed in less than O(nlogn) steps? publication-title: American Mathematical Monthly doi: 10.2307/2318871 – ident: 10.1016/j.comgeo.2011.12.001_br0030 – ident: 10.1016/j.comgeo.2011.12.001_br0050 doi: 10.1007/978-3-540-92182-0_40 – volume: 43 start-page: 243 year: 2010 ident: 10.1016/j.comgeo.2011.12.001_br0070 article-title: A (slightly) faster algorithm for Kleeʼs measure problem publication-title: Computational Geometry: Theory and Applications doi: 10.1016/j.comgeo.2009.01.007 – ident: 10.1016/j.comgeo.2011.12.001_br0150 – volume: 31 start-page: 207 year: 2004 ident: 10.1016/j.comgeo.2011.12.001_br0080 article-title: Binary space partitions for axis-parallel segments, rectangles, and hyperrectangles publication-title: Discrete & Computational Geometry doi: 10.1007/s00454-003-0729-3 – volume: 20 start-page: 1034 year: 1991 ident: 10.1016/j.comgeo.2011.12.001_br0140 article-title: New upper bounds in Kleeʼs measure problem publication-title: SIAM J. Comput. doi: 10.1137/0220065 – ident: 10.1016/j.comgeo.2011.12.001_br0040 doi: 10.1145/220279.220288 |
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| SubjectTerms | Algorithms Computational geometry Cubes Dimensional measurements Geometric data structures Run time (computers) Union of cubes Unions |
| Title | An improved algorithm for Kleeʼs measure problem on fat boxes |
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