Reporting intersecting pairs of polytopes in two and three dimensions

Pankaj K. Agarwal; Mark de Berg; Sariel Har-Peled; Mark H. Overmars; Micha Sharir; Jan Vahrenhold

doi:10.1007/3-540-44634-6_12

Reporting intersecting pairs of polytopes in two and three dimensions

Pankaj K. Agarwal, Mark de Berg, Sariel Har-Peled, Mark H. Overmars, Micha Sharir, Jan Vahrenhold

School of Computer Science

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

Abstract

Let P = {P1,…, Pm} be a set of m convex polytopes in R^d, for d = 2, 3, with a total of n vertices. We present output-sensitive algorithms for reporting all k pairs of indices (i, j) such that P_iintersects P_j. For the planar case we describe a simple algorithm with running time O(n^4/3log n + k), and an improved randomized algorithm with expected running time O((n logm + k)α(n) log n) (which is faster for small values of k). For d = 3, we present an O(n^8/5+ε+ k)-time algorithm, for any ε > 0. Our algorithms can be modified to count the number of intersecting pairs in O(n^4/3logO(1) n) time for the planar case, and in O(n^8/5+ε) time and R³.

Original language	English
Title of host publication	Algorithms and Data Structures - 7th International Workshop, WADS 2001, Proceedings
Editors	Frank Dehne, Jorg-Rudiger Sack, Roberto Tamassia
Publisher	Springer Verlag
Pages	122-134
Number of pages	13
ISBN (Print)	3540424237, 9783540424239
DOIs	https://doi.org/10.1007/3-540-44634-6_12
State	Published - 2001
Event	7th International Workshop on Algorithms and Data Structures, WADS 2001 - Providence, United States Duration: 8 Aug 2001 → 10 Aug 2001

Publication series

Name	Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume	2125
ISSN (Print)	0302-9743
ISSN (Electronic)	1611-3349

Conference

Conference	7th International Workshop on Algorithms and Data Structures, WADS 2001
Country/Territory	United States
City	Providence
Period	8/08/01 → 10/08/01

Funding

Funders	Funder number
National Science Foundation	CCR-9732787, EIA-9870724, CCR-97-32101, EIA-997287

Access to Document

10.1007/3-540-44634-6_12

Cite this

Agarwal, P. K., de Berg, M., Har-Peled, S., Overmars, M. H., Sharir, M., & Vahrenhold, J. (2001). Reporting intersecting pairs of polytopes in two and three dimensions. In F. Dehne, J.-R. Sack, & R. Tamassia (Eds.), Algorithms and Data Structures - 7th International Workshop, WADS 2001, Proceedings (pp. 122-134). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 2125). Springer Verlag. https://doi.org/10.1007/3-540-44634-6_12

Agarwal, Pankaj K. ; de Berg, Mark ; Har-Peled, Sariel et al. / Reporting intersecting pairs of polytopes in two and three dimensions. Algorithms and Data Structures - 7th International Workshop, WADS 2001, Proceedings. editor / Frank Dehne ; Jorg-Rudiger Sack ; Roberto Tamassia. Springer Verlag, 2001. pp. 122-134 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).

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abstract = "Let P = {P1,…, Pm} be a set of m convex polytopes in Rd, for d = 2, 3, with a total of n vertices. We present output-sensitive algorithms for reporting all k pairs of indices (i, j) such that Piintersects Pj. For the planar case we describe a simple algorithm with running time O(n4/3log n + k), and an improved randomized algorithm with expected running time O((n logm + k)α(n) log n) (which is faster for small values of k). For d = 3, we present an O(n8/5+ε+ k)-time algorithm, for any ε > 0. Our algorithms can be modified to count the number of intersecting pairs in O(n4/3logO(1) n) time for the planar case, and in O(n8/5+ε) time and R3.",

author = "Agarwal, {Pankaj K.} and {de Berg}, Mark and Sariel Har-Peled and Overmars, {Mark H.} and Micha Sharir and Jan Vahrenhold",

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Agarwal, PK, de Berg, M, Har-Peled, S, Overmars, MH, Sharir, M & Vahrenhold, J 2001, Reporting intersecting pairs of polytopes in two and three dimensions. in F Dehne, J-R Sack & R Tamassia (eds), Algorithms and Data Structures - 7th International Workshop, WADS 2001, Proceedings. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 2125, Springer Verlag, pp. 122-134, 7th International Workshop on Algorithms and Data Structures, WADS 2001, Providence, United States, 8/08/01. https://doi.org/10.1007/3-540-44634-6_12

Reporting intersecting pairs of polytopes in two and three dimensions. / Agarwal, Pankaj K.; de Berg, Mark; Har-Peled, Sariel et al.
Algorithms and Data Structures - 7th International Workshop, WADS 2001, Proceedings. ed. / Frank Dehne; Jorg-Rudiger Sack; Roberto Tamassia. Springer Verlag, 2001. p. 122-134 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 2125).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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AU - de Berg, Mark

AU - Har-Peled, Sariel

AU - Overmars, Mark H.

AU - Sharir, Micha

AU - Vahrenhold, Jan

N1 - Publisher Copyright: © Springer-Verlag Berlin Heidelberg 2001.

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N2 - Let P = {P1,…, Pm} be a set of m convex polytopes in Rd, for d = 2, 3, with a total of n vertices. We present output-sensitive algorithms for reporting all k pairs of indices (i, j) such that Piintersects Pj. For the planar case we describe a simple algorithm with running time O(n4/3log n + k), and an improved randomized algorithm with expected running time O((n logm + k)α(n) log n) (which is faster for small values of k). For d = 3, we present an O(n8/5+ε+ k)-time algorithm, for any ε > 0. Our algorithms can be modified to count the number of intersecting pairs in O(n4/3logO(1) n) time for the planar case, and in O(n8/5+ε) time and R3.

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ER -

Agarwal PK, de Berg M, Har-Peled S, Overmars MH, Sharir M, Vahrenhold J. Reporting intersecting pairs of polytopes in two and three dimensions. In Dehne F, Sack JR, Tamassia R, editors, Algorithms and Data Structures - 7th International Workshop, WADS 2001, Proceedings. Springer Verlag. 2001. p. 122-134. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). doi: 10.1007/3-540-44634-6_12

Reporting intersecting pairs of polytopes in two and three dimensions

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