On Neighbors in Geometric Permutations

Micha Sharir; Shakhar Smorodinsky

doi:10.1007/3-540-45471-3_14

On Neighbors in Geometric Permutations

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

Abstract

We introduce a new notion of ‘neighbors’ in geometric permutations. We conjecture that the maximum number of neighbors in a set S of n pairwise disjoint convex bodies in ℝ^d is O(n), and we prove this conjecture for d = 2. We show that if the set of pairs of neighbors in a set S is of size N, then S admits at most O(N^d−1) geometric permutations. Hence we obtain an alternative proof of a linear upper bound on the number of geometric permutations for any finite family of pairwise disjoint convex bodies in the plane.

Original language	English
Title of host publication	Algorithm Theory, SWAT 2002
Subtitle of host publication	8th Scandinavian Workshop on Algorithm Theory, Proceedings
Editors	Martti Penttonen, Erik Meineche Schmidt
Publisher	Springer Verlag
Pages	131-139
Number of pages	9
ISBN (Print)	9783540438663
DOIs	https://doi.org/10.1007/3-540-45471-3_14
State	Published - 2002
Externally published	Yes
Event	8th Scandinavian Workshop on Algorithm Theory, SWAT 2002 - Turku, Finland Duration: 3 Jul 2002 → 5 Jul 2002

Publication series

Name	Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume	2368
ISSN (Print)	0302-9743
ISSN (Electronic)	1611-3349

Conference

Conference	8th Scandinavian Workshop on Algorithm Theory, SWAT 2002
Country/Territory	Finland
City	Turku
Period	3/07/02 → 5/07/02

ASJC Scopus subject areas

Theoretical Computer Science
General Computer Science

Access to Document

10.1007/3-540-45471-3_14

Cite this

Sharir, M., & Smorodinsky, S. (2002). On Neighbors in Geometric Permutations. In M. Penttonen, & E. M. Schmidt (Eds.), Algorithm Theory, SWAT 2002: 8th Scandinavian Workshop on Algorithm Theory, Proceedings (pp. 131-139). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 2368). Springer Verlag. https://doi.org/10.1007/3-540-45471-3_14

Sharir, Micha ; Smorodinsky, Shakhar. / On Neighbors in Geometric Permutations. Algorithm Theory, SWAT 2002: 8th Scandinavian Workshop on Algorithm Theory, Proceedings. editor / Martti Penttonen ; Erik Meineche Schmidt. Springer Verlag, 2002. pp. 131-139 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).

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title = "On Neighbors in Geometric Permutations",

abstract = "We introduce a new notion of {\textquoteleft}neighbors{\textquoteright} in geometric permutations. We conjecture that the maximum number of neighbors in a set S of n pairwise disjoint convex bodies in ℝd is O(n), and we prove this conjecture for d = 2. We show that if the set of pairs of neighbors in a set S is of size N, then S admits at most O(Nd−1) geometric permutations. Hence we obtain an alternative proof of a linear upper bound on the number of geometric permutations for any finite family of pairwise disjoint convex bodies in the plane.",

author = "Micha Sharir and Shakhar Smorodinsky",

note = "Funding Information: Work on this paper has been supported by NSF Grants CCR-97-32101 and CCR-00-98246, by a grant from the US–Israeli Binational Science Foundation, by a grant from the Israeli Academy of Sciences for a Center of Excellence in Geometric Computing at Tel Aviv University, and by the Hermann Minkowski—MINERVA Center for Geometry at Tel Aviv University. Publisher Copyright: {\textcopyright} Springer-Verlag Berlin Heidelberg 2002.; 8th Scandinavian Workshop on Algorithm Theory, SWAT 2002 ; Conference date: 03-07-2002 Through 05-07-2002",

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doi = "10.1007/3-540-45471-3_14",

language = "English",

isbn = "9783540438663",

series = "Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)",

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Sharir, M & Smorodinsky, S 2002, On Neighbors in Geometric Permutations. in M Penttonen & EM Schmidt (eds), Algorithm Theory, SWAT 2002: 8th Scandinavian Workshop on Algorithm Theory, Proceedings. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 2368, Springer Verlag, pp. 131-139, 8th Scandinavian Workshop on Algorithm Theory, SWAT 2002, Turku, Finland, 3/07/02. https://doi.org/10.1007/3-540-45471-3_14

On Neighbors in Geometric Permutations. / Sharir, Micha; Smorodinsky, Shakhar.
Algorithm Theory, SWAT 2002: 8th Scandinavian Workshop on Algorithm Theory, Proceedings. ed. / Martti Penttonen; Erik Meineche Schmidt. Springer Verlag, 2002. p. 131-139 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 2368).

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

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N1 - Funding Information: Work on this paper has been supported by NSF Grants CCR-97-32101 and CCR-00-98246, by a grant from the US–Israeli Binational Science Foundation, by a grant from the Israeli Academy of Sciences for a Center of Excellence in Geometric Computing at Tel Aviv University, and by the Hermann Minkowski—MINERVA Center for Geometry at Tel Aviv University. Publisher Copyright: © Springer-Verlag Berlin Heidelberg 2002.

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AB - We introduce a new notion of ‘neighbors’ in geometric permutations. We conjecture that the maximum number of neighbors in a set S of n pairwise disjoint convex bodies in ℝd is O(n), and we prove this conjecture for d = 2. We show that if the set of pairs of neighbors in a set S is of size N, then S admits at most O(Nd−1) geometric permutations. Hence we obtain an alternative proof of a linear upper bound on the number of geometric permutations for any finite family of pairwise disjoint convex bodies in the plane.

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Sharir M, Smorodinsky S. On Neighbors in Geometric Permutations. In Penttonen M, Schmidt EM, editors, Algorithm Theory, SWAT 2002: 8th Scandinavian Workshop on Algorithm Theory, Proceedings. Springer Verlag. 2002. p. 131-139. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). doi: 10.1007/3-540-45471-3_14

On Neighbors in Geometric Permutations

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