Improved upper bounds on the reflexivity of point sets

Eyal Ackerman; Oswin Aichholzer; Balázs Keszegh

doi:10.1016/j.comgeo.2008.05.004

Improved upper bounds on the reflexivity of point sets

Eyal Ackerman, Oswin Aichholzer, Balázs Keszegh

Research output: Contribution to journal › Article › peer-review

Abstract

Given a set S of n points in the plane, the reflexivity of S, ρ(S), is the minimum number of reflex vertices in a simple polygonalization of S. Arkin et al. [E.M. Arkin, S.P. Fekete, F. Hurtado, J.S.B. Mitchell, M. Noy, V. Sacristán, S. Sethia, On the reflexivity of point sets, in: B. Aronov, S. Basu, J. Pach M. Sharir (Eds.), Discrete and Computational Geometry: The Goodman-Pollack Festschrift, Springer, 2003, pp. 139-156] proved that ρ(S)≤⌈n/2⌉ for any set S, and conjectured that the tight upper bound is ⌊n/4⌋. We show that the reflexivity of any set of n points is at most 37n+O(1)≈0.4286n. Using computer-aided abstract order type extension the upper bound can be further improved to 512n+O(1)≈0.4167n. We also present an algorithm to compute polygonalizations with at most this number of reflex vertices in O(nlogn) time.

Original language	English
Pages (from-to)	241-249
Number of pages	9
Journal	Computational Geometry: Theory and Applications
Volume	42
Issue number	3
DOIs	https://doi.org/10.1016/j.comgeo.2008.05.004
State	Published - Apr 2009
Externally published	Yes

Bibliographical note

Funding Information:
E-mail addresses: [email protected] (E. Ackerman), [email protected] (O. Aichholzer), [email protected] (B. Keszegh). 1 Supported by the Austrian FWF Joint Research Project ‘Industrial Geometry’ S9205-N12. 2 This work was done while visiting the School of Computing Science at Simon Fraser University.

Keywords

Polygonalization
Reflexivity
Steiner points

ASJC Scopus subject areas

Computer Science Applications
Geometry and Topology
Control and Optimization
Computational Theory and Mathematics
Computational Mathematics

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10.1016/j.comgeo.2008.05.004

Cite this

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title = "Improved upper bounds on the reflexivity of point sets",

abstract = "Given a set S of n points in the plane, the reflexivity of S, ρ(S), is the minimum number of reflex vertices in a simple polygonalization of S. Arkin et al. [E.M. Arkin, S.P. Fekete, F. Hurtado, J.S.B. Mitchell, M. Noy, V. Sacrist{\'a}n, S. Sethia, On the reflexivity of point sets, in: B. Aronov, S. Basu, J. Pach M. Sharir (Eds.), Discrete and Computational Geometry: The Goodman-Pollack Festschrift, Springer, 2003, pp. 139-156] proved that ρ(S)≤⌈n/2⌉ for any set S, and conjectured that the tight upper bound is ⌊n/4⌋. We show that the reflexivity of any set of n points is at most 37n+O(1)≈0.4286n. Using computer-aided abstract order type extension the upper bound can be further improved to 512n+O(1)≈0.4167n. We also present an algorithm to compute polygonalizations with at most this number of reflex vertices in O(nlogn) time.",

keywords = "Polygonalization, Reflexivity, Steiner points",

author = "Eyal Ackerman and Oswin Aichholzer and Bal{\'a}zs Keszegh",

note = "Funding Information: E-mail addresses: [email protected] (E. Ackerman), [email protected] (O. Aichholzer), [email protected] (B. Keszegh). 1 Supported by the Austrian FWF Joint Research Project {\textquoteleft}Industrial Geometry{\textquoteright} S9205-N12. 2 This work was done while visiting the School of Computing Science at Simon Fraser University.",

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doi = "10.1016/j.comgeo.2008.05.004",

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AU - Ackerman, Eyal

AU - Aichholzer, Oswin

AU - Keszegh, Balázs

N1 - Funding Information: E-mail addresses: [email protected] (E. Ackerman), [email protected] (O. Aichholzer), [email protected] (B. Keszegh). 1 Supported by the Austrian FWF Joint Research Project ‘Industrial Geometry’ S9205-N12. 2 This work was done while visiting the School of Computing Science at Simon Fraser University.

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N2 - Given a set S of n points in the plane, the reflexivity of S, ρ(S), is the minimum number of reflex vertices in a simple polygonalization of S. Arkin et al. [E.M. Arkin, S.P. Fekete, F. Hurtado, J.S.B. Mitchell, M. Noy, V. Sacristán, S. Sethia, On the reflexivity of point sets, in: B. Aronov, S. Basu, J. Pach M. Sharir (Eds.), Discrete and Computational Geometry: The Goodman-Pollack Festschrift, Springer, 2003, pp. 139-156] proved that ρ(S)≤⌈n/2⌉ for any set S, and conjectured that the tight upper bound is ⌊n/4⌋. We show that the reflexivity of any set of n points is at most 37n+O(1)≈0.4286n. Using computer-aided abstract order type extension the upper bound can be further improved to 512n+O(1)≈0.4167n. We also present an algorithm to compute polygonalizations with at most this number of reflex vertices in O(nlogn) time.

AB - Given a set S of n points in the plane, the reflexivity of S, ρ(S), is the minimum number of reflex vertices in a simple polygonalization of S. Arkin et al. [E.M. Arkin, S.P. Fekete, F. Hurtado, J.S.B. Mitchell, M. Noy, V. Sacristán, S. Sethia, On the reflexivity of point sets, in: B. Aronov, S. Basu, J. Pach M. Sharir (Eds.), Discrete and Computational Geometry: The Goodman-Pollack Festschrift, Springer, 2003, pp. 139-156] proved that ρ(S)≤⌈n/2⌉ for any set S, and conjectured that the tight upper bound is ⌊n/4⌋. We show that the reflexivity of any set of n points is at most 37n+O(1)≈0.4286n. Using computer-aided abstract order type extension the upper bound can be further improved to 512n+O(1)≈0.4167n. We also present an algorithm to compute polygonalizations with at most this number of reflex vertices in O(nlogn) time.

KW - Polygonalization

KW - Reflexivity

KW - Steiner points

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Improved upper bounds on the reflexivity of point sets

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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