An almost optimal bound on the number of intersections of two simple polygons

Eyal Ackerman; Balázs Keszegh; Günter Rote

doi:10.4230/LIPIcs.SoCG.2020.1

An almost optimal bound on the number of intersections of two simple polygons

Eyal Ackerman, Balázs Keszegh, Günter Rote

Faculty of Natural Sciences

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

Abstract

What is the maximum number of intersections of the boundaries of a simple m-gon and a simple n-gon, assuming general position? This is a basic question in combinatorial geometry, and the answer is easy if at least one of m and n is even. If both m and n are odd, the best known construction has mn − (m + n) + 3 intersections, and it is conjectured that this is the maximum. However, the best known upper bound is only mn − (m + dⁿ₆ e), for m ≥ n. We prove a new upper bound of mn − (m + n) + C for some constant C, which is optimal apart from the value of C.

Original language	English
Title of host publication	36th International Symposium on Computational Geometry, SoCG 2020
Editors	Sergio Cabello, Danny Z. Chen
Publisher	Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)	9783959771436
DOIs	https://doi.org/10.4230/LIPIcs.SoCG.2020.1
State	Published - 1 Jun 2020
Event	36th International Symposium on Computational Geometry, SoCG 2020 - Zurich, Switzerland Duration: 23 Jun 2020 → 26 Jun 2020

Publication series

Name	Leibniz International Proceedings in Informatics, LIPIcs
Volume	164
ISSN (Print)	1868-8969

Conference

Conference	36th International Symposium on Computational Geometry, SoCG 2020
Country/Territory	Switzerland
City	Zurich
Period	23/06/20 → 26/06/20

Bibliographical note

Funding Information:
Funding Eyal Ackerman: The main part of this work was performed during a visit to Freie Universität Berlin which was supported by the Freie Universität Alumni Program. Balázs Keszegh: Research supported by the Lendület program of the Hungarian Academy of Sciences (MTA), under the grant LP2017-19/2017 and by the National Research, Development and Innovation Office – NKFIH under the grant K 116769.

Funding Information:
Eyal Ackerman: The main part of this work was performed during a visit to Freie Universit?t Berlin which was supported by the Freie Universit?t Alumni Program. We thank the reviewers for helpful suggestions.

Publisher Copyright:
© Eyal Ackerman, Balázs Keszegh, and Günter Rote; licensed under Creative Commons License CC-BY 36th International Symposium on Computational Geometry (SoCG 2020).

Keywords

Combinatorial geometry
Ramsey theory
Simple polygon

ASJC Scopus subject areas

Software

Access to Document

10.4230/LIPIcs.SoCG.2020.1

Cite this

Ackerman, E., Keszegh, B., & Rote, G. (2020). An almost optimal bound on the number of intersections of two simple polygons. In S. Cabello, & D. Z. Chen (Eds.), 36th International Symposium on Computational Geometry, SoCG 2020 Article LIPIcs-SoCG-2020-1 (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 164). Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.SoCG.2020.1

Ackerman, Eyal ; Keszegh, Balázs ; Rote, Günter. / An almost optimal bound on the number of intersections of two simple polygons. 36th International Symposium on Computational Geometry, SoCG 2020. editor / Sergio Cabello ; Danny Z. Chen. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 2020. (Leibniz International Proceedings in Informatics, LIPIcs).

@inproceedings{2d6ec6f2abbc4675aabb539b728343e6,

title = "An almost optimal bound on the number of intersections of two simple polygons",

abstract = "What is the maximum number of intersections of the boundaries of a simple m-gon and a simple n-gon, assuming general position? This is a basic question in combinatorial geometry, and the answer is easy if at least one of m and n is even. If both m and n are odd, the best known construction has mn − (m + n) + 3 intersections, and it is conjectured that this is the maximum. However, the best known upper bound is only mn − (m + dn6 e), for m ≥ n. We prove a new upper bound of mn − (m + n) + C for some constant C, which is optimal apart from the value of C.",

keywords = "Combinatorial geometry, Ramsey theory, Simple polygon",

author = "Eyal Ackerman and Bal{\'a}zs Keszegh and G{\"u}nter Rote",

note = "Funding Information: Funding Eyal Ackerman: The main part of this work was performed during a visit to Freie Universit{\"a}t Berlin which was supported by the Freie Universit{\"a}t Alumni Program. Bal{\'a}zs Keszegh: Research supported by the Lend{\"u}let program of the Hungarian Academy of Sciences (MTA), under the grant LP2017-19/2017 and by the National Research, Development and Innovation Office – NKFIH under the grant K 116769. Funding Information: Eyal Ackerman: The main part of this work was performed during a visit to Freie Universit?t Berlin which was supported by the Freie Universit?t Alumni Program. We thank the reviewers for helpful suggestions. Publisher Copyright: {\textcopyright} Eyal Ackerman, Bal{\'a}zs Keszegh, and G{\"u}nter Rote; licensed under Creative Commons License CC-BY 36th International Symposium on Computational Geometry (SoCG 2020).; 36th International Symposium on Computational Geometry, SoCG 2020 ; Conference date: 23-06-2020 Through 26-06-2020",

year = "2020",

month = jun,

day = "1",

doi = "10.4230/LIPIcs.SoCG.2020.1",

language = "English",

series = "Leibniz International Proceedings in Informatics, LIPIcs",

publisher = "Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing",

editor = "Sergio Cabello and Chen, {Danny Z.}",

booktitle = "36th International Symposium on Computational Geometry, SoCG 2020",

address = "Germany",

}

Ackerman, E, Keszegh, B & Rote, G 2020, An almost optimal bound on the number of intersections of two simple polygons. in S Cabello & DZ Chen (eds), 36th International Symposium on Computational Geometry, SoCG 2020., LIPIcs-SoCG-2020-1, Leibniz International Proceedings in Informatics, LIPIcs, vol. 164, Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 36th International Symposium on Computational Geometry, SoCG 2020, Zurich, Switzerland, 23/06/20. https://doi.org/10.4230/LIPIcs.SoCG.2020.1

An almost optimal bound on the number of intersections of two simple polygons. / Ackerman, Eyal; Keszegh, Balázs; Rote, Günter.
36th International Symposium on Computational Geometry, SoCG 2020. ed. / Sergio Cabello; Danny Z. Chen. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 2020. LIPIcs-SoCG-2020-1 (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 164).

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

TY - GEN

T1 - An almost optimal bound on the number of intersections of two simple polygons

AU - Ackerman, Eyal

AU - Keszegh, Balázs

AU - Rote, Günter

N1 - Funding Information: Funding Eyal Ackerman: The main part of this work was performed during a visit to Freie Universität Berlin which was supported by the Freie Universität Alumni Program. Balázs Keszegh: Research supported by the Lendület program of the Hungarian Academy of Sciences (MTA), under the grant LP2017-19/2017 and by the National Research, Development and Innovation Office – NKFIH under the grant K 116769. Funding Information: Eyal Ackerman: The main part of this work was performed during a visit to Freie Universit?t Berlin which was supported by the Freie Universit?t Alumni Program. We thank the reviewers for helpful suggestions. Publisher Copyright: © Eyal Ackerman, Balázs Keszegh, and Günter Rote; licensed under Creative Commons License CC-BY 36th International Symposium on Computational Geometry (SoCG 2020).

PY - 2020/6/1

Y1 - 2020/6/1

N2 - What is the maximum number of intersections of the boundaries of a simple m-gon and a simple n-gon, assuming general position? This is a basic question in combinatorial geometry, and the answer is easy if at least one of m and n is even. If both m and n are odd, the best known construction has mn − (m + n) + 3 intersections, and it is conjectured that this is the maximum. However, the best known upper bound is only mn − (m + dn6 e), for m ≥ n. We prove a new upper bound of mn − (m + n) + C for some constant C, which is optimal apart from the value of C.

AB - What is the maximum number of intersections of the boundaries of a simple m-gon and a simple n-gon, assuming general position? This is a basic question in combinatorial geometry, and the answer is easy if at least one of m and n is even. If both m and n are odd, the best known construction has mn − (m + n) + 3 intersections, and it is conjectured that this is the maximum. However, the best known upper bound is only mn − (m + dn6 e), for m ≥ n. We prove a new upper bound of mn − (m + n) + C for some constant C, which is optimal apart from the value of C.

KW - Combinatorial geometry

KW - Ramsey theory

KW - Simple polygon

UR - http://www.scopus.com/inward/record.url?scp=85086500938&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.SoCG.2020.1

DO - 10.4230/LIPIcs.SoCG.2020.1

M3 - Conference contribution

AN - SCOPUS:85086500938

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 36th International Symposium on Computational Geometry, SoCG 2020

A2 - Cabello, Sergio

A2 - Chen, Danny Z.

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 36th International Symposium on Computational Geometry, SoCG 2020

Y2 - 23 June 2020 through 26 June 2020

ER -

Ackerman E, Keszegh B, Rote G. An almost optimal bound on the number of intersections of two simple polygons. In Cabello S, Chen DZ, editors, 36th International Symposium on Computational Geometry, SoCG 2020. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. 2020. LIPIcs-SoCG-2020-1. (Leibniz International Proceedings in Informatics, LIPIcs). doi: 10.4230/LIPIcs.SoCG.2020.1

An almost optimal bound on the number of intersections of two simple polygons

Abstract

Publication series

Conference

Bibliographical note

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Cite this