Improved Upper Bounds on the Growth Constants of Polyominoes and Polycubes

Gill Barequet; Mira Shalah

doi:https://doi.org/10.1007/978-3-030-61792-9_42

Improved Upper Bounds on the Growth Constants of Polyominoes and Polycubes

Gill Barequet, Mira Shalah

Computer Science

Technion - Israel Institute of Technology

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

Abstract

A d-dimensional polycube is a face-connected set of cells on Z^d. Let A_d(n) denote the number of d-dimensional polycubes (distinct up to translations) with n cubes, and λ_d denote their growth constant limn→∞Ad(n+1)Ad(n). We revisit and extend the method for the best known upper bound on A₂(n). Our contributions: We (1) prove that λ₂≤ 4.5252 ; (2) prove that λ_d≤ (2 d- 2 ) e+ o(1 ) for d≥ 2 (already improving significantly the upper bound on λ₃ to 9.8073); and (3) implement an iterative process in 3D, improving further the upper bound on λ₃ to 9.3835.

Original language	English
Title of host publication	LATIN 2020
Subtitle of host publication	Theoretical Informatics - 14th Latin American Symposium 2021, Proceedings
Editors	Yoshiharu Kohayakawa, Flávio Keidi Miyazawa
Publisher	Springer Science and Business Media Deutschland GmbH
Pages	532-545
Number of pages	14
ISBN (Print)	9783030617912
DOIs	https://doi.org/10.1007/978-3-030-61792-9_42
State	Published - 2020
Event	14th Latin American Symposium on Theoretical Informatics, LATIN 2020 - Sao Paulo, Brazil Duration: 5 Jan 2021 → 8 Jan 2021

Publication series

Name	Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume	12118 LNCS

Conference

Conference	14th Latin American Symposium on Theoretical Informatics, LATIN 2020
Country/Territory	Brazil
City	Sao Paulo
Period	5/01/21 → 8/01/21

Keywords

Cubical lattice
Klarner’s constant
Square lattice

All Science Journal Classification (ASJC) codes

Theoretical Computer Science
General Computer Science

Access to Document

https://doi.org/10.1007/978-3-030-61792-9_42

Cite this

Barequet, G., & Shalah, M. (2020). Improved Upper Bounds on the Growth Constants of Polyominoes and Polycubes. In Y. Kohayakawa, & F. K. Miyazawa (Eds.), LATIN 2020: Theoretical Informatics - 14th Latin American Symposium 2021, Proceedings (pp. 532-545). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 12118 LNCS). Springer Science and Business Media Deutschland GmbH. https://doi.org/10.1007/978-3-030-61792-9_42

Barequet, Gill ; Shalah, Mira. / Improved Upper Bounds on the Growth Constants of Polyominoes and Polycubes. LATIN 2020: Theoretical Informatics - 14th Latin American Symposium 2021, Proceedings. editor / Yoshiharu Kohayakawa ; Flávio Keidi Miyazawa. Springer Science and Business Media Deutschland GmbH, 2020. pp. 532-545 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).

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abstract = "A d-dimensional polycube is a face-connected set of cells on Zd. Let Ad(n) denote the number of d-dimensional polycubes (distinct up to translations) with n cubes, and λd denote their growth constant limn→∞Ad(n+1)Ad(n). We revisit and extend the method for the best known upper bound on A2(n). Our contributions: We (1) prove that λ2≤ 4.5252 ; (2) prove that λd≤ (2 d- 2 ) e+ o(1 ) for d≥ 2 (already improving significantly the upper bound on λ3 to 9.8073); and (3) implement an iterative process in 3D, improving further the upper bound on λ3 to 9.3835.",

keywords = "Cubical lattice, Klarner{\textquoteright}s constant, Square lattice",

author = "Gill Barequet and Mira Shalah",

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Barequet, G & Shalah, M 2020, Improved Upper Bounds on the Growth Constants of Polyominoes and Polycubes. in Y Kohayakawa & FK Miyazawa (eds), LATIN 2020: Theoretical Informatics - 14th Latin American Symposium 2021, Proceedings. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 12118 LNCS, Springer Science and Business Media Deutschland GmbH, pp. 532-545, 14th Latin American Symposium on Theoretical Informatics, LATIN 2020, Sao Paulo, Brazil, 5/01/21. https://doi.org/10.1007/978-3-030-61792-9_42

Improved Upper Bounds on the Growth Constants of Polyominoes and Polycubes. / Barequet, Gill; Shalah, Mira.
LATIN 2020: Theoretical Informatics - 14th Latin American Symposium 2021, Proceedings. ed. / Yoshiharu Kohayakawa; Flávio Keidi Miyazawa. Springer Science and Business Media Deutschland GmbH, 2020. p. 532-545 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 12118 LNCS).

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

TY - GEN

T1 - Improved Upper Bounds on the Growth Constants of Polyominoes and Polycubes

AU - Barequet, Gill

AU - Shalah, Mira

PY - 2020

Y1 - 2020

N2 - A d-dimensional polycube is a face-connected set of cells on Zd. Let Ad(n) denote the number of d-dimensional polycubes (distinct up to translations) with n cubes, and λd denote their growth constant limn→∞Ad(n+1)Ad(n). We revisit and extend the method for the best known upper bound on A2(n). Our contributions: We (1) prove that λ2≤ 4.5252 ; (2) prove that λd≤ (2 d- 2 ) e+ o(1 ) for d≥ 2 (already improving significantly the upper bound on λ3 to 9.8073); and (3) implement an iterative process in 3D, improving further the upper bound on λ3 to 9.3835.

AB - A d-dimensional polycube is a face-connected set of cells on Zd. Let Ad(n) denote the number of d-dimensional polycubes (distinct up to translations) with n cubes, and λd denote their growth constant limn→∞Ad(n+1)Ad(n). We revisit and extend the method for the best known upper bound on A2(n). Our contributions: We (1) prove that λ2≤ 4.5252 ; (2) prove that λd≤ (2 d- 2 ) e+ o(1 ) for d≥ 2 (already improving significantly the upper bound on λ3 to 9.8073); and (3) implement an iterative process in 3D, improving further the upper bound on λ3 to 9.3835.

KW - Cubical lattice

KW - Klarner’s constant

KW - Square lattice

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DO - https://doi.org/10.1007/978-3-030-61792-9_42

M3 - منشور من مؤتمر

SN - 9783030617912

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 532

EP - 545

BT - LATIN 2020

A2 - Kohayakawa, Yoshiharu

A2 - Miyazawa, Flávio Keidi

PB - Springer Science and Business Media Deutschland GmbH

T2 - 14th Latin American Symposium on Theoretical Informatics, LATIN 2020

Y2 - 5 January 2021 through 8 January 2021

ER -

Barequet G, Shalah M. Improved Upper Bounds on the Growth Constants of Polyominoes and Polycubes. In Kohayakawa Y, Miyazawa FK, editors, LATIN 2020: Theoretical Informatics - 14th Latin American Symposium 2021, Proceedings. Springer Science and Business Media Deutschland GmbH. 2020. p. 532-545. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). doi: https://doi.org/10.1007/978-3-030-61792-9_42

Improved Upper Bounds on the Growth Constants of Polyominoes and Polycubes

Abstract

Publication series

Conference

Keywords

All Science Journal Classification (ASJC) codes

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