Improved Upper Bounds on the Growth Constants of Polyominoes and Polycubes

Gill Barequet; Mira Shalah

doi:10.1007/s00453-022-00948-6

Improved Upper Bounds on the Growth Constants of Polyominoes and Polycubes

Gill Barequet, Mira Shalah

Computer Science

Technion - Israel Institute of Technology

Research output: Contribution to journal › Article › 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 include the following. 1.We prove that λ₂≤ 4.5252 ;2.We prove that λ_d≤ (2 d- 2) e+ o(1) for d≥ 2 (already improving significantly the known upper bound on λ₃ to 9.8073); and3.We implement an iterative process in three dimensions, improving further the upper bound on λ₃ to 9.3835.

Original language	English
Pages (from-to)	3559-3586
Number of pages	28
Journal	Algorithmica
Volume	84
Issue number	12
DOIs	https://doi.org/10.1007/s00453-022-00948-6
State	Published - Dec 2022

Keywords

Cubical lattice
Klarner’s constant
Square lattice

All Science Journal Classification (ASJC) codes

General Computer Science
Applied Mathematics
Computer Science Applications

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10.1007/s00453-022-00948-6

Cite this

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title = "Improved Upper Bounds on the Growth Constants of Polyominoes and Polycubes",

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 include the following. 1.We prove that λ2≤ 4.5252 ;2.We prove that λd≤ (2 d- 2) e+ o(1) for d≥ 2 (already improving significantly the known upper bound on λ3 to 9.8073); and3.We implement an iterative process in three dimensions, 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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doi = "10.1007/s00453-022-00948-6",

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TY - JOUR

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

AU - Barequet, Gill

AU - Shalah, Mira

PY - 2022/12

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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 include the following. 1.We prove that λ2≤ 4.5252 ;2.We prove that λd≤ (2 d- 2) e+ o(1) for d≥ 2 (already improving significantly the known upper bound on λ3 to 9.8073); and3.We implement an iterative process in three dimensions, 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 include the following. 1.We prove that λ2≤ 4.5252 ;2.We prove that λd≤ (2 d- 2) e+ o(1) for d≥ 2 (already improving significantly the known upper bound on λ3 to 9.8073); and3.We implement an iterative process in three dimensions, improving further the upper bound on λ3 to 9.3835.

KW - Cubical lattice

KW - Klarner’s constant

KW - Square lattice

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U2 - 10.1007/s00453-022-00948-6

DO - 10.1007/s00453-022-00948-6

M3 - مقالة

SN - 0178-4617

VL - 84

SP - 3559

EP - 3586

JO - Algorithmica

JF - Algorithmica

IS - 12

ER -

Improved Upper Bounds on the Growth Constants of Polyominoes and Polycubes

Abstract

Keywords

All Science Journal Classification (ASJC) codes

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