Minimal-perimeter polyominoes: chains, roots, and algorithms

Gill Barequet; Gil Ben-Shachar

doi:https://doi.org/10.1007/978-3-030-11509-8_10

Minimal-perimeter polyominoes: chains, roots, and algorithms

Gill Barequet, Gil Ben-Shachar

Computer Science

Technion - Israel Institute of Technology

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

Abstract

A polyomino is a set of edge-connected squares on the square lattice. We investigate the combinatorial and geometric properties of minimal-perimeter polyominoes. We explore the behavior of minimal-perimeter polyominoes when they are “inflated,” i.e., expanded by all empty cells neighboring them, and show that inflating all minimal-perimeter polyominoes of a given area create the set of all minimal-perimeter polyominoes of some larger area. We characterize the roots of the infinite chains of sets of minimal-perimeter polyominoes which are created by inflating polyominoes of another set of minimal-perimeter polyominoes, and show that inflating any polyomino for a sufficient amount of times results in a minimal-perimeter polyomino. In addition, we devise two efficient algorithms for counting the number of minimal-perimeter polyominoes of a given area, compare the algorithms and analyze their running times, and provide the counts of polyominoes which they produce.

Original language	English
Title of host publication	Algorithms and Discrete Applied Mathematics - 5th International Conference, CALDAM 2019, Proceedings
Editors	Sudebkumar Prasant Pal, Ambat Vijayakumar
Pages	109-123
Number of pages	15
DOIs	https://doi.org/10.1007/978-3-030-11509-8_10
State	Published - 2019
Event	5th International Conference on Algorithms and Discrete Applied Mathematics, CALDAM 2019 - Kharagpur, India Duration: 14 Feb 2019 → 16 Feb 2019

Publication series

Name	Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume	11394 LNCS

Conference

Conference	5th International Conference on Algorithms and Discrete Applied Mathematics, CALDAM 2019
Country/Territory	India
City	Kharagpur
Period	14/02/19 → 16/02/19

All Science Journal Classification (ASJC) codes

Theoretical Computer Science
General Computer Science

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https://doi.org/10.1007/978-3-030-11509-8_10

Cite this

Barequet, G., & Ben-Shachar, G. (2019). Minimal-perimeter polyominoes: chains, roots, and algorithms. In S. P. Pal, & A. Vijayakumar (Eds.), Algorithms and Discrete Applied Mathematics - 5th International Conference, CALDAM 2019, Proceedings (pp. 109-123). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 11394 LNCS). https://doi.org/10.1007/978-3-030-11509-8_10

Barequet, Gill ; Ben-Shachar, Gil. / Minimal-perimeter polyominoes : chains, roots, and algorithms. Algorithms and Discrete Applied Mathematics - 5th International Conference, CALDAM 2019, Proceedings. editor / Sudebkumar Prasant Pal ; Ambat Vijayakumar. 2019. pp. 109-123 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).

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abstract = "A polyomino is a set of edge-connected squares on the square lattice. We investigate the combinatorial and geometric properties of minimal-perimeter polyominoes. We explore the behavior of minimal-perimeter polyominoes when they are “inflated,” i.e., expanded by all empty cells neighboring them, and show that inflating all minimal-perimeter polyominoes of a given area create the set of all minimal-perimeter polyominoes of some larger area. We characterize the roots of the infinite chains of sets of minimal-perimeter polyominoes which are created by inflating polyominoes of another set of minimal-perimeter polyominoes, and show that inflating any polyomino for a sufficient amount of times results in a minimal-perimeter polyomino. In addition, we devise two efficient algorithms for counting the number of minimal-perimeter polyominoes of a given area, compare the algorithms and analyze their running times, and provide the counts of polyominoes which they produce.",

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Barequet, G & Ben-Shachar, G 2019, Minimal-perimeter polyominoes: chains, roots, and algorithms. in SP Pal & A Vijayakumar (eds), Algorithms and Discrete Applied Mathematics - 5th International Conference, CALDAM 2019, Proceedings. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 11394 LNCS, pp. 109-123, 5th International Conference on Algorithms and Discrete Applied Mathematics, CALDAM 2019, Kharagpur, India, 14/02/19. https://doi.org/10.1007/978-3-030-11509-8_10

Minimal-perimeter polyominoes: chains, roots, and algorithms. / Barequet, Gill; Ben-Shachar, Gil.
Algorithms and Discrete Applied Mathematics - 5th International Conference, CALDAM 2019, Proceedings. ed. / Sudebkumar Prasant Pal; Ambat Vijayakumar. 2019. p. 109-123 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 11394 LNCS).

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

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N2 - A polyomino is a set of edge-connected squares on the square lattice. We investigate the combinatorial and geometric properties of minimal-perimeter polyominoes. We explore the behavior of minimal-perimeter polyominoes when they are “inflated,” i.e., expanded by all empty cells neighboring them, and show that inflating all minimal-perimeter polyominoes of a given area create the set of all minimal-perimeter polyominoes of some larger area. We characterize the roots of the infinite chains of sets of minimal-perimeter polyominoes which are created by inflating polyominoes of another set of minimal-perimeter polyominoes, and show that inflating any polyomino for a sufficient amount of times results in a minimal-perimeter polyomino. In addition, we devise two efficient algorithms for counting the number of minimal-perimeter polyominoes of a given area, compare the algorithms and analyze their running times, and provide the counts of polyominoes which they produce.

AB - A polyomino is a set of edge-connected squares on the square lattice. We investigate the combinatorial and geometric properties of minimal-perimeter polyominoes. We explore the behavior of minimal-perimeter polyominoes when they are “inflated,” i.e., expanded by all empty cells neighboring them, and show that inflating all minimal-perimeter polyominoes of a given area create the set of all minimal-perimeter polyominoes of some larger area. We characterize the roots of the infinite chains of sets of minimal-perimeter polyominoes which are created by inflating polyominoes of another set of minimal-perimeter polyominoes, and show that inflating any polyomino for a sufficient amount of times results in a minimal-perimeter polyomino. In addition, we devise two efficient algorithms for counting the number of minimal-perimeter polyominoes of a given area, compare the algorithms and analyze their running times, and provide the counts of polyominoes which they produce.

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Barequet G, Ben-Shachar G. Minimal-perimeter polyominoes: chains, roots, and algorithms. In Pal SP, Vijayakumar A, editors, Algorithms and Discrete Applied Mathematics - 5th International Conference, CALDAM 2019, Proceedings. 2019. p. 109-123. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). doi: https://doi.org/10.1007/978-3-030-11509-8_10

Minimal-perimeter polyominoes: chains, roots, and algorithms

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