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Minimal-Perimeter Polyominoes: Chains, Roots, and Algorithms

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Algorithms and Discrete Applied Mathematics (CALDAM 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11394))

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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.

Work on this paper by both authors has been supported in part by ISF Grant 575/15 and by BSF (joint with NSF) Grant 2017684.

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Authors and Affiliations

  1. Department of Computer Science, Technion—IIT, Haifa, Israel

    Gill Barequet & Gil Ben-Shachar

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  1. Gill Barequet
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  2. Gil Ben-Shachar
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Correspondence to Gil Ben-Shachar .

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Editors and Affiliations

  1. Indian Institute of Technology, Kharagpur, India

    Sudebkumar Prasant Pal

  2. Cochin University of Science and Technology, Cochin, India

    Ambat Vijayakumar

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Barequet, G., Ben-Shachar, G. (2019). Minimal-Perimeter Polyominoes: Chains, Roots, and Algorithms. In: Pal, S., Vijayakumar, A. (eds) Algorithms and Discrete Applied Mathematics. CALDAM 2019. Lecture Notes in Computer Science(), vol 11394. Springer, Cham. https://doi.org/10.1007/978-3-030-11509-8_10

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

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