research-article

Guarding polyominoes

Authors:

Mohammad T. Irfan,

Joseph S.B. MitchellAuthors Info & Claims

SoCG '11: Proceedings of the twenty-seventh annual symposium on Computational geometry

Pages 387 - 396

https://doi.org/10.1145/1998196.1998261

Published: 13 June 2011 Publication History

Abstract

We explore the art gallery problem for the special case that the domain (gallery) P is an m-polyomino, a polyform whose cells are m unit squares. We study the combinatorics of guarding polyominoes in terms of the parameter m, in contrast with the traditional parameter n, the number of vertices of P; in particular, we show that floor((m+1)/3) point guards are always sufficient and sometimes necessary to cover an m-polyomino. When m d 3n/4 - 4, the point guard sufficiency condition yields a strictly lower guard number than floor(n/4), given by the art gallery theorem for orthogonal polygons. When pixels behave themselves like guards (pixel guards), we prove that floor(3m/11) + 1 guards are sufficient and sometimes necessary to cover an m-polyomino. We also study the algorithmic complexity of computing optimal guard sets for polyominoes. We prove that determining the guard number of a given m-polyomino is NP-hard. We provide polynomial-time algorithms to solve exactly some special cases in which the polyomino is "thin".

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Cited By

Filtser OKrohn ENilsson BRieck CSchmidt C(2024)Guarding Polyominoes Under k-Hop VisibilityLATIN 2024: Theoretical Informatics10.1007/978-3-031-55598-5_19(288-302)Online publication date: 18-Mar-2024
https://dl.acm.org/doi/10.1007/978-3-031-55598-5_19
Rieck CScheffer C(2023)The Dispersive Art Gallery ProblemComputational Geometry10.1016/j.comgeo.2023.102054(102054)Online publication date: Oct-2023
https://doi.org/10.1016/j.comgeo.2023.102054
Cruz VTomás A(2022)On r-Guarding SCOTs – A New Family of Orthogonal PolygonsLATIN 2022: Theoretical Informatics10.1007/978-3-031-20624-5_43(713-729)Online publication date: 29-Oct-2022
https://doi.org/10.1007/978-3-031-20624-5_43
Show More Cited By

Index Terms

Guarding polyominoes
1. Mathematics of computing
  1. Discrete mathematics
    1. Combinatorics
2. Theory of computation
  1. Randomness, geometry and discrete structures
    1. Computational geometry

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cover image ACM Conferences

SoCG '11: Proceedings of the twenty-seventh annual symposium on Computational geometry

June 2011

532 pages

ISBN:9781450306829

DOI:10.1145/1998196

Program Chairs:
Ferran Hurtado
Universitat Politècnica de Catalunya
,
Marc van Kreveld
Utrecht University

Copyright © 2011 ACM.

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Publication History

Published: 13 June 2011

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SoCG '11

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SoCG '11: Symposium on Computational Geometry

June 13 - 15, 2011

Paris, France

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Overall Acceptance Rate 625 of 1,685 submissions, 37%

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Cited By

Filtser OKrohn ENilsson BRieck CSchmidt C(2024)Guarding Polyominoes Under k-Hop VisibilityLATIN 2024: Theoretical Informatics10.1007/978-3-031-55598-5_19(288-302)Online publication date: 18-Mar-2024
https://dl.acm.org/doi/10.1007/978-3-031-55598-5_19
Rieck CScheffer C(2023)The Dispersive Art Gallery ProblemComputational Geometry10.1016/j.comgeo.2023.102054(102054)Online publication date: Oct-2023
https://doi.org/10.1016/j.comgeo.2023.102054
Cruz VTomás A(2022)On r-Guarding SCOTs – A New Family of Orthogonal PolygonsLATIN 2022: Theoretical Informatics10.1007/978-3-031-20624-5_43(713-729)Online publication date: 29-Oct-2022
https://doi.org/10.1007/978-3-031-20624-5_43
Alpert HRoldán É(2021)Art Gallery Problem with Rook and Queen VisionGraphs and Combinatorics10.1007/s00373-020-02272-8Online publication date: 21-Jan-2021
https://doi.org/10.1007/s00373-020-02272-8
Hoorfar HBagheri A(2020)NP-completeness of chromatic orthogonal art gallery problemThe Journal of Supercomputing10.1007/s11227-020-03379-8Online publication date: 23-Jul-2020
https://doi.org/10.1007/s11227-020-03379-8
Biedl TIrfan MIwerks JKim JMitchell J(2019)The Art Gallery Theorem for PolyominoesDiscrete & Computational Geometry10.5555/3116673.311714248:3(711-720)Online publication date: 1-Jan-2019
https://dl.acm.org/doi/10.5555/3116673.3117142
Iwamoto CKume T(2014)Computational Complexity of the $$r$$-visibility Guard Set Problem for PolyominoesDiscrete and Computational Geometry and Graphs10.1007/978-3-319-13287-7_8(87-95)Online publication date: 21-Nov-2014
https://doi.org/10.1007/978-3-319-13287-7_8
Tomás A(2013)Guarding thin orthogonal polygons is hardProceedings of the 19th international conference on Fundamentals of Computation Theory10.1007/978-3-642-40164-0_29(305-316)Online publication date: 19-Aug-2013
https://dl.acm.org/doi/10.1007/978-3-642-40164-0_29
Biedl TIrfan MIwerks JKim JMitchell J(2012)The Art Gallery Theorem for PolyominoesDiscrete & Computational Geometry10.1007/s00454-012-9429-148:3(711-720)Online publication date: 9-May-2012
https://doi.org/10.1007/s00454-012-9429-1

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