Abstract
Given a binary matrix, deciding wether it can be decomposed into three hv-convex matrices is an \(\cal NP\)-complete problem, whereas its decomposition into two hv-convex matrices or two hv-polyominoes can be performed in polynomial time. In this paper we give a polynomial time algorithm that decomposes a binary matrix into three hv-polyominoes, if such a decomposition exists. These problems are motivated by the Intensity Modulated Radiation Therapy (IMRT).
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Baatar, D., Hamacher, H.W., Ehrgott, M., Woeginger, G.J.: Decomposition of integer matrices and multileaf collimator sequencing. Discrete Applied Mathematics 152(1-3), 6–34 (2005)
Barcucci, E., Del Lungo, A., Nivat, M., Pinzani, R.: Reconstructing convex polyominoes from horizontal and vertical projections. Theoretical Computer Science 155, 321–347 (1996)
Barcucci, E., Frosini, A., Rinaldi, S.: An algorithm for the reconstruction of discrete sets from two projections in presence of absorption. Discrete Applied Mathematics 151(1-3), 21–35 (2005)
Battaglino, D., Fedou, J.M., Frosini, A., Rinaldi, S.: Encoding Centered Polyominoes by Means of a Regular Language. In: Mauri, G., Leporati, A. (eds.) DLT 2011. LNCS, vol. 6795, pp. 464–465. Springer, Heidelberg (2011)
Boland, N., Hamacher, H., Lenzen, F.: Minimizing beam-on time in cancer radiation treatment using multileaf collimators. Networks 43(4), 226–240 (2003)
Bortfeld, T., Boyer, A., Kahler, D., Waldron, T.: X-ray field compensation with multileaf collimators. International Journal of Radiation Oncology, Biology, Physics 28(3), 723–730 (1994)
Bousquet-Mélou, M.: A method for the enumeration of various classes of column-convex polygons. Discrete Mathematics 154, 1–25 (1996)
Castiglione, G., Frosini, A., Munarini, E., Restivo, A., Rinaldi, S.: Enumeration of L-convex polyominoes. II. Bijection and area. In: Proceedings of FPSAC 2005, #49, pp. 531–541 (2005)
Castiglione, G., Restivo, A.: Reconstruction of L-convex Polyominoes. Electronic Notes in Discrete Mathematics 12, 290–301 (2003)
Delest, M., Viennot, X.: Algebraic languages and polyominoes enumeration. Theoretical Computer Science 34, 169–206 (1984)
Del Lungo, A., Duchi, E., Frosini, A., Rinaldi, S.: On the generation and enumeration of some classes of convex polyominoes. The Electronic Journal of Combinatorics 11, #R60 (2004)
Ehrgott, M., Hamacher, H.W., Nußbaum, M.: Decomposition of Matrices and Static Multileaf Collimators: A Survey. In: Optimization in Medicine. Optimization and Its Applications, vol. 12, pp. 25–46 (2008)
Frosini, A., Nivat, M.: Binary Matrices under the Microscope: A Tomographical Problem. Theoretical Computer Science 370, 201–217 (2007)
Golomb, S.W.: Checker boards and polyominoes. American Mathematical Monthly 61(10), 675–682 (1954)
Herman, G.T., Kuba, A. (eds.): Discrete tomography: Foundations algorithms and applications. Birkhauser, Boston (1999)
Hochstätter, W., Loebl, M., Moll, C.: Generating convex polyominoes at random. In: Proceeding of the 5th FPSAC, Discrete Mathematics, vol. 153, pp. 165–176 (1996)
Jarray, F., Picouleau, C.: Minimum decomposition in convex binary matrices. Discrete Applied Mathematics 160, 1164–1175 (2012)
Shepard, D., Ferris, M., Olivera, G., Mackie, T.: Optimizing the delivery of radiation therapy to cancer patients. SIAM Review 41(4), 721–744 (1999)
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Frosini, A., Picouleau, C. (2013). How to Decompose a Binary Matrix into Three hv-convex Polyominoes. In: Gonzalez-Diaz, R., Jimenez, MJ., Medrano, B. (eds) Discrete Geometry for Computer Imagery. DGCI 2013. Lecture Notes in Computer Science, vol 7749. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-37067-0_27
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DOI: https://doi.org/10.1007/978-3-642-37067-0_27
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