Abstract
In the field of Discrete Tomography, the 2-color problem consists in determining a matrix whose elements are of two different types, starting from its horizontal and vertical projections. It is known that the one color problem has a polynomial time reconstruction algorithm, while, with k ≥ 2, the k-color problem is NP-complete. Thus, the 2-color problem constitutes an interesting example of a problem just in the frontier between hard and easy problems.
In this paper we define a linear time algorithm to solve a set of its instances, where some values of the horizontal and vertical projections are constant, while the others are upper bounded by a positive number proportional to the dimension of the problem. Our algorithm relies on classical studies for the solution of the one color problem.
Chapter PDF
Similar content being viewed by others
References
Brocchi, S., Frosini, A., Picouleau, C.: Reconstruction of binary matrices under fixed size neighborhood constraints. Theoretical Computer Science 406, 1-2, 43-54 (2008)
Chrobak, M., Durr, C.: Reconstructing polyatomic structures from discrete X-rays: NP-completeness proof for three atoms. Theoretical computer science 259, 81–98 (2001)
Costa, M.C., de Werra, D., Picouleau, C., Schindl, D.: A solvable case of image reconstruction in discrete tomography. Discrete Applied Mathematics 148(3), 240–245 (2005)
Costa, M.C., de Werra, D., Picouleau, C.: Using graphs for some discrete tomography problems, CEDRIC report (2004), http://cedric.cnam/fr/
Dũrr, C., Guiñez, F., Matamala, M.: Reconstructing 3-colored grids from horizontal and vertical projections is NP-hard. In: Fiat, A., Sanders, P. (eds.) ESA 2009. LNCS, vol. 5757, pp. 776–788. Springer, Heidelberg (2009)
Gardner, R.J., Gritzmann, P., Pranenberg, D.: On the computational complexity of reconstructing lattice sets from their X-rays. Discrete Mathematics 202(1-3), 45–71 (1999)
Gardner, R.J., Gritzmann, P., Pranenberg, D.: On the computational complexity of determining polyatomic structures by X-rays. Theoretical computer science 233, 91–106 (2000)
Herman, G., Kuba, A.: Discrete Tomography: Foundations, Algorithms and Applications. Birkhauser, Basel (1999)
Kisielowski, C., Schwander, P., Baumann, F.H., Seibt, M., Kim, Y., Ourmazd, A.: An approach to quantitative high-resolution transmission electron microscopy of crystalline materials. Ultramicroscopy 58, 131–155 (1995)
Ryser, H.J.: Combinatorial properties of matrices of zeros and ones. Canad. J. Math. 9, 371–377 (1957)
Schwander, P., Kisielowski, C., Seibt, M., Baumann, F.H., Kim, Y., Ourmazd, A.: Mapping projected potential interfacial roughness, and composition in general crystalline solids by quantitative transmission electron microscopy. Phys. Rev. Lett. 71, 4150–4153 (1993)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Brocchi, S., Frosini, A., Rinaldi, S. (2009). Solving Some Instances of the 2-Color Problem. In: Brlek, S., Reutenauer, C., Provençal, X. (eds) Discrete Geometry for Computer Imagery. DGCI 2009. Lecture Notes in Computer Science, vol 5810. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04397-0_43
Download citation
DOI: https://doi.org/10.1007/978-3-642-04397-0_43
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-04396-3
Online ISBN: 978-3-642-04397-0
eBook Packages: Computer ScienceComputer Science (R0)