Abstract
A semimagic square of order n is an n × n matrix containing the integers 0, ..., n 2–1 arranged in such a way that each row and column add up to the same value. We generalize this notion to that of a zero k × k-discrepancy matrix by replacing the requirement that the sum of each row and each column be the same by that of requiring that the sum of the entries in each k × k square contiguous submatrix be the same. We show that such matrices exist if k and n are both even, and do not if k and n are are relatively prime. Further, the existence is also guaranteed whenever n = k m, for some integers k,m ≥ 2. We present a space-efficient algorithm for constructing such a matrix.
Another class that we call constant-gap matrices arises in this construction. We give a characterization of such matrices.
An application to digital halftoning is also mentioned.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Aggarwal, A., Klawe, M., Moran, S., Shor, P., Wilber, R.: Geometric Applications of a Matrix Searching Algorithm. In: Proc. 2nd ACM Symposium on Computational Geometry, pp. 285–292 (1986)
Asano, T., Katoh, N., Obokata, K., Tokuyama, T.: Combinatorial and geometric problems related to digital halftoning. In: Asano, T., Klette, R., Ronse, C. (eds.) Geometry, Morphology, and Computational Imaging. LNCS, vol. 2616, pp. 58–71. Springer, Heidelberg (2003)
Asano, T.: Digital Halftoning: Algorithm Engineering Challenges. IEICE Trans. on Inf. and Syst., E86-D 2, 159–178 (2003)
Asano, T., Obokata, K., Katoh, N., Tokuyama, T.: Matrix rounding under the L p -discrepancy measure and its application to digital halftoning. In: Proc. ACM-SIAM Symposium on Discrete Algorithms, San Francisco, pp. 896–904 (2002)
Bayer, B.E.: An optimum method for two-level rendition of continuous-tone pictures. In: Conference Record, IEEE International Conference on Communications, vol. 1, pp. (26-11)–(26-15) (1973)
Chazelle, B.: The Discrepancy Method: Randomness and Complexity. Cambridge University Press, Cambridge (2000)
Floyd, R.W., Steinberg, L.: An adaptive algorithm for spatial gray scale. In: SID 75 Digest, Society for Information Display, pp. 36–37 (1975)
Heinz, H.D.: Magic Squares, Magic Stars & Other Patterns. web site, http://www.geocities.com/CapeCanaveral/Launchpad/4057/
Graham, R.L., Lubachevsky, B.D., Nurmela, K.J., Östergård, P.R.J.: Dense packings of congruent circles in a circle. Discrete Math 181, 139–154 (1998)
Knuth, D.E.: The Art of Computer Programming. In: Fundamental Algorithms, 3rd edn., vol. 1, Addison-Wesley, Reading (1997)
Kodera Ed.: Practical Design and Evaluation of Halftoned Images (in Japanese) Trikepps (2000)
Matoušek, J.: Geometric Discrepancy. Springer, Heidelberg (1991)
Mitsa, T., Parker, K.J.: Digital halftoning technique using a blue-noise mask. J. Opt. Soc. Am. 9(11), 1920–1929 (1992)
Nurmela, K.J., Östergård, P.R.J.: Packing up to 50 Equal Circles in a Square. Discrete Comput. Geom. 18, 111–120 (1997)
Ulichney, R.A.: Dithering with blue noise. Proc. IEEE 76(1), 56–79 (1988)
Ulichney, R.: The void-and-cluster method for dither array generation. In: Allebach, J.W. (ed.) IS&T/SPIE Symposium on Electronic Imaging Science and Technology, Proceedings of Conf. Human Vision, Visual Processing and Digital Display IV. SPIE, pp. 332–343 (1993)
Yao, M., Parker, K.J.: Modified approach to the construction of a blue noise mask. J. Electronic Imaging 3, 92–97 (1994)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2004 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Aronov, B., Asano, T., Kikuchi, Y., Nandy, S.C., Sasahara, S., Uno, T. (2004). A Generalization of Magic Squares with Applications to Digital Halftoning. In: Fleischer, R., Trippen, G. (eds) Algorithms and Computation. ISAAC 2004. Lecture Notes in Computer Science, vol 3341. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-30551-4_10
Download citation
DOI: https://doi.org/10.1007/978-3-540-30551-4_10
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-24131-7
Online ISBN: 978-3-540-30551-4
eBook Packages: Computer ScienceComputer Science (R0)