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A Generalization of Magic Squares with Applications to Digital Boris Aronov1 , Tetsuo Asano2 , Yosuke Kikuchi3, Subhas C. Nandy4,
 

Summary: A Generalization of Magic Squares with Applications to Digital
Halftoning
Boris Aronov1 , Tetsuo Asano2 , Yosuke Kikuchi3, Subhas C. Nandy4,
Shinji Sasahara5, and Takeaki Uno6
1 Polytechnic University, Brooklyn, NY 11201-3840, USA, http://cis.poly.edu/~aronov
2 JAIST, Nomi, 923-1292 Japan, t-asano@jaist.ac.jp
3 ERATO QCI Project, JST, Tokyo 113-0033, Japan, kikuchi@qci.jst.go.jp
4 Indian Statistical Institute, Kolkata 700 108, India, nandysc@isical.ac.in
5 Fuji Xerox Co., Ltd., Kanagawa 259-0157, Japan. shinji.sasahara@fujixerox.co.jp
6 National Institute of Informatics (NII), Tokyo, 101-8430 Japan, uno@nii.jp
Abstract. A semimagic square of order n is an n n matrix containing the integers 0 n2 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 kk 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 km, 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.

  

Source: Aronov, Boris - Department of Computer Science and Engineering, Polytechnic Institute of New York University

 

Collections: Computer Technologies and Information Sciences