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Please use this identifier to cite or link to this item: http://hdl.handle.net/10119/4923

Title: A Generalization of Magic Squares with Applications to Digital Halftoning
Authors: Aronov, Boris
Asano, Tetsuo
Kikuchi, Yosuke
Nandy, Subhas C.
Sasahara, Shinji
Uno, Takeaki
Keywords: digital halftoning
discrepancy
latin square
magic square
matrix
Issue Date: 2008-02
Publisher: Springer
Magazine name: Theory of Computing Systems
Volume: 42
Number: 2
Start page: 143
End page: 156
DOI: 10.1007/s00224-007-9005-x
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 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.
Rights: This is the author-created version of Springer, Boris Aronov, Tetsuo Asano, Yosuke Kikuchi, Subhas C. Nandy, Shinji Sasahara and Takeaki Uno, Theory of Computing Systems, 42(2), 2008, 143-156. The original publication is available at www.springerlink.com, http://dx.doi.org/10.1007/s00224-007-9005-x
URI: http://hdl.handle.net/10119/4923
Material Type: author
Appears in Collections:b10-1. 雑誌掲載論文 (Journal Articles)

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