A generalization of magic squares with applications to digital halftoning

Boris Aronov, Tetsuo Asano, Yosuke Kikuchi, Subhas C. Nandy, Shinji Sasahara, Takeaki Uno

    Research output: Contribution to journalArticle

    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 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 = 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.

    Original languageEnglish (US)
    Pages (from-to)89-100
    Number of pages12
    JournalLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
    Volume3341
    StatePublished - 2004

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    Digital Halftoning
    Magic
    Magic square
    Relatively prime
    Integer
    Discrepancy
    Generalization
    Efficient Algorithms
    Generalise
    Requirements
    Zero

    ASJC Scopus subject areas

    • Computer Science(all)
    • Biochemistry, Genetics and Molecular Biology(all)
    • Theoretical Computer Science

    Cite this

    A generalization of magic squares with applications to digital halftoning. / Aronov, Boris; Asano, Tetsuo; Kikuchi, Yosuke; Nandy, Subhas C.; Sasahara, Shinji; Uno, Takeaki.

    In: Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), Vol. 3341, 2004, p. 89-100.

    Research output: Contribution to journalArticle

    Aronov, Boris ; Asano, Tetsuo ; Kikuchi, Yosuke ; Nandy, Subhas C. ; Sasahara, Shinji ; Uno, Takeaki. / A generalization of magic squares with applications to digital halftoning. In: Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). 2004 ; Vol. 3341. pp. 89-100.
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    AU - Sasahara, Shinji

    AU - Uno, Takeaki

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