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,...,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×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)143-156
    Number of pages14
    JournalTheory of Computing Systems
    Volume42
    Issue number2
    DOIs
    StatePublished - Feb 2008

    Fingerprint

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

    Keywords

    • Digital halftoning
    • Discrepancy
    • Latin square
    • Magic square
    • Matrix

    ASJC Scopus subject areas

    • Theoretical Computer Science
    • Computational Theory and Mathematics
    • Mathematics(all)

    Cite this

    Aronov, B., Asano, T., Kikuchi, Y., Nandy, S. C., Sasahara, S., & Uno, T. (2008). A generalization of magic squares with applications to digital halftoning. Theory of Computing Systems, 42(2), 143-156. https://doi.org/10.1007/s00224-007-9005-x

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

    In: Theory of Computing Systems, Vol. 42, No. 2, 02.2008, p. 143-156.

    Research output: Contribution to journalArticle

    Aronov, B, Asano, T, Kikuchi, Y, Nandy, SC, Sasahara, S & Uno, T 2008, 'A generalization of magic squares with applications to digital halftoning', Theory of Computing Systems, vol. 42, no. 2, pp. 143-156. https://doi.org/10.1007/s00224-007-9005-x
    Aronov, Boris ; Asano, Tetsuo ; Kikuchi, Yosuke ; Nandy, Subhas C. ; Sasahara, Shinji ; Uno, Takeaki. / A generalization of magic squares with applications to digital halftoning. In: Theory of Computing Systems. 2008 ; Vol. 42, No. 2. pp. 143-156.
    @article{afa6e5a0670e4feeaf55b7c5ccfbaa2d,
    title = "A generalization of magic squares with applications to digital halftoning",
    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×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.",
    keywords = "Digital halftoning, Discrepancy, Latin square, Magic square, Matrix",
    author = "Boris Aronov and Tetsuo Asano and Yosuke Kikuchi and Nandy, {Subhas C.} and Shinji Sasahara and Takeaki Uno",
    year = "2008",
    month = "2",
    doi = "10.1007/s00224-007-9005-x",
    language = "English (US)",
    volume = "42",
    pages = "143--156",
    journal = "Theory of Computing Systems",
    issn = "1432-4350",
    publisher = "Springer New York",
    number = "2",

    }

    TY - JOUR

    T1 - A generalization of magic squares with applications to digital halftoning

    AU - Aronov, Boris

    AU - Asano, Tetsuo

    AU - Kikuchi, Yosuke

    AU - Nandy, Subhas C.

    AU - Sasahara, Shinji

    AU - Uno, Takeaki

    PY - 2008/2

    Y1 - 2008/2

    N2 - 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×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.

    AB - 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×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.

    KW - Digital halftoning

    KW - Discrepancy

    KW - Latin square

    KW - Magic square

    KW - Matrix

    UR - http://www.scopus.com/inward/record.url?scp=37749055058&partnerID=8YFLogxK

    UR - http://www.scopus.com/inward/citedby.url?scp=37749055058&partnerID=8YFLogxK

    U2 - 10.1007/s00224-007-9005-x

    DO - 10.1007/s00224-007-9005-x

    M3 - Article

    AN - SCOPUS:37749055058

    VL - 42

    SP - 143

    EP - 156

    JO - Theory of Computing Systems

    JF - Theory of Computing Systems

    SN - 1432-4350

    IS - 2

    ER -