### 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 language | English (US) |
---|---|

Pages (from-to) | 143-156 |

Number of pages | 14 |

Journal | Theory of Computing Systems |

Volume | 42 |

Issue number | 2 |

DOIs | |

State | Published - Feb 2008 |

### Fingerprint

### Keywords

- Digital halftoning
- Discrepancy
- Latin square
- Magic square
- Matrix

### ASJC Scopus subject areas

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

### Cite this

*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.

Research output: Contribution to journal › Article

*Theory of Computing Systems*, vol. 42, no. 2, pp. 143-156. https://doi.org/10.1007/s00224-007-9005-x

}

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

VL - 42

SP - 143

EP - 156

JO - Theory of Computing Systems

JF - Theory of Computing Systems

SN - 1432-4350

IS - 2

ER -