### 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 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.

Original language | English (US) |
---|---|

Pages (from-to) | 89-100 |

Number of pages | 12 |

Journal | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |

Volume | 3341 |

State | Published - 2004 |

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### ASJC Scopus subject areas

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

### Cite this

*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)*,

*3341*, 89-100.

**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

*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)*, vol. 3341, pp. 89-100.

}

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 - 2004

Y1 - 2004

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=35048813548&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=35048813548&partnerID=8YFLogxK

M3 - Article

VL - 3341

SP - 89

EP - 100

JO - Lecture Notes in Computer Science

JF - Lecture Notes in Computer Science

SN - 0302-9743

ER -