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) |
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Pages (from-to) | 143-156 |
Number of pages | 14 |
Journal | Theory of Computing Systems |
Volume | 42 |
Issue number | 2 |
DOIs | |
State | Published - Feb 2008 |
Keywords
- Digital halftoning
- Discrepancy
- Latin square
- Magic square
- Matrix
ASJC Scopus subject areas
- Theoretical Computer Science
- Computational Theory and Mathematics