A Generalization of Magic Squares with Applications to Digital Halftoning

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 ¢kdiscrepancy 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.

• Type: Conference
• Year: 2004
• Where: ISAAC
• Authors: Boris Aronov, Tetsuo Asano, Yosuke Kikuchi, Subhas C. Nandy, Shinji Sasahara, Takeaki Uno
• Keywords: ISAAC 2004, Semimagic Square, Square Contiguous Submatrix, ¢n Matrix