Generalized Semimagic Squares for Digital Halftoning

A semimagic square is a square matrix whose entries are consecutive integers and which has equal row and column sums. One way to generalize this millennia-old concept is to specify the sums on regions other than rows and columns. Ingenious constructions of squares satisfying various sum constraints have been described by many professional and amateur mathematicians. While most of them are interested in adding more and more constraints to make their squares impressive, one can generally consider sum conditions on any set of regions.

Aronov et al. [1] took up this problem for square regions: is there an n × n matrix with entries 0, . . . , n 2 -1 such that every k × k region has the same sum? It is amusing to note that this variant of the classical problem is motivated by an engineering question of finding good dither matrices for digital halftoning, a method to approximate a continuous-tone image by a binary image for printing (see their paper for details). They showed [1, Theorem 1], using what they call constant-gap matrices, that the answer is yes if k and n are even or if n is an integer power of k, and no if k and n are relatively prime or if k is odd and n is even. We will solve this problem completely by determining all (n, k) for which such matrices exist (Section 1). Our construction of the matrices is much simpler even for the cases that have already been settled positively. We also give counterexamples to Asano et al.’s conjecture on the smallest possible discrepancy when n is odd and k = 2 (Section 2).

Definitions. For a positive integer N, we write [N] = {0, 1, . . . , N -1}. The remainder when an integer x is divided by N belongs to [N] and is denoted by x mod N.

We consider the slightly generalized setting where the matrices and regions are rectangles instead of squares. Let m and n be positive integers. For an m × n matrix D and index (i, j) ∈ [m] × [n], we denote the (i, j)th entry of D by D(i, j).

The sum of the numbers on R is denoted by D(R) = (i,j)∈R D(i, j). The discrepancy of D with respect to a set R of regions is the difference between the maximum and minimum D(R) as R varies in R. When it is zero, D is said to be R-uniform.

The translate of R by (a, b) ∈ Z 2 is denoted by

The set of all translates of R is denoted by By an m × n table we mean an m × n matrix in which each element of [mn] appears exactly once. We are interested in tables with small (or zero) discrepancy with respect to [k] × [l], the set of all k-by-l rectangles (Figure 1).

The greatest common divisor of positive integers x and y is denoted by gcd(x, y). The goal of this section is to show the following: Theorem 1. Let m, n, k, l be positive integers with k < m and l < n. Let k ′ = gcd(k, m) and l ′ = gcd(l, n). Then there exists a [k] × [l]-uniform m × n table if and only if k ′ and l ′ are greater than 1 and k ′ l ′ (mn -1) is even. This is an immediate consequence of the following Lemmas 2 and 3.

where k ′ = gcd(k, m). We get the conclusion of the lemma by repeating the same argument with rows and columns switched.

For each (i, j)

all have the same sum on D.

Since the rightmost side is a constant independent of (i, j) by

Lemma 3. Let m and n be positive integers, and let k < m and l < n be their positive divisors, respectively. Then there exists a [k] × [l]-uniform m × n table if and only if k and l are greater than 1 and kl(mn -1) is even. One direction is a simple generalization of [1, Theorem 1 (b, c)]:

. Since D(R) must be an integer, the second claim follows. For the first claim, assume k = 1 for contradiction

) and hence D(0, 0) = D(0, l), contradicting the assumption that D is a table.

For the converse, we use the building blocks provided by the following lemma:

Lemma 4. Let k > 1 and l > 0 be integers and let n be a positive multiple of l. If kl(n -1) is even, then there exists a

]-uniform for any multiple l ′ of l. Therefore, it suffices to construct the desired matrix P for the cases (k, l) = (2, 1), (3, 1) and (3, 2) (Figure 2). If (k, l) = (2, 1), let

), then n is odd by the assumption; let

Proof of the “if " part of Lemma 3. We may assume without loss of generality that l(mn -1) is even. In this case, both kl(n -1) and l(m/k -1) are even, so by Lemma 4, there are a (Figure 3). Since P and T are

. By ( 7) and (8) we see that

Since P ’s rows are permutations, the first and the third equation imply that b = b ′ . Since Q’s rows are permutations, this and the second equation imply that a = a ′ .

In the above, we constructed the uniform table as a linear combination of two uniform matrices with smaller entries. This idea is due to Euler [3] who gave a construction of a semimagic square (that is, a (

)-uniform matrices called Latin squares.

In this section, we confine ourselves, as Asano et al. [2] did, to the case where k = l = 2 and m = n. Theorem 1 states that in this case a uniform table exists if and only if n is even. For odd n’s, they construct a table with discrepancy 2n, an

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