Combinatorics and algorithms for low-discrepancy roundings of a real sequence

We discuss the problem of computing all the integer sequences obtained by rounding an input sequence of n real numbers such that the discrepancy between the input sequence and each output binary sequence is less than one. The problem arises in the design of digital halftoning methods in computer graphics. We show that the number of such roundings is at most n+1 if we consider the discrepancy with respect to the set of all subintervals, and give an efficient algorithm to report all of them. Then, we give an optimal method to construct a compact graph to represent the set of global roundings satisfying a weaker discrepancy condition.