Abstract

Consider to construct an infinite sequence, or an infinite word, from a finite set of letters such as each letter is distributed with “good balance,” that is, as evenly as possible, when the densities of letters are provided. Such words have been applied to many scheduling and routing problems in various areas. Concerning the balancedness of words, the notions of regularity and balanced words have been exploited. However, it is known that there does not always exist a balanced word for given densities of letters. In this paper, we introduce a new notion called m-balanced words, which gives a measure of “well balancedness” for any words with any densities of letters. We derive some properties of m-balanced words and give a set of algorithms generating well balanced words. We further give a few examples of applications to simple network scheduling problems.

Author Keywords: Balanced words; Sturmian conditions; Billiard words; Network scheduling problems