m-Balanced words: A generalization of balanced words

• Published in: Theoretical computer science Vol. 314; no. 1; pp. 97 - 120
• Main Authors: Sano, Shinya, Miyoshi, Naoto, Kataoka, Ryohei
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.02.2004; Elsevier
• Subjects: Algorithmics. Computability. Computer arithmetics; Applied sciences; Artificial intelligence; Balanced words; Billiard words; Computer science; control theory; systems; Computer systems and distributed systems. User interface; Exact sciences and technology; Network scheduling problems; Problem solving, game playing; Software; Sturmian conditions; Theoretical computing; Sturmian conditions; Balanced words; Billiard words; Network scheduling problems; Letter; Computer theory; Computer network; Balanced word; Finite word; Routing; Scheduling; Infinite word; Algorithm; Billiard word; Sturmian condition
• ISSN: 0304-3975; 1879-2294
• DOI: 10.1016/j.tcs.2003.11.021

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.