Balanced words

m-Balanced words: A generalization of balanced words

Consider to construct an in5nite sequence, or an in5nite word, from a 5nite 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...

Balanced Words and majorization

Write |w| = m, the length of w, and |w|1 = card({1 ≤ i ≤ m : wi = 1}) its 1-length. Define the cyclic shift σ : {0, 1}m → {0, 1}m by σ(w1 . . . wm) = w2 . . . wmw1. A cyclic subword of w is any length-q prefix of some σi−1(w), 1 ≤ i, q ≤ m. To any word w = w1 . . . wm we associate its orbit O(w), the vector O(w) = (O1(w), . . . ,Om(w)) consisting of the iterated cyclic shifts w, σ(w), . . . , σ...

Optimizing Properties of Balanced Words

Firstly, we recall that a finite or infinite 0-1 word w = w1w2 . . . is called balanced if for every pair of finite subwords u,v such that |u| = |v|, we necessarily have ||u|1 −|v|1| ≤ 1, where |u|1 = #{ j : u j = 1} stands for the 1-length of u. An infinite balanced word which is not eventually periodic is called Sturmian. There are several equivalent definitions of Sturmian sequences. Let w =...

Critical exponents of infinite balanced words

Over an alphabet of size 3 we construct an infinite balanced word with critical exponent 2 + √ 2/2. Over an alphabet of size 4 we construct an infinite balanced word with critical exponent (5 + √ 5)/4. Over larger alphabets, we give some candidates for balanced words (found computationally) having small critical exponents. We also explore a method for proving these results using the automated t...

Characterisations of balanced words via orderings