The maximal number of cubic runs in a word

Abstract

International audienceA run is an inclusion maximal occurrence in a word (as a subinterval) of a factor in which the period repeats at least twice. The maximal number of runs in a word of length n has been thoroughly studied, and is known to be between 0.944n and 1.029n. The proofs are very technical. In this paper we investigate cubic runs, in which the period repeats at least three times. We show the upper bound on their maximal number, cubic-runs(n), in a word of length n: cubic-runs(n) < 0.5n. The proof of linearity of cubic-runs(n) utilizes only simple properties of Lyndon words and is considerably simpler than the corresponding proof for general runs. For binary words, we provide a better upper bound cubic-runs2(n) < 0.48n which requires computer-assisted verification of a large number of cases. We also construct an infinite sequence of words over a binary alphabet for which the lower bound is 0.41n

The “Runs” Theorem

“…Despite the simplicity of its definition, Lyndon words have many deep and interesting combinatorial properties [36] and have been applied to a wide range of problems [36,43,35,15,5,19,34,4,27,41,18]. Lyndon words have recently been considered in the context of runs [12,13], since any run with period p must contain a length-p substring that is a Lyndon word, called an L-root of the run. Concerning the number of cubic runs (runs with exponent at least 3), Crochemore et al [12] gave a very simple proof that it can be no more than 0.5n.…”

On the density of Lyndon roots in factors

“…The last statement is proved in [6] employing the notion of Critical position, which is discussed in the next section.…”

“…Therefore, since a run has length at least twice as long as its root, the first occurrence of its Lyndon root is followed by its first letter. This notion of Lyndon root is the basis of the proof of the 0.5n upper bound on the number of cubic runs given in [6]. Recall that a run is said to be cubic if its length is at least three times larger than its period.…”

Cartesian and Lyndon trees

“…They also have several interesting algorithmic applications and attracted much interest in connection with the detection of runs (maximal periodicities) in words. The notion of Lyndon roots of runs, introduced for cubic runs in [5], has led to the property that there is linear number of square runs in a word. Originally conjectured by Kolpakov and Kucherov [11], it has eventually been proved by Bannai et al [1].…”

“…Note the Lyndon factorisation of y can be recovered by following the longest decreasing sequence of ranks from the first rank. It is (7, 4, 3, 1, 0) in the above example, corresponding to positions (0, 3,5,7,15) and to the Lyndon factorisation abb · ab · ab · aababbab · a.…”