On the Maximal Number of Cubic Runs in a String
نویسندگان
چکیده
A run is an inclusion maximal occurrence in a string (as a subinterval) of a factor in which the period repeats at least twice. The maximal number of runs in a string 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 string 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 strings, 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 binary alphabet for which the lower bound is 0.41n. 1
منابع مشابه
Maximal-exponent factors in strings
The exponent of a string is the quotient of its length over its smallest period. The exponent and the period of a string can be computed in time proportional to the string length. We design an algorithm to compute the maximal exponent of all factors of an overlap-free string. Our algorithm runs in lineartime on a fixed-size alphabet, while a naive solution of the question would run in cubic tim...
متن کاملOn the maximal number of highly periodic runs in a string
A run is a maximal occurrence of a repetition v with a period p such that 2p ≤ |v|. The maximal number of runs in a string of length n was studied by several authors and it is known to be between 0.944n and 1.029n. We investigate highly periodic runs, in which the shortest period p satisfies 3p ≤ |v|. We show the upper bound 0.5n on the maximal number of such runs in a string of length n and co...
متن کاملThe Number of Runs in a String: Improved Analysis of the Linear Upper Bound
A run (or a maximal repetition) in a string is an inclusionmaximal periodic segment in a string. Let ρ(n) be the maximal number of runs in a string of length n. It has been shown in [8] that ρ(n) = O(n), the proof was very complicated and the constant coefficient in O(n) has not been given explicitly. We propose a new approach to the analysis of runs based on the properties of subperiods: the p...
متن کاملThe "Runs" Theorem
We give a new characterization of maximal repetitions (or runs) in strings based on Lyndon words. The characterization leads to a proof of the so-called runs conjecture (Kolpakov & Kucherov (FOCS ’99)), which states that the maximum number of runs ρ(n) in a string of length n is less than n. The proof is remarkably simple, considering the numerous endeavors to tackle this problem in the last 15...
متن کاملEfficient computation of maximal anti-exponent in palindrome-free strings
A palindrome is a string x = a1 · · · an which is equal to its reversal x̃ = an · · · a1. We consider gapped palindromes which are strings of the form uvũ, where u, v are strings, |v| ≥ 2, and ũ is the reversal of u. Replicating the standard notion of string exponent, we define the antiexponent of a gapped palindrome uvũ as the quotient of |uvũ| by |uv|. To get an efficient computation of maxima...
متن کامل