Avoiding approximate repetitions with respect to the longest common subsequence distance

Abstract

Ochem, Rampersad, and Shallit gave various examples of infinite words avoiding what they called approximate repetitions. An approximate repetition is a factor of the form xx ′ , where x and x ′ are close to being identical. In their work, they measured the similarity of x and x ′ using either the Hamming distance or the edit distance. In this paper, we show the existence of words avoiding approximate repetitions, where the measure of similarity between adjacent factors is based on the length of the longest common subsequence. Our principal technique is the so-called "entropy compression" method, which has its origins in Moser and Tardos's algorithmic version of the Lovász local lemma.(2010 Mathematics Subject Classification: 68R15)(k−1) 2 to be o(k M ). We can disregard the term k n+2 M √ πM (k−1) 2 , as it approaches 0 as M

Moser–Tardos resampling algorithm, entropy compression method and the subset gas

“…Esperet and Parreau suggested, through examples and applications, that their algorithm could be adapted to treat most of the applications in graph coloring problems covered by the LLL. Indeed, this was confirmed in several successive papers [8,14,15,25,27,40,43,44,46], where the Esperet-Parreau scheme was applied to various graph coloring problems and beyond, generally improving previous results obtained via the LLL/CELL (sometimes the improvement is more noticeable, sometimes less). However, in all papers mentioned above, the Esperet-Parreau algorithmic scheme, usually called entropy compression method (the name is probably due to Tao [49]), has been commonly utilized as a set of ad hoc instructions to be implemented on a case-by-case basis.…”

Moser-Tardos resampling algorithm, entropy compression method and the subset gas

“…Esperet and Parreau suggested, through further examples and applications, that their algorithm could be adapted to treat most of the applications in graph coloring problems covered by the LLL. Indeed, this was confirmed in several successive papers [26,42,39,43,15,14,45,24,8], where the Esperet-Parreau scheme has been applied to various graph coloring problems and beyond, generally improving previous results obtained via the LLL/CELL (sometimes the improvement is more sensible, sometimes less). However, in all papers mentioned above the Esperet-Parreau algorithmic scheme, usually called entropy compression method (the name is probably due to Tao [48]), has been commonly utilized as a set of ad hoc instructions to be implemented on a case-by-case basis.…”

Synchronization Strings: Highly Efficient Deterministic Constructions over Small Alphabets

“…Finally, in Appendix A, we provide experimental evidence for the existence of synchronization strings over ternary alphabets by finding lower-bounds for ε for which ε-synchronization strings over alphabets of size 3, 4, 5, and 6 might exist. Similar experiments have been provided for related combinatorial objects in the previous work [20,5].…”

“…Finally, Camungol and Rampersad [5] study approximate squares with respect to edit distance, which is equivalent to the ε-synchronization string notion except that the edit distance property is only required to hold for pairs of consecutive substrings of equal length. [5] employs a technique based on entropy compression to prove that such strings exist over alphabets that are constant in terms of string length but exponentially large in terms of ε −1 . We note that the previous result of Haeupler and Shahrasbi [15] already improves this dependence to O(ε −4 ).…”

“…One interesting question that has been commonly addressed by previous work on similar combinatorial objects is the size of the smallest alphabet over which one can find such objects. Along the lines of [26,28,20,5], we study the existence of synchronization strings over alphabets with minimal constant size.…”