[Paper Review] Quasiperiodicities in Fibonacci strings

Michalis Christou, Maxime Crochemore+1 more|arXiv (Cornell University)|Jan 30, 2012

semigroups and automata theory6 references3 citations

TL;DR

This paper identifies all left seeds, right seeds, seeds, and covers of finite Fibonacci strings, as well as all covers of circular Fibonacci strings, under the constraint that these quasiperiodicities must be substrings of the original string. Using structural properties of Fibonacci strings and their borders, periods, and overlaps, the authors derive exact characterizations of these quasiperiodicities through combinatorial analysis and recursive string decomposition, providing a complete classification of such structures in Fibonacci words.

ABSTRACT

We consider the problem of finding quasiperiodicities in a Fibonacci string. A factor u of a string y is a cover of y if every letter of y falls within some occurrence of u in y. A string v is a seed of y, if it is a cover of a superstring of y. A left seed of a string y is a prefix of y that it is a cover of a superstring of y. Similarly a right seed of a string y is a suffix of y that it is a cover of a superstring of y. In this paper, we present some interesting results regarding quasiperiodicities in Fibonacci strings, we identify all covers, left/right seeds and seeds of a Fibonacci string and all covers of a circular Fibonacci string.

Motivation & Objective

• To characterize all left seeds, right seeds, seeds, and covers of finite Fibonacci strings.
• To determine all covers of circular Fibonacci strings, where the circular string is formed by joining the first and last characters of a Fibonacci string.
• To provide a complete classification of quasiperiodicities in Fibonacci strings under the constraint that they must be substrings of the original string.
• To offer a new combinatorial approach distinct from prior work, particularly Iliopoulos et al. [10], for identifying covers in circular Fibonacci strings.
• To establish theoretical foundations useful for testing string algorithms and extending results to Sturmian words.

Proposed method

• Leveraging the recursive structure of Fibonacci strings, defined as $F_0 = b$, $F_1 = a$, $F_n = F_{n-1}F_{n-2}$ for $n \geq 2$, to analyze quasiperiodicities.
• Using properties of borders, periods, and overlaps in Fibonacci strings to determine when a factor can serve as a cover or seed.
• Applying superposition techniques with overlap lengths to model how substrings can cover superstrings of Fibonacci strings.
• Analyzing left and right seeds as prefixes and suffixes that cover extended strings of the form $yv$ and $vy$, respectively.
• Characterizing seeds via their ability to cover superstrings of the form $xF_ny$, where $x$ is a suffix and $y$ is a prefix of $F_n$, and verifying coverage using recursive Fibonacci decompositions.
• Deriving closed-form descriptions of all valid covers and seeds by considering all possible overlaps and length constraints, particularly for $m \in \{3, \dots, n-1\}$.

Experimental results

Research questions

• RQ1 What are all the left seeds of a Fibonacci string $F_n$?
• RQ2 What are all the right seeds of a Fibonacci string $F_n$?
• RQ3 What are all the covers of a Fibonacci string $F_n$?
• RQ4 What are all the seeds of a Fibonacci string $F_n$?
• RQ5 What are all the covers of a circular Fibonacci string $C(F_n)$, formed by joining the first and last character of $F_n$?

Key findings

• All left seeds of $F_n$ are prefixes of the form $F_n[1\mathinner{.}.|F_k|+i]$ for $k \in \{3, \dots, n-1\}$ and $i \in \{0, \dots, |F_{k-1}|-2\}$, provided they cover appropriate superstrings via overlap.
• All right seeds of $F_n$ that are covers of $C(F_n)$ are limited to the covers of $F_n$ itself, as other right seeds fail to cover extended strings beyond their length.
• All covers of $C(F_n)$ include $F_n$ for $n \in \{0,1,2,3\}$, $F_n$ and $F_{n-1}$ for $n=4$, and additional covers of the form $F_m x$, $xF_my$, and $xF_{m-1}F_my$ for $m \in \{3, \dots, n-1\}$, with specific length and overlap constraints.
• Seeds of the form $xF_my$ with $x$ a suffix of $F_m$, $y$ a prefix of $F_{m-1}$, and $|x|+|y| \geq |F_m|$, are valid covers of $C(F_n)$ if they cover superstrings of $F_n$ via appropriate overlaps.
• For $n \geq 5$, the covers of $C(F_n)$ include $F_n$, $F_{n-1}$, and various combinations involving $F_m$ and $F_{m-1}$ with partial overlaps, provided the total length of the seed and its overlap satisfies $|x|+|y| \geq |F_m|$.
• The paper proves that $F_n[1\mathinner{.}.|F_n|-1]$ is not a cover of $C(F_n)$, as it fails to cover a prefix of $F_nF_n$ longer than $|F_n|-1$, due to structural limitations in Fibonacci string expansion.