Circular Words in Finite and Infinite Sequences: Theory and Applications Marinella Sciortino University of Palermo, Italy AutoMathA 2015 Leipzig, May 6 – 9, 2015 M. Sciortino Circular Words in Finite and Infinite Sequences: Theory and Applications
Central topic of the talk A circular word is an equivalence class under conjugation of a finite word. M. Sciortino Circular Words in Finite and Infinite Sequences: Theory and Applications
Central topic of the talk A circular word is an equivalence class under conjugation of a finite word. Some preliminaries: Let Σ be a finite alphabet. Two finite words u , v ∈ Σ ∗ are conjugate if there exist words w 1 , w 2 such that u = w 1 w 2 and v = w 2 w 1 . Example: the words ababba and babbaa are conjugate. The conjugacy relation (denoted by ∼ ) is an equivalence over Σ ∗ , whose classes are called conjugacy classes . M. Sciortino Circular Words in Finite and Infinite Sequences: Theory and Applications
Central topic of the talk A circular word is an equivalence class under conjugation of a finite word. Some preliminaries: Let Σ be a finite alphabet. Two finite words u , v ∈ Σ ∗ are conjugate if there exist words w 1 , w 2 such that u = w 1 w 2 and v = w 2 w 1 . Example: the words ababba and babbaa are conjugate. The conjugacy relation (denoted by ∼ ) is an equivalence over Σ ∗ , whose classes are called conjugacy classes . If Σ is a total ordered alphabet, a word w is Lyndon if it is the smallest conjugate (w.r.t lexicographic order) in its conjugacy class. M. Sciortino Circular Words in Finite and Infinite Sequences: Theory and Applications
Central topic of the talk A circular word is an equivalence class under conjugation of a finite word. Some preliminaries: Let Σ be a finite alphabet. Two finite words u , v ∈ Σ ∗ are conjugate if there exist words w 1 , w 2 such that u = w 1 w 2 and v = w 2 w 1 . Example: the words ababba and babbaa are conjugate. The conjugacy relation (denoted by ∼ ) is an equivalence over Σ ∗ , whose classes are called conjugacy classes . If Σ is a total ordered alphabet, a word w is Lyndon if it is the smallest conjugate (w.r.t lexicographic order) in its conjugacy class. We denote by ( w ) the circular word corresponding to all the conjugates of the word w . We say that the circular word is primitive if the word in the conjugacy class is primitive. M. Sciortino Circular Words in Finite and Infinite Sequences: Theory and Applications
Circular words or necklaces A circular word is also called necklace and represented on a circle (read clockwise). a a a a a a a a b b b b a a a a b b b b a a b b b b a b b b Figure : The six primitive necklaces of length 5 on the alphabet { a , b } . M. Sciortino Circular Words in Finite and Infinite Sequences: Theory and Applications
Necklaces can represent real structures in several context (Single or multiple) circular structure of DNA of viruses, bacteria, eukaryotic cells, and archaea M. Sciortino Circular Words in Finite and Infinite Sequences: Theory and Applications
Necklaces can represent real structures in several context (Single or multiple) circular structure of DNA of viruses, bacteria, eukaryotic cells, and archaea Figures in computational geometry Circular structures in astronomical data ... M. Sciortino Circular Words in Finite and Infinite Sequences: Theory and Applications
Goals of the talk Necklaces from several points of view: As a tool to characterize finite words To measure the complexity of infinite words To construct indexing structures for circular matching of a pattern in a text M. Sciortino Circular Words in Finite and Infinite Sequences: Theory and Applications
How many primitive necklaces Let τ ( n , k ) be the number of primitive necklaces of length n over a k -letters alphabet. Proposition (Witt’s Formula) The number of primitive necklaces of length n on k letters is τ ( n , k ) = 1 n ∑ µ ( n / d ) k d , d | n where µ is the M¨ obius function defined by µ (1) = 1 and for n > 1 � ( − 1) i if n is the product of i distinct prime numbers µ ( n ) = 0 otherwise Example Let Σ = { a , b } . The number of primitive necklaces of length n over the alpfabet Σ is: τ ( n , 2)=2 , 1 , 2 , 3 , 6 , 9 , 18 , 30 , 56 , 99 , 186 , 335 , 630 , 1161 , 2182 , 4080 ,... M. Sciortino Circular Words in Finite and Infinite Sequences: Theory and Applications
How many necklaces Let ν ( n , k ) be the number of necklaces of length n over a k -letters alphabet. Proposition The number of necklaces of length n on k letters is ν ( n , k ) = 1 n ∑ ϕ ( n / d ) k d . d | n where ϕ is the Euler’s totient function. Example Let Σ = { a , b } . The number of necklaces of length n over the alpfabet Σ is: ν ( n , 2)=2 , 3 , 4 , 6 , 8 , 14 , 20 , 36 , 60 , 108 , 188 , 352 , 632 , 1182 , 2192 , 4116 ,... M. Sciortino Circular Words in Finite and Infinite Sequences: Theory and Applications
Some combinatorial properties of necklaces A finite word v ∈ Σ ∗ is a factor of a necklace ( w ) if v occurs in some conjugate of w . A finite word u ∈ Σ ∗ is a special factor of ( w ) if both ux and uy are factors of ( w ), with x , y ∈ Σ, x � = y . A necklace ( w ) is called balanced if for each u , v factors of ( w ), with | u | = | v | , and for each a ∈ Σ one has that || u | a −| v | a | ≤ 1. Example: ( baab ) is not balanced, ( abaab ) is balanced. Analogous definition of balanceness in a finite or infinite word. M. Sciortino Circular Words in Finite and Infinite Sequences: Theory and Applications
Some combinatorial properties of necklaces A finite word v ∈ Σ ∗ is a factor of a necklace ( w ) if v occurs in some conjugate of w . A finite word u ∈ Σ ∗ is a special factor of ( w ) if both ux and uy are factors of ( w ), with x , y ∈ Σ, x � = y . A necklace ( w ) is called balanced if for each u , v factors of ( w ), with | u | = | v | , and for each a ∈ Σ one has that || u | a −| v | a | ≤ 1. Example: ( baab ) is not balanced, ( abaab ) is balanced. Analogous definition of balanceness in a finite or infinite word. Proposition (Borel and Reutenauer, 2006) Let w be a finite word of length n ≥ 2 . The following statements are equivalent: ( w ) is primitive; 1 for k = 0 ,... n − 1 the necklace ( w ) has at least k +1 factors of 2 length k; ( w ) has n factors of length n − 1 . 3 M. Sciortino Circular Words in Finite and Infinite Sequences: Theory and Applications
Sturmian Necklaces A finite word w on a binary alphabet is called a Christoffel word if it is obtained by discretizing a segment in the lattice N × N . Given the pair of coprime integers p and q and the segment from the point 0 , 0 to the point p , q , the (lower) Christoffel word is obtained by considering the path under the segment and by coding by a a horizontal step and by b a vertical step. Such words are conjugate of standard sturmian words (used to construct infinite Sturmian words). a a 5 5 a a b b a a a a b b a a a b b a a a b b a a a b b a a 8 8 M. Sciortino Circular Words in Finite and Infinite Sequences: Theory and Applications
Sturmian Necklaces A finite word w on a binary alphabet is called a Christoffel word if it is obtained by discretizing a segment in the lattice N × N . Given the pair of coprime integers p and q and the segment from the point 0 , 0 to the point p , q , the (lower) Christoffel word is obtained by considering the path under the segment and by coding by a a horizontal step and by b a vertical step. Such words are conjugate of standard sturmian words (used to construct infinite Sturmian words). a a 5 5 a a b b a a a a b b a a a b b a a a b b a a a b b a a 8 8 A necklace is Sturmian if some word in its conjugacy class is a Christoffel word. For instance, ( baaba ) is a Sturmian necklace. M. Sciortino Circular Words in Finite and Infinite Sequences: Theory and Applications
Combinatorial Properties of Sturmian Necklaces Proposition (Jenkinson and Zamboni, 2004. Borel and Reutenauer, 2006) Let w be a word of length n ≥ 2 . The following statements are equivalent: ( w ) is a Sturmian necklace; 1 for k = 0 ,... n − 1 the necklace ( w ) has exactly k +1 factors of 2 length k; ( w ) has n − 1 factors of length n − 2 and w is primitive; 3 ( w ) is balanced. 4 Example Let us consider the Sturmian necklace ( abaababaabaab ) . One can verify the properties. M. Sciortino Circular Words in Finite and Infinite Sequences: Theory and Applications
Combinatorial Properties of Sturmian Necklaces Proposition (Castiglione, Restivo, S., 2009) The necklace ( w ) is Sturmian if and only if for each k = 0 ,..., n − 2 1 there exists a unique special factor of ( w ) of length k. If v is a Christoffel word and v R its reverse, then ( v ) = ( v R ) . 2 If ( w ) is a Sturmian necklace with | w | a > | w | b then either w = a or 3 there exists an integer p > 0 such that ( w ) is a concatenation of ba p and ba p +1 (analogously if | w | a > | w | b , by exchanging a and b). Example Let us consider the Sturmian necklace ( abaababaabaab ) . One can verify the properties. M. Sciortino Circular Words in Finite and Infinite Sequences: Theory and Applications
Necklaces and Finite Words Let Σ be a finite alphabet. Let M be the family of multisets of primitive necklaces (circular words) of Σ ∗ . Theorem ( Gessel and Reutenauer , 1993.) There exists a bijection between Σ ∗ and M. Example Let Σ = { a , b , c } . ccbbbcacaaabba M. Sciortino Circular Words in Finite and Infinite Sequences: Theory and Applications
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