Subword Complexity Research Articles

On the asymptotic abelian complexity of morphic words

The subword complexity of an infinite word counts the number of subwords of a given length, while the abelian complexity counts this number up to letter permutation. Although a lot of research has been done on the subword complexity of morphic words, i.e., words obtained as fixed points of iterated morphisms, little is known on their abelian complexity. In this paper, we undertake the classification of the asymptotic growths of the abelian complexities of fixed points of binary morphisms. Some general results we obtain stem from the concept of factorization of morphisms. We give an algorithm that yields all canonical factorizations of a given morphism, describe how to use it to check quickly whether a binary morphism is Sturmian, discuss how to fully factorize the Parry morphisms, and finally derive a complete classification of the abelian complexities of fixed points of uniform binary morphisms.

Denominator vectors and compatibility degrees in cluster algebras of finite type

We present two simple descriptions of the denominator vectors of the cluster variables of a cluster algebra of finite type, with respect to any initial cluster seed: one in terms of the compatibility degrees between almost positive roots defined by S. Fomin and A. Zelevinsky, and the other in terms of the root function of a certain subword complex. These descriptions only rely on linear algebra. They provide two simple proofs of the known fact that the d d -vector of any non-initial cluster variable with respect to any initial cluster seed has non-negative entries and is different from zero.

Subword complexes and 2-truncated cubes

For a Coxeter element $c$ of a finite Coxeter group, we consider a family of subword complexes parameterized by reduced expressions of the longest element. This family generalizes $c-$cluster complexes. We describe vertices of these complexes in terms of roots of the corresponding root system. We prove that dual polytopes of all such complexes are combinatorial 2-truncated cubes.

Bott-Samelson Varieties, Subword Complexes and Brick Polytopes

Bott-Samelson varieties factor the flag variety $G/B$ into a product of $\mathbb{C}\mathbb{P}^1$'s with a map into $G/B$. These varieties are mostly studied in the case in which the map into $G/B$ is birational; however in this paper we study fibers of this map when it is not birational. We will see that in some cases this fiber is a toric variety. In order to do so we use the moment map of a Bott-Samelson variety to translate this problem into a purely combinatorial one in terms of a subword complex. These simplicial complexes, defined by Knutson and Miller, encode a lot of information about reduced words in a Coxeter system. Pilaud and Stump realized certain subword complexes as the dual to the boundary of a polytope which generalizes the brick polytope defined by Pilaud and Santos. For a nice family of words, the brick polytope is the generalized associahedron realized by Hohlweg and Lange. These stories connect in a nice way: the moment polytope of a fiber of the Bott-Samelson map is the Brick polytope. In particular, we give a nice description of the toric variety of the associahedron.

Subword complexity and Sturmian colorings of regular trees

AbstractIn this article, we discuss subword complexity of colorings of regular trees. We characterize colorings of bounded subword complexity and study Sturmian colorings, which are colorings of minimal unbounded subword complexity. We classify Sturmian colorings using their type sets. We show that any Sturmian coloring is a lifting of a coloring on a quotient graph of the tree which is a geodesic or a ray, with loops possibly attached, thus a lifting of an ‘infinite word’. We further give a complete characterization of the quotient graph for eventually periodic colorings.

Extremal sequences of polynomial complexity

AbstractThe joint spectral radius of a bounded set of d × d real matrices is defined to be the maximum possible exponential growth rate of products of matrices drawn from that set. For a fixed set of matrices, a sequence of matrices drawn from that set is called extremal if the associated sequence of partial products achieves this maximal rate of growth. An influential conjecture of J. Lagarias and Y. Wang asked whether every finite set of matrices admits an extremal sequence which is periodic. This is equivalent to the assertion that every finite set of matrices admits an extremal sequence with bounded subword complexity. Counterexamples were subsequently constructed which have the property that every extremal sequence has at least linear subword complexity. In this paper we extend this result to show that for each integer p ≥ 1, there exists a pair of square matrices of dimension 2p(2p+1 − 1) for which every extremal sequence has subword complexity at least 2−p2np.

A note on constructing infinite binary words with polynomial subword complexity

Most of the constructions of infinite words having polynomial subword complexity are quite complicated, e.g. , sequences of Toeplitz, sequences defined by billiards in the cube, etc. In this paper, we describe a simple method for constructing infinite words w over a binary alphabet { a,b } with polynomial subword complexity p w . Assuming w contains an infinite number of a ’s, our method is based on the gap function which gives the distances between consecutive b ’s. It is known that if the gap function is injective, we can obtain at most quadratic subword complexity, and if the gap function is blockwise injective, we can obtain at most cubic subword complexity. Here, we construct infinite binary words w such that p w (n ) = Θ (n β ) for any real number β > 1.

Subword complexes, cluster complexes, and generalized multi-associahedra

In this paper, we use subword complexes to provide a uniform approach to finite-type cluster complexes and multi-associahedra. We introduce, for any finite Coxeter group and any nonnegative integer k, a spherical subword complex called multi-cluster complex. For k=1, we show that this subword complex is isomorphic to the cluster complex of the given type. We show that multi-cluster complexes of types A and B coincide with known simplicial complexes, namely with the simplicial complexes of multi-triangulations and centrally symmetric multi-triangulations, respectively. Furthermore, we show that the multi-cluster complex is universal in the sense that every spherical subword complex can be realized as a link of a face of the multi-cluster complex.

EL-labelings and canonical spanning trees for subword complexes

We describe edge labelings of the increasing flip graph of a subword complex on a finite Coxeter group, and study applications thereof. On the one hand, we show that they provide canonical spanning trees of the facet-ridge graph of the subword complex, describe inductively these trees, and present their close relations to greedy facets. Searching these trees yields an efficient algorithm to generate all facets of the subword complex, which extends the greedy flip algorithm for pointed pseudotriangulations. On the other hand, when the increasing flip graph is a Hasse diagram, we show that the edge labeling is indeed an EL-labeling and derive further combinatorial properties of paths in the increasing flip graph. These results apply in particular to Cambrian lattices, in which case a similar EL-labeling was recently studied by M. Kallipoliti and H. Mühle.

Ancestors graph and an upper bound for the subword complexity function

A general upper bound for the subword complexity function of a fixed point of a primitive substitution is presented. The approach relies on the introduction and discussion of the ancestors graph of a substitution. More precisely, we characterize the language of a fixed point of a primitive substitution as the strongly connected component of the letters in its ancestors graph; we propose algorithms for the local construction of this graph; we give an upper bound for the subword complexity function in terms of the Hilbert series of some quotient of a noncommutative polynomial ring by a sum of automatic bilateral ideals.

ALGORITHMIC COMBINATORICS ON PARTIAL WORDS

Algorithmic combinatorics on partial words, or sequences of symbols over a finite alphabet that may have some do-not-know symbols or holes, has been developing in the past few years. Applications can be found, for instance, in molecular biology for the sequencing and analysis of DNA, in bio-inspired computing where partial words have been considered for identifying good encodings for DNA computations, and in data compression. In this paper, we focus on two areas of algorithmic combinatorics on partial words, namely, pattern avoidance and subword complexity. We discuss recent contributions as well as a number of open problems. In relation to pattern avoidance, we classify all binary patterns with respect to partial word avoidability, we classify all unary patterns with respect to hole sparsity, and we discuss avoiding abelian powers in partial words. In relation to subword complexity, we generate and count minimal Sturmian partial words, we construct de Bruijn partial words, and we construct partial words with subword complexities not achievable by full words (those without holes).

Generalized associahedra via brick polytopes

We generalize the brick polytope of V. Pilaud and F. Santos to spherical subword complexes for finite Coxeter groups. This construction provides polytopal realizations for a certain class of subword complexes containing all cluster complexes of finite types. For the latter, the brick polytopes turn out to coincide with the known realizations of generalized associahedra, thus opening new perspectives on these constructions. This new approach yields in particular the vertex description and a relevant Minkowski sum decomposition of generalized associahedra.

Sublinear Algorithms for Approximating String Compressibility

We raise the question of approximating the compressibility of a string with respect to a fixed compression scheme, in sublinear time. We study this question in detail for two popular lossless compression schemes: run-length encoding (RLE) and a variant of Lempel-Ziv (LZ77), and present sublinear algorithms for approximating compressibility with respect to both schemes. We also give several lower bounds that show that our algorithms for both schemes cannot be improved significantly. Our investigation of LZ77 yields results whose interest goes beyond the initial questions we set out to study. In particular, we prove combinatorial structural lemmas that relate the compressibility of a string with respect to LZ77 to the number of distinct short substrings contained in it (its ℓth subword complexity , for small ℓ). In addition, we show that approximating the compressibility with respect to LZ77 is related to approximating the support size of a distribution.

Constructing partial words with subword complexities not achievable by full words

Partial words are sequences over a finite alphabet that may contain wildcard symbols, called holes, which match, or are compatible with, all letters in the alphabet ((full) words are just partial words without holes). The subword complexity function of a partial word w over a finite alphabet A assigns to each positive integer, n, the number, pw(n), of distinct full words over A that are compatible with factors of length n of w. In this paper, with the help of our so-called hole functions, we construct infinite partial words w such that pw(n)=Θ(nα) for any real number α>1. In addition, these partial words have the property that there exist infinitely many non-negative integers m satisfying pw(m+1)−pw(m)≥mα. Combining these results with earlier ones on full words, we show that this represents a class of subword complexity functions not achievable by full words. We also construct infinite partial words with intermediate subword complexity, that is, between polynomial and exponential.

Generalized associahedra via brick polytopes

We generalize the brick polytope of V. Pilaud and F. Santos to spherical subword complexes for finite Coxeter groups. This construction provides polytopal realizations for a certain class of subword complexes containing all cluster complexes of finite types. For the latter, the brick polytopes turn out to coincide with the known realizations of generalized associahedra, thus opening new perspectives on these constructions. This new approach yields in particular the vertex description and a relevant Minkowski sum decomposition of generalized associahedra.

Eulerian entropy and non-repetitive subword complexity

We consider continuous self-maps of compact metric spaces, and for each point of the space we define the notion of eulerian entropy by considering the exponential growth rate of complexity in the initial chunks of the orbit of the point. We show that eulerian entropy is constant on a residual subset for transitive dynamical systems. For elements in the shift dynamical system we define an equivalent notion named non-repetitive subword complexity, and show that for a large class of mixing subshifts of finite type, the set of points for which the non-repetitive subword complexity is equal to the topological entropy is residual. If f is either a transitive interval map or an infinite transitive subshift of finite type, we establish that there is t ∈ N such that the eulerian entropy of f t is a positive constant that is attained on a residual set of points.

Recurrent Partial Words

Partial words are sequences over a finite alphabet that may contain wildcard symbols, called holes, which match or are compatible with all letters; partial words without holes are said to be full words (or simply words). Given an infinite partial word w, the number of distinct full words over the alphabet that are compatible with factors of w of length n, called subwords of w, refers to a measure of complexity of infinite partial words so-called subword complexity. This measure is of particular interest because we can construct partial words with subword complexities not achievable by full words. In this paper, we consider the notion of recurrence over infinite partial words, that is, we study whether all of the finite subwords of a given infinite partial word appear infinitely often, and we establish connections between subword complexity and recurrence in this more general framework.

Subword complexity of uniform D0L words over finite groups

We deal with the subword complexity of uniform D0L words obtained from group substitutions. Our main interest is whether the subword complexity is “almost proportional” to the length of the factor. We find necessary and sufficient conditions for that. For some cases we show that this is impossible.

On minimal Sturmian partial words

Partial words, which are sequences that may have some undefined positions called holes, can be viewed as sequences over an extended alphabet A ♢ = A ∪ { ♢ } , where ♢ stands for a hole and matches (or is compatible with) every letter in A . The subword complexity of a partial word w , denoted by p w ( n ) , is the number of distinct full words (those without holes) over the alphabet that are compatible with factors of length n of w . A function f : N → N is ( k , h ) - feasible if for each integer N ≥ 1 , there exists a k -ary partial word w with h holes such that p w ( n ) = f ( n ) for all n such that 1 ≤ n ≤ N . We show that when dealing with feasibility in the context of finite binary partial words, the only affine functions that need investigation are f ( n ) = n + 1 and f ( n ) = 2 n . It turns out that both are ( 2 , h ) -feasible for all non-negative integers h . We classify all minimal partial words with h holes of order N with respect to f ( n ) = n + 1 , called Sturmian, computing their lengths as well as their numbers, except when h = 0 in which case we describe an algorithm that generates all minimal Sturmian full words. We show that up to reversal and complement, any minimal Sturmian partial word with one hole is of the form a i ♢ a j b a l , where i , j , l are integers satisfying some restrictions, that all minimal Sturmian partial words with two holes are one-periodic, and that up to complement, ♢ ( a N − 1 ♢ ) h − 1 is the only minimal Sturmian partial word with h ≥ 3 holes. Finally, we give upper bounds on the lengths of minimal partial words with respect to f ( n ) = 2 n , showing them tight for h = 0 , 1 or 2 .

An improvement of subword complexity

In this article we propose a simple method to estimate the complexity of a finite word written over a finite alphabet. We use the notion of subword complexity (which is equal to the number of different subwords in the word) as a starting point and show the computation difficulties connected with the usage of subword complexity. To avoid them we propose a new simple measure of a word's complexity, which turned out to be not only easy to compute, but also more precise than classical subword complexity.