Parikh Vector Research Articles

Primitivity and Hurwitz Primitivity of Nonnegative Matrix Tuples: A Unified Approach

For an -tuple of nonnegative matrices , primitivity/Hurwitz primitivity means the existence of a positive product/Hurwitz product, respectively (all products are with repetitions permitted). The Hurwitz product with a Parikh vector is the sum of all products with multipliers , . Ergodicity/Hurwitz ergodicity means the existence of the corresponding product with a positive row. We give a unified proof for the Protasov–Vonyov characterization (2012) of primitive tuples of matrices without zero rows and columns and for the Protasov characterization (2013) of Hurwitz primitive tuples of matrices without zero rows. By establishing a connection with synchronizing automata, we, under the aforementioned conditions, find an -time algorithm to decide primitivity and an -time algorithm to construct a Hurwitz primitive vector of weight . We also report results on ergodic and Hurwitz ergodic matrix tuples.

2D Words and Generalized Parikh Matrices

The extension of Parikh vector is Parikh matrix which is a useful tool in arithmetizing words by numbers. Another interesting problem is the extension of Parikh matrix of a words to 2D words. However, Parikh matrix fails to tell about the number of subword occurrences in 2D word. Therefore, in this paper we introduce the notion of generalized Parikh matrix of 2D words which fulfill the number of subwords occurrences in an 2D words and also we obtain the properties of the generalized Parikh matrices.

Surface and Curve Languages

In the area of combinatorial studies of string languages under formal grammars, the concept of Generalized Parikh Vector (GPV) gives the positions of symbols in linear strings. It has been proved that GPVs of the strings of the same length lie on a hyper plane. The studies on GPV of strings over a binary alphabet gave rise to the concept of line languages. The concept has been extended to surface languages in this paper.

The geometric properties of an infinitary line and plane languages

Formal grammar is an area of discrete mathematics dealing with syntactic techniques of generation of words and their properties. The concept of Parikh vector (PV) and Generalized Parikh vector (GPV) play important role in the combinatorial properties of words under formal grammars. The concept of Generalized Parikh Vector introduced by Siromoney et al. gives the positions of symbols in linear strings. It has been proved that GPVs of the strings of the same length lie on a hyper plane. As any formal language over binary letters are represented geometrically, the study done in this paper explore geometrical properties of lines induced by corresponding languages. The study is extended over three alphabet set giving rise to plane objects.

Enhanced string factoring from alphabet orderings

In this note we consider the concept of alphabet ordering in the context of string factoring. We propose a greedy algorithm that produces Lyndon factorizations with small numbers of factors which can be modified to produce large numbers of factors. For the technique we introduce the Exponent Parikh vector. Applications and research directions derived from circ-UMFFs are discussed.

On prefix normal words and prefix normal forms

A 1-prefix normal word is a binary word with the property that no factor has more 1s than the prefix of the same length; a 0-prefix normal word is defined analogously. These words arise in the context of indexed binary jumbled pattern matching, where the aim is to decide whether a word has a factor with a given number of 1s and 0s (a given Parikh vector). Each binary word has an associated set of Parikh vectors of the factors of the word. Using prefix normal words, we provide a characterization of the equivalence class of binary words having the same set of Parikh vectors of their factors.We prove that the language of prefix normal words is not context-free and is strictly contained in the language of pre-necklaces, which are prefixes of powers of Lyndon words. We give enumeration results on pnw(n), the number of prefix normal words of length n, showing that, for sufficiently large n,2n−4nlg⁡n≤pnw(n)≤2n−lg⁡n+1.For fixed density (number of 1s), we show that the ordinary generating function of the number of prefix normal words of length n and density d is a rational function. Finally, we give experimental results on pnw(n), discuss further properties, and state open problems.

Aperiodic pseudorandom number generators based on infinite words

In this paper we study how certain families of aperiodic infinite words can be used to produce aperiodic pseudorandom number generators (PRNGs) with good statistical behavior. We introduce the well distributed occurrences (WELLDOC) combinatorial property for infinite words, which guarantees absence of the lattice structure defect in related pseudorandom number generators. An infinite word u on a d-ary alphabet has the WELLDOC property if, for each factor w of u, positive integer m, and vector v∈Zmd, there is an occurrence of w such that the Parikh vector of the prefix of u preceding such occurrence is congruent to v modulo m. (The Parikh vector of a finite word v over an alphabet A has its i-th component equal to the number of occurrences of the i-th letter of A in v.) We prove that Sturmian words, and more generally Arnoux–Rauzy words and some morphic images of them, have the WELLDOC property. Using the TestU01 [11] and PractRand [5] statistical tests, we moreover show that not only the lattice structure is absent, but also other important properties of PRNGs are improved when linear congruential generators are combined using infinite words having the WELLDOC property.

Efficient Indexes for Jumbled Pattern Matching with Constant-Sized Alphabet

We introduce efficient indexes for a problem in non-standard stringology: jumbled pattern matching. An index is a data structure constructed for a text of length n over an alphabet of size sigma that can answer queries asking if the text contains a fragment which is jumbled (Abelian) equivalent to a pattern, specified by its so-called Parikh vector. We denote the length of the pattern by m. Moosa and Rahman (J Discrete Algorithms 10:5–9, 2012) gave an index for the case of binary alphabets with mathcal {O}left( frac{n^2}{(log n)^2}right) -time construction in the word-RAM model. Several earlier papers stated as an open problem the existence of an efficient solution for larger alphabets. In this paper we develop an index for any constant-sized alphabet. The construction involves a trade-off parameter, which in particular lets us achieve the following complexities: mathcal {O}(n^{2-delta }) space and mathcal {O}(m^{(2sigma -1)delta }) query time for any 0<delta <1, or mathcal {O}left( frac{n^2 (log log n)^2}{log n}right) space and polylogarithmic, o(log ^{2sigma -1} m), query time. The construction time in both cases is subquadratic: mathcal {O}left( frac{n^2 (log log n)^2}{log n}right) in the word-RAM model (using bit-parallelism). Our construction algorithms are randomized (Las Vegas, running time w.h.p.), which is due to the usage of perfect hashing. On the other hand, all queries are answered deterministically. A preliminary version of this work appeared at ESA 2013 (Kociumaka et al. in Algorithms, ESA 2013. LNCS, vol 8125. Springer, Berlin, pp. 625–636, 2013). Here we improve it in several ways. We achieve mathcal {O}(n^2)-time construction of the index with mathcal {O}(n^{2-delta }) space and mathcal {O}(m^{(2sigma -1)delta }) query time, which was not present in the preliminary version. We also extend the index so that the position of the leftmost occurrence of the query pattern is provided at no additional cost in the complexity; this required rather nontrivial changes in the construction algorithm.

Generalized Parikh Prime 2D Binary Arrays

The words over finite alphabets have been studied using various properties. One such branch of study that deals with numerical properties of words includes Parikh vector and the Parikh matrices of words. The Generalized Parikh vectors and Generalized Parikh matrices of words have added more clarity to this. The extension of these to 2D binary arrays has been of recent interest and this paper throws light on the Parikh prime and Generalized Parikh prime 2D binary arrays.

An abelian periodicity lemma

We write x≺y when x and y are vectors with each element of x less than or equal to the corresponding element of y and P(w) for the Parikh vector of a word w. A word w has abelian period p if it has the form u0u1⋯umum+1 with |u0|≤p, |ui|=p for i=1,…m and |um+1|≤p, and P(u0)≺P, P(u0)=P for i=1,…,m and P(um+1)≺P for some vector P. Blanchet-Sadri et al. conjectured that if a word has abelian periods pd and qd, where gcd⁡(p,q)=1, and length at least 2pqd−1 then the number of distinct letters appearing in the word is at most d, and that under certain conditions the bound may be reduced to 2pqd−2. We prove their conjecture.

Fast computation of abelian runs

Given a word w and a Parikh vector P, an abelian run of period P in w is a maximal occurrence of a substring of w having abelian period P. Our main result is an online algorithm that, given a word w of length n over an alphabet of cardinality σ and a Parikh vector P, returns all the abelian runs of period P in w in time O(n) and space O(σ+p), where p is the norm of P, i.e., the sum of its components. We also present an online algorithm that computes all the abelian runs with periods of norm p in w in time O(np), for any given norm p. Finally, we give an O(n2)-time offline randomized algorithm for computing all the abelian runs of w. Its deterministic counterpart runs in O(n2log⁡σ) time.

Another generalization of abelian equivalence: Binomial complexity of infinite words

The binomial coefficient of two words u and v is the number of times v occurs as a subsequence of u. Based on this classical notion, we introduce the m-binomial equivalence of two words refining the abelian equivalence. Two words x and y are m-binomially equivalent, if, for all words v of length at most m, the binomial coefficients of x and v and respectively, y and v are equal. The m-binomial complexity of an infinite word x maps an integer n to the number of m-binomial equivalence classes of factors of length n occurring in x. We study the first properties of m-binomial equivalence. We compute the m-binomial complexity of two classes of words: Sturmian words and (pure) morphic words that are fixed points of Parikh-constant morphisms like the Thue–Morse word, i.e., images by the morphism of all the letters have the same Parikh vector. We prove that the frequency of each symbol of an infinite recurrent word with bounded 2-binomial complexity is rational.

Approach for Flow Shop Scheduling Based on Petri Net

A flow shop scheduling problem based on the controlled Petri net and GASA is presented with multi-workstation operation. Firstly, the math model of this flow shop scheduling problem is constructed. Secondly, Petri net controller is designed based on Parikh vector, and the Petri net model is constructed. And then, GASA is applied based on the math model and the controlled Petri net model. Finally, one example is applied to test the effectiveness of the method.

Approach for Multi-Objective Flexible Job Shop Scheduling

A multi-objective scheduling method based on the controlled Petri net and GA is proposed to the flexible job shop scheduling problem (FJSP). Function objectives of the proposed method are to minimize the completion time and the total expense and workload of machines. Firstly, a Parikh vector based approach for Petri net controller is introduced, and based on this method, the Petri net model is constructed for FSP with machine breaking down. Then, the genetic algorithm (GA) is applied based on the controlled Petri net model and Pareto. Finally, simulation results based on an example show that the method is efficient.

ALGORITHMS FOR JUMBLED PATTERN MATCHING IN STRINGS

The Parikh vector p(s) of a string s over a finite ordered alphabet Σ = {a1, …, aσ} is defined as the vector of multiplicities of the characters, p(s) = (p1, …, pσ), where pi = |{j | sj = ai}|. Parikh vector q occurs in s if s has a substring t with p(t) = q. The problem of searching for a query q in a text s of length n can be solved simply and worst-case optimally with a sliding window approach in O(n) time. We present two novel algorithms for the case where the text is fixed and many queries arrive over time. The first algorithm only decides whether a given Parikh vector appears in a binary text. It uses a linear size data structure and decides each query in O(1) time. The preprocessing can be done trivially in Θ(n2) time. The second algorithm finds all occurrences of a given Parikh vector in a text over an arbitrary alphabet of size σ ≥ 2 and has sub-linear expected time complexity. More precisely, we present two variants of the algorithm, both using an O(n) size data structure, each of which can be constructed in O(n) time. The first solution is very simple and easy to implement and leads to an expected query time of [Formula: see text], where m = ∑i qi is the length of a string with Parikh vector q. The second uses wavelet trees and improves the expected runtime to [Formula: see text], i.e., by a factor of log m.

PRODUCT OF PARIKH MATRICES AND COMMUTATIVITY

The Parikh vector of a word enumerates the symbols of the alphabet that occur in the word. The Parikh matrix of a word which has been recently introduced, is an extension of the notion of Parikh vector and gives more numerical information about the word in terms of certain subwords. Intensive investigation on various theoretical properties of Parikh matrices has taken place. This paper deals with the problem of finding properties of words so that their Parikh matrices commute.

Controller Design for Petri Net with Uncontrollable and Unobservable Transitions

A controller design method for Petri net with uncontrollable and unobservable transitions that enforces the conjunction of a set of linear inequalities on the Parikh vector is proposed. The method is based on the theory that each place can be described with a Parikh vector inequality. Constraints are classified into admissible and inadmissible constraints. An inadmissible constraint cannot be directly enforced on a plant because of the uncontrollability or unobservability of certain transitions. Construct the controller though transforming the inadmissible constraint into admissible one. The method eases the design of controller, because it is based on part net design, and it only considers the direct or indirect transitions related to the constraints. So the computation required to find the Petri net controller is quite simple. Finally, the method is proved to be simple and efficient through one example.

On Approximate Jumbled Pattern Matching in Strings

Given a string s, the Parikh vector of s, denoted p(s), counts the multiplicity of each character in s. Searching for a match of a Parikh vector q in the text s requires finding a substring t of s with p(t)=q. This can be viewed as the task of finding a jumbled (permuted) version of a query pattern, hence the term Jumbled Pattern Matching. We present several algorithms for the approximate version of the problem: Given a string s and two Parikh vectors u,v (the query bounds), find all maximal occurrences in s of some Parikh vector q such that u≤q≤v. This definition encompasses several natural versions of approximate Parikh vector search. We present an algorithm solving this problem in sub-linear expected time using a wavelet tree of s, which can be computed in time O(n) in a preprocessing phase. We then discuss a Scrabble-like variation of the problem, in which a weight function on the letters of s is given and one has to find all occurrences in s of a substring t with maximum weight having Parikh vector p(t)≤v. For the case of a binary alphabet, we present an algorithm which solves the decision version of the Approximate Jumbled Pattern Matching problem in constant time, by indexing the string in subquadratic time.

On graphs of central episturmian words

Episturmian sequences are a natural extension of Sturmian sequences to the case of finite alphabets of arbitrary cardinality. In this paper, we are interested in central episturmian words, or simply, epicentral words, i.e., the palindromic prefixes of standard episturmian sequences. An epicentral word admits a variety of faithful representations including as a directive word, as a certain type of period vector, as a Parikh vector, as a certain type of Fine and Wilf extremal word, as a suitable modular matrix, and as a labeled graph. Various interconnections between the different representations of an epicentral word are analyzed. In particular, we investigate the structure of the graphs of epicentral words proving some curious and surprising properties.

ON PARIKH MATRICES

Mateescu et al (2000) introduced an interesting new tool, called Parikh matrix, to study in terms of subwords, the numerical properties of words over an alphabet. The Parikh matrix gives more information than the well-known Parikh vector of a word which counts only occurrences of symbols in a word. In this note a property of two words u, υ, called "ratio property", is introduced. This property is a sufficient condition for the words uυ and υu to have the same Parikh matrix. Thus the ratio property gives information on the M–ambiguity of certain words and certain sets of words. In fact certain regular, context-free and context-sensitive languages that have the same set of Parikh matrices are exhibited. In the study of fair words, Cerny (2006) introduced another kind of matrix, called the p–matrix of a word. Here a "weak-ratio property" of two words u, υ is introduced. This property is a sufficient condition for the words uυ and υu to have the same p–matrix. Also the words uυ and υu are fair whenever u, υ are fair and have the weak ratio property.