Indeterminate Strings Research Articles

A new approach to regular & indeterminate strings

In this paper we propose a new, more appropriate definition of regular and indeterminate strings. A regular string is one that is “isomorphic” to a string whose entries all consist of a single letter, but which nevertheless may itself include entries containing multiple letters. A string that is not regular is said to be indeterminate. We begin by proposing a new model for the representation of strings, regular or indeterminate, then go on to describe a linear time algorithm to determine whether or not a string x=x[1..n] is regular and, if so, to replace it by a lexicographically least (lex-least) string y whose entries are all single letters. Furthermore, we connect the regularity of a string to the transitive closure problem on a graph, which in our special case can be efficiently solved. We then introduce the idea of a feasible palindrome array MP of a string, and prove that every feasible MP corresponds to some (regular or indeterminate) string. We describe an algorithm that constructs a string x corresponding to given feasible MP, while ensuring that whenever possible x is regular and if so, then lex-least. A final section outlines new research directions suggested by this changed perspective on regular and indeterminate strings.

An improved upper bound and algorithm for clique covers

Indeterminate strings have received considerable attention in the recent past; see for example [1] and [3]. This attention is due to their applicability in bioinformatics, and to the natural correspondence with undirected graphs. One aspect of this correspondence is the fact that the minimum alphabet size of indeterminates representing any given undirected graph equals the size of the minimal clique cover of this graph. This paper first considers a related problem proposed in [3]: characterize Θn(m), which is the size of the largest possible minimal clique cover (i.e., an exact upper bound), and hence alphabet size of the corresponding indeterminate, of any graph on n vertices and m edges. We provide improvements to the known upper bound for Θn(m) in section 3.3. [3] also presents an algorithm which finds clique covers in polynomial time. We build on this result with a heuristic for vertex sorting which significantly improves their algorithm's results, particularly in dense graphs.

Covering problems for partial words and for indeterminate strings

Indeterminate strings are a subclass of non-standard words having non-deterministic nature. In a classic string every position contains exactly one symbol—we say it is a solid symbol—while in an indeterminate string a position may contain a set of symbols (possible at this position); such sets are called non-solid symbols. The most important subclass of indeterminate strings are partial words, where each non-solid symbol is the whole alphabet; in this case non-solid symbols are also called don't care symbols. We consider the problem of finding a shortest cover of an indeterminate string, i.e., finding a shortest solid string whose occurrences cover the whole indeterminate string. We show that this classical problem becomes NP-complete for indeterminate strings and even for partial words. The proof of this fact is one of the main results of this paper. Our other main results focus on design of algorithms efficient with respect to certain parameters of the input (so called FPT algorithms) for the shortest cover problem. For the indeterminate string covering problem we obtain an O(nk2+2kk3)-time algorithm, where k is the number of non-solid symbols, while for the partial word covering problem we obtain a running time of O(nk2+2O(klog⁡k)). Additionally, we prove that, unless the Exponential Time Hypothesis is false, no 2o(k)nO(1)-time solution exists for either problem, which shows that our algorithm for partial words is close to optimal. We also present an algorithm for both problems parameterized both by k and the alphabet size with a simple implementation.A preliminary version of this article was presented at the 25th International Symposium on Algorithms and Computation (ISAAC 2014), LNCS, vol. 8889, pp. 220–232, Springer (2014) [12].

New Bounds and Extended Relations Between Prefix Arrays, Border Arrays, Undirected Graphs, and Indeterminate Strings

We extend earlier works on the relation of prefix arrays of indeterminate strings to undirected graphs and border arrays. If integer array y is the prefix array for indeterminate string w, then we say w satisfies y. We use a graph theoretic approach to construct a string on a minimum alphabet size which satisfies a given prefix array. We relate the problem of finding a minimum alphabet size to finding edge clique covers of a particular graph, allowing us to bound the minimum alphabet size by n+n$n+\sqrt {n}$ for indeterminate strings, where n is the size of the prefix array. When we restrict ourselves to prefix arrays for partial words, we bound the minimum alphabet size by 2n$\left \lceil \sqrt {2n} \right \rceil $. Moreover, we show that this bound is tight up to a constant multiple by using Sidon sets. We also study the relationship between prefix arrays and border arrays. We give necessary and sufficient conditions for a border array and prefix array to be satisfied by the same indeterminate string. We show that the slowly-increasing property completely characterizes border arrays for indeterminate strings, whence there are exactly Cn distinct border arrays of size n for indeterminate strings (here Cn is the nth Catalan number). We give an algorithm to enumerate all prefix arrays for partial words of a given size, n. Our algorithm has a time complexity of n3 times the output size, that is, the number of valid prefix arrays for partial words of length n. We also bound the number of prefix arrays for partial words of a given size using Stirling numbers of the second kind.

Indeterminate strings, prefix arrays & undirected graphs

An integer array y=y[1..n] is said to be feasible if and only if y[1]=n and, for every i∈2..n, i≤i+y[i]≤n+1. A string is said to be indeterminate if and only if at least one of its elements is a subset of cardinality greater than one of a given alphabet Σ; otherwise it is said to be regular. A feasible array y is said to be regular if and only if it is the prefix array of some regular string. We show using a graph model that every feasible array of integers is a prefix array of some (indeterminate or regular) string, and for regular strings corresponding to y, we use the model to provide a lower bound on the alphabet size. We show further that there is a 1–1 correspondence between labelled simple graphs and indeterminate strings, and we show how to determine the minimum alphabet size σ of an indeterminate string x based on its associated graphGx. Thus, in this sense, indeterminate strings are a more natural object of combinatorial interest than the strings on elements of Σ that have traditionally been studied.

Computing covers using prefix tables

An indeterminate stringx=x[1..n] on an alphabet Σ is a sequence of nonempty subsets of Σ; x is said to be regular if every subset is of size one. A proper substring u of regular x is said to be a cover of x iff for every i∈1..n, an occurrence of u in x includes x[i]. The cover arrayγ=γ[1..n] of x is an integer array such that γ[i] is the longest cover of x[1..i]. Fifteen years ago a complex, though nevertheless linear-time, algorithm was proposed to compute the cover array of regular x based on prior computation of the border array of x. In this paper we first describe a linear-time algorithm to compute the cover array of regular x based on the prefix table of x. We then extend this result to indeterminate strings.

Identifying gene clusters by discovering common intervals in indeterminate strings

BackgroundComparative analyses of chromosomal gene orders are successfully used to predict gene clusters in bacterial and fungal genomes. Present models for detecting sets of co-localized genes in chromosomal sequences require prior knowledge of gene family assignments of genes in the dataset of interest. These families are often computationally predicted on the basis of sequence similarity or higher order features of gene products. Errors introduced in this process amplify in subsequent gene order analyses and thus may deteriorate gene cluster prediction.ResultsIn this work, we present a new dynamic model and efficient computational approaches for gene cluster prediction suitable in scenarios ranging from traditional gene family-based gene cluster prediction, via multiple conflicting gene family annotations, to gene family-free analysis, in which gene clusters are predicted solely on the basis of a pairwise similarity measure of the genes of different genomes. We evaluate our gene family-free model against a gene family-based model on a dataset of 93 bacterial genomes.ConclusionsOur model is able to detect gene clusters that would be also detected with well-established gene family-based approaches. Moreover, we show that it is able to detect conserved regions which are missed by gene family-based methods due to wrong or deficient gene family assignments.

Indeterminate string inference algorithms

Regularities in indeterminate strings have recently been a matter of interest because of their use in the fields of molecular biology, musical text analysis, cryptanalysis and so on. In this paper, we study the problem of reconstructing an indeterminate string from a border array. We present two efficient algorithms to reconstruct an indeterminate string from a valid border array – one using an unbounded alphabet and the other using minimum sized alphabet. We also propose an O ( n 2 ) algorithm for reconstructing an indeterminate string from suffix array and LCP array.

AN ADAPTIVE HYBRID PATTERN-MATCHING ALGORITHM ON INDETERMINATE STRINGS

We describe a hybrid pattern-matching algorithm that works on both regular and indeterminate strings. This algorithm is inspired by the recently proposed hybrid algorithm FJS and its indeterminate successor. However, as discussed in this paper, because of the special properties of indeterminate strings, it is not straightforward to directly migrate FJS to an indeterminate version. Our new algorithm combines two fast pattern-matching algorithms, ShiftAnd and BMS (the Sunday variant of the Boyer-Moore algorithm), and is highly adaptive to the nature of the text being processed. It avoids using the border array, therefore avoids some of the cases that are awkward for indeterminate strings. Although not always the fastest in individual test cases, our new algorithm is superior in overall performance to its two component algorithms — perhaps a general advantage of hybrid algorithms.

Fast pattern-matching on indeterminate strings

In a string x on an alphabet Σ, a position i is said to be indeterminate iff x [ i ] may be any one of a specified subset { λ 1 , λ 2 , … , λ j } of Σ, 2 ⩽ j ⩽ | Σ | . A string x containing indeterminate positions is therefore also said to be indeterminate. Indeterminate strings can arise in DNA and amino acid sequences as well as in cryptological applications and the analysis of musical texts. In this paper we describe fast algorithms for finding all occurrences of a pattern p = p [ 1 . . m ] in a given text x = x [ 1 . . n ] , where either or both of p and x can be indeterminate. Our algorithms are based on the Sunday variant of the Boyer–Moore pattern-matching algorithm, one of the fastest exact pattern-matching algorithms known. The methodology we describe applies more generally to all variants of Boyer–Moore (such as Horspool's, for example) that depend only on calculation of the δ (“rightmost shift”) array: our method therefore assumes that Σ is indexed (essentially, an integer alphabet), a requirement normally satisfied in practice.