Binary Alphabet Research Articles

Non-approximability of weighted multiple sequence alignment for arbitrary metrics

We prove that the multiple sequence alignment problem with weighted sum-of-pairs score is APX-hard for arbitrary metric scoring functions over the binary alphabet. This holds even when the weights are restricted to zero and one.

FFT-based algorithms for the string matching with mismatches problem

The string matching with mismatches problem requires finding the Hamming distance between a pattern P of length m and every length m substring of text T with length n. Fischer and Paterson's FFT-based algorithm solves the problem without error in O ( σ n log m ) , where σ is the size of the alphabet Σ [SIAM–AMS Proc. 7 (1973) 113–125]. However, this in the worst case reduces to O ( n m log m ) . Atallah, Chyzak and Dumas used the idea of randomly mapping the letters of the alphabet to complex roots of unity to estimate the score vector in time O ( n log m ) [Algorithmica 29 (2001) 468–486]. We show that the algorithm's score variance can be substantially lowered by using a bijective mapping, and specifically to zero in the case of binary and ternary alphabets. This result is extended via alphabet remappings to deterministically solve the string matching with mismatches problem with a constant factor of 2 improvement over Fischer–Paterson's method.

Parameterized Intractability of Distinguishing Substring Selection

A central question in computational biology is the design of genetic markers to distinguish between two given sets of (DNA) sequences. This question is formalized as the NP-complete Distinguishing Substring Selection problem (DSSS for short) which asks, given a set of "good" strings and a set of "bad" strings, for a solution string which is, with respect to the Hamming metric, "away" from the good strings and "close" to the bad strings. More precisely, given integers dg, db, and L, we ask for a length-L string s such that s has Hamming distance at least dg to every length-L substring of the good strings and such that every bad string has a length-L substring with Hamming distance at most db to s. Studying the parameterized complexity of DSSS, we show that, already for binary alphabet, DSSS is W[1]-hard with respect to its natural parameters. This, in particular, implies that a recently given polynomial-time approximation scheme (PTAS) by Deng et al. cannot be replaced by a so-called efficient polynomial-time approximation scheme (EPTAS) unless an unlikely collapse in parameterized complexity theory occurs. This is seemingly the first computational biology problem for which such a border between PTAS (which exists) and EPTAS (which is unlikely to exist) could be established. By way of contrast, for a special case of DSSS, we present an exact fixed-parameter algorithm solving the problem efficiently. In this way we also exhibit a sharp border between fixed-parameter tractability and intractability results.

First-order expressibility of languages with neutral letters or: The Crane Beach conjecture

A language L over an alphabet A is said to have a neutral letter if there is a letter e ∈ A such that inserting or deleting e's from any word in A * does not change its membership or non-membership in L. The presence of a neutral letter affects the definability of a language in first-order logic. It was conjectured that it renders all numerical predicates apart from the order predicate useless, i.e., that if a language L with a neutral letter is not definable in first-order logic with linear order, then it is not definable in first-order logic with any set N of numerical predicates. Named after the location of its first, flawed, proof this conjecture is called the Crane Beach conjecture (CBC, for short). The CBC is closely related to uniformity conditions in circuit complexity theory and to collapse results in database theory. We investigate the CBC in detail, showing that it fails for N = { + , × } , or, possibly stronger, for any set N that allows counting up to the m times iterated logarithm, for any constant m. On the positive side, we prove the conjecture for the case of all monadic numerical predicates, for the addition predicate + , for the fragment BC ( Σ 1 ) of first-order logic, for regular languages, and for languages over a binary alphabet. We explain the precise relation between the CBC and so-called natural-generic collapse results in database theory. Furthermore, we introduce a framework that gives better understanding of what exactly may cause a failure of the conjecture.

State complexity of some operations on binary regular languages

We investigate the state complexity of some operations on binary regular languages. In particular, we consider the concatenation of languages represented by deterministic finite automata, and the reversal and complementation of languages represented by nondeterministic finite automata. We prove that the upper bounds on the state complexity of these operations, which were known to be tight for larger alphabets, are tight also for binary alphabets.

Type inference for light affine logic via constraints on words

Light Affine Logic (LAL) is a system due to Girard and Asperti capturing the complexity class P in a proof-theoretical approach based on Linear Logic. LAL provides a typing for lambda-calculus which guarantees that a well-typed program is executable in polynomial time on any input. We prove that the LAL type inference problem for lambda-calculus is decidable (for propositional LAL). To establish this result we reformulate the type-assignment system into an equivalent one which makes use of subtyping and is more flexible. We then use a reduction to a satisfiability problem for a system of inequations on words over a binary alphabet, for which we provide a decision procedure.

Hardness results for the center and median string problems under the weighted and unweighted edit distances

Given a finite set of strings, the Median String problem consists in finding a string that minimizes the sum of the edit distances to the strings in the set. Approximations of the median string are used in a very broad range of applications where one needs a representative string that summarizes common information to the strings of the set. It is the case in classification, in speech and pattern recognition, and in computational biology. In the latter, Median String is related to the key problem of multiple alignment. In the recent literature, one finds a theorem stating the NP-completeness of the Median String for unbounded alphabets. However, in the above mentioned areas, the alphabet is often finite. Thus, it remains a crucial question whether the Median String problem is NP-complete for bounded and even binary alphabets. In this work, we provide an answer to this question and also give the complexity of the related Center String problem. Moreover, we study the parameterized complexity of both problems with respect to the number of input strings. In addition, we provide an algorithm to compute an optimal center under a weighted edit distance in polynomial time when the number of input strings is fixed.

On deterministic finite automata and syntactic monoid size

We investigate the relationship between regular languages and syntactic monoid size. In particular, we consider the transformation monoids of n-state (minimal) deterministic finite automata. We show tight upper and lower bounds on the syntactic monoid size depending on the number of generators (input alphabet size) used. It turns out, that the two generator case is the most involved one. There we show a lower bound of n n 1 - 2 n for the size of the syntactic monoid of a language accepted by an n-state deterministic finite automaton with binary input alphabet. Moreover, we prove that for every prime n ⩾ 7 , the maximal size semigroup w.r.t. its size among all (transformation) semigroups which can be generated with two generators, is generated by a permutation with two cycles (of appropriate lengths) and a non-bijective mapping merging elements from these two cycles. As a by-product of our investigations we determine the maximal size among all semigroups generated by two transformations, where one is a permutation with a single cycle and the other is a non-bijective mapping.

Randomness relative to Cantor expansions

Imagine a sequence in which the first letter comes from a binary alphabet, the second letter can be chosen on an alphabet with 10 elements, the third letter can be chosen on an alphabet with 3 elements and so on. Such sequences occur in various physical contexts, in which the coding of experimental outcome varies with scale. When can such a sequence be called random? In this paper we offer a solution to the above question using the approach to randomness proposed by Algorithmic Information Theory.

On the Capacity Loss Due to Separation of Detection and Decoding

The performance loss due to separation of detection and decoding on the binary-input additive white Gaussian noise (AWGN) channel is quantified in terms of mutual information. Results are reported for both the code-division multiple-access (CDMA) channel in the large system limit and the intersymbol interference (ISI) channel. The results for CDMA rely on the replica method developed in statistical mechanics. It is shown that a previous result of Shamai and Verdu found for Gaussian input alphabet holds also for binary input alphabets. For the ISI channel, the performance loss is calculated via the Bahl-Cocke-Jelinek-Raviv (BCJR) algorithm. Comparisons are made to the capacity of separate detection and decoding using suboptimum detectors such as a decision-feedback equalizer.

Properties of the complexity function for finite words

The subword complexity function \(p_{w}\) of a finite word \(w\) over a finite alphabet \(A\) with \(\operatorname*{card}A=q\geq1\) is defined by \(p_{w}(n)=\operatorname*{card}(F(w)\cap A^{n})\) for \(n\in\mathbb{N},\) where \(F(w)\) represents the set of all the subwords or factors of \(w\). The shape of the complexity function, especially its piecewise monotonicity, is studied in detail.The function \(h\) defined as \(h(n)=\min\left\{ q^{n},N-n+1\right\} \) for \(n\in\{0,1,\) \(...,N\}\) has values greater than or equal to those of the complexity function \(p_{w}\) for any \(w\in A^{N}\), i.e., \(p_{w}(n)\leq h(n)\) for all \(n\in\{0,1,...,N\}\). As a first result regarding \(h\), it is proved that for each \(N\in\mathbb{N}\) there exist words of length \(N\) for which the maximum of their complexity function is equal to the maximum of the function \(h\); a way to construct such words is described. This result gives rise to a further question: for a given \(N\), is there a word of length \(N\) whose complexity function coincides with \(h\) for each \(n\in\{0,1,...,N\}?\) The problem is answered in affirmative, with different constructive proofs for binary alphabets (\(q=2\)) and for those with \(q>2.\) This means that for each \(N\in\mathbb{N},\) there exist words \(w\) of length \(N\) whose complexity function is equal to the function \(h\). Such words are constructed using the de Bruijn graphs.

Markov Types and Minimax Redundancy for Markov Sources

Redundancy of universal codes for a class of sources determines by how much the actual code length exceeds the optimal code length. In the minimax scenario, one designs the best code for the worst source within the class. Such minimax redundancy comes in two flavors: average minimax or worst case minimax. We study the worst case minimax redundancy of universal block codes for Markovian sources of any order. We prove that the maximal minimax redundancy for Markov sources of order r is asymptotically equal to 1/2m/sup r/(m-1)log/sub 2/n+log/sub 2/A/sub m//sup r/-(lnlnm/sup 1/(m-1)/)/lnm+o(1), where n is the length of a source sequence, m is the size of the alphabet, and A/sub m//sup r/ is an explicit constant (e.g., we find that for a binary alphabet m=2 and Markov of order r=1 the constant A/sub 2//sup 1/=16/spl middot/G/spl ap/14.655449504 where G is the Catalan number). Unlike previous attempts, we view the redundancy problem as an asymptotic evaluation of certain sums over a set of matrices representing Markov types. The enumeration of Markov types is accomplished by reducing it to counting Eulerian paths in a multigraph. In particular, we propose exact and asymptotic formulas for the number of strings of a given Markov type. All of these findings are obtained by analytic and combinatorial tools of analysis of algorithms.

Patterns in words and languages

A word p, over the alphabet of variables E, is a pattern of a word w over A if there exists a non-erasing morphism h from E ∗ to A ∗ such that h( p)= w. If we take E= A, given two words u,v∈A ∗ , we write u⩽ v if u is a pattern of v. The restriction of ⩽ to aA ∗ , where A is the binary alphabet { a, b}, is a partial order relation. We introduce, given a word v, the set P( v) of all words u such that u⩽ v. P( v), with the relation ⩽, is a poset and it is called the pattern poset of v. The first part of the paper is devoted to investigate the relationships between the structure of the poset P( v) and the combinatorial properties of the word v. In the last section, for a given language L, we consider the language P( L) of all patterns of words in L. The main result of this section shows that, if L is a regular language, then P( L) is a regular language too.

On enumeration problems in Lie–Butcher theory

The algebraic structure underlying non-commutative Lie–Butcher series is the free Lie algebra over ordered trees. In this paper we present a characterization of this algebra in terms of balanced Lyndon words over a binary alphabet. This yields a systematic manner of enumerating terms in non-commutative Lie–Butcher series.

Algorithmic complexity of protein identification: combinatorics of weighted strings

We investigate a problem which arises in computational biology: Given a constant-size alphabet A with a weight function μ : A→ N , find an efficient data structure and query algorithm solving the following problem: For a string σ over A and a weight M∈ N , decide whether σ contains a substring with weight M, where the weight of a string is the sum of the weights of its letters (O NE-S TRING M ASS F INDING P ROBLEM). If the answer is yes, then we may in addition require a witness, i.e., indices i⩽ j such that the substring beginning at position i and ending at position j has weight M. We allow preprocessing of the string and measure efficiency in two parameters: storage space required for the preprocessed data and running time of the query algorithm for given M. We are interested in data structures and algorithms requiring subquadratic storage space and sublinear query time, where we measure the input size as the length n of the input string σ. Among others, we present two non-trivial efficient algorithms: L OOKUP solves the problem with O( n) storage space and O(n/ log n) time; I NTERVAL solves the problem for binary alphabets with O( n) storage space in O( log n) query time. We introduce other variants of the problem and sketch how our algorithms may be extended for these variants. Finally, we discuss combinatorial properties of weighted strings.

Developing an Infrastructure for Sharing Environmental Models

The Internet and the World Wide Web offer a new solution to the problem of sharing scientific knowledge. Unlike traditional libraries based on print media, these new technologies facilitate the sharing of any information that can be expressed in a binary alphabet. Environmental models expressed as computer codes are instances of such information objects, and codes are the last of a four-stage process of model formulation. The transition to digital technologies changes the relative importance of the four stages. We present a six-stage model of the process of searching for information in distributed digital libraries. Search is made possible by metadata, which serve several distinct purposes, including description of the contents of information objects and their fitness for specified use, and the details needed to make use of information objects once accessed and retrieved. We present a straw-man structure for model metadata, explain the process used in its development, and invite its evaluation.

Prediction of secondary protein structure with binary coding patterns of amino acid and nucleotide physicochemical properties

AbstractWe present binary coding algorithm for the α‐ and β‐protein fold prediction. The method links amino acid molecular polarity patterns and physicochemical properties of nucleotide bases coded by means of a binary addresses. Primary sequences that define secondary protein structure were analyzed with respect to the symbolic oligopeptides (SO) obtained by the reduction of the 20 amino acid letter alphabet into a binary alphabet of nonpolar group 0 (W, C, I, F, M, V, L, Y) and polar group 1 (Q, R, H, K, N, E, D, S, G, T, A, P). The groups were extracted from the Grantham polarity scale with the clustering around medoids procedure. The transformation of protein strings into binary coding patterns of the polar and nonpolar amino acid groups reduced analyzed elements within the protein motif of length n by the factor of 10n. SMO learning algorithm for the support vector machines was applied to classify α‐helices and β‐strands. It was shown that the relative frequencies of binary hexapeptides classify all 174 nonhomologous α‐ and β‐protein folds from the Jpred database with 100% accuracy. The results of 10‐fold cross‐validation and leave‐one‐out test were 86.78%. Classification tree confirmed the results of SMO analysis and correctly classified 100% of the folds by means of 9 binary hexapeptides. Linear block triple‐check code was proposed for the description of hexapeptide patterns. The presented method enables simple, quick, and accurate prediction of α‐ and β‐protein folding types from the primary amino acid and nucleotide sequences on a personal computer. Our results imply that few amino acid polarity patterns specified by the nucleotide physicochemical properties describe basic protein folding types with >90% accuracy. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003

On the dimensions of the spectral measure of symmetric binary substitutions

Substitution rules defined on the binary alphabet and invariant under the interchange of both participating symbols are considered. Typically, the resulting symbolic sequences are weakly disordered, and their Fourier spectra are singular continuous. Exact relations are derived which express the correlation dimension of the multifractal spectral measure in terms of the entries in the substitution pattern. Recurrence relations are obtained for the finite-length spectral sums and for the autocorrelation function.

Bounds for parametric sequence comparison

We consider the problem of computing a global alignment between two or more sequences subject to varying mismatch and indel penalties. We prove a tight 3(n/2π) 2/3+O(n 1/3 log n) bound on the worst-case number of distinct optimum alignments for two sequences of length n as the parameters are varied. This refines a O( n 2/3) upper bound by Gusfield et al., answering a question posed by Pevzner and Waterman. Our lower bound requires an unbounded alphabet. For strings over a binary alphabet, we prove a Ω(n 1/2) lower bound. For the parametric global alignment of k⩾2 sequences under sum-of-pairs scoring we prove a 3(( k 2 )n/2π) 2/3+O(k 2/3n 1/3 log n) upper bound on the number of distinct optimality regions and a Ω(n 2/3) lower bound, partially answering a problem of Pevzner. Based on experimental evidence, we conjecture that for two random sequences, the number of optimality regions is approximately n with high probability.

On-Line Approximate String Searching Algorithms: Survey and Experimental Results

The problem of approximate string searching comprises two classes of problems: string searching with k mismatches and string searching with k differences. In this paper we present a short survey and experimental results for well known sequential approximate string searching algorithms. We consider algorithms based on different approaches including dynamic programming, deterministic finite automata, filtering, counting and bit parallelism. We compare these algorithms in terms of running time against pattern length and for several values of k for four different kinds of text: binary alphabet, alphabet of size 8, English alphabet and DNA alphabet. Finally, we compare the experimental results of the algorithms with their theoretical complexities.