Small Alphabet Research Articles

Fast index based algorithms and software for matching position specific scoring matrices.

BackgroundIn biological sequence analysis, position specific scoring matrices (PSSMs) are widely used to represent sequence motifs in nucleotide as well as amino acid sequences. Searching with PSSMs in complete genomes or large sequence databases is a common, but computationally expensive task.ResultsWe present a new non-heuristic algorithm, called ESAsearch, to efficiently find matches of PSSMs in large databases. Our approach preprocesses the search space, e.g., a complete genome or a set of protein sequences, and builds an enhanced suffix array that is stored on file. This allows the searching of a database with a PSSM in sublinear expected time. Since ESAsearch benefits from small alphabets, we present a variant operating on sequences recoded according to a reduced alphabet. We also address the problem of non-comparable PSSM-scores by developing a method which allows the efficient computation of a matrix similarity threshold for a PSSM, given an E-value or a p-value. Our method is based on dynamic programming and, in contrast to other methods, it employs lazy evaluation of the dynamic programming matrix. We evaluated algorithm ESAsearch with nucleotide PSSMs and with amino acid PSSMs. Compared to the best previous methods, ESAsearch shows speedups of a factor between 17 and 275 for nucleotide PSSMs, and speedups up to factor 1.8 for amino acid PSSMs. Comparisons with the most widely used programs even show speedups by a factor of at least 3.8. Alphabet reduction yields an additional speedup factor of 2 on amino acid sequences compared to results achieved with the 20 symbol standard alphabet. The lazy evaluation method is also much faster than previous methods, with speedups of a factor between 3 and 330.ConclusionOur analysis of ESAsearch reveals sublinear runtime in the expected case, and linear runtime in the worst case for sequences not shorter than ||m + m - 1, where m is the length of the PSSM and a finite alphabet. In practice, ESAsearch shows superior performance over the most widely used programs, especially for DNA sequences. The new algorithm for accurate on-the-fly calculations of thresholds has the potential to replace formerly used approximation approaches. Beyond the algorithmic contributions, we provide a robust, well documented, and easy to use software package, implementing the ideas and algorithms presented in this manuscript.

An improvement of the tree code construction

An improvement of the randomized construction for tree codes is presented. This construction, in contrary to the original one, almost surely gives a good code and uses smaller alphabet.

PAMA: A FAST STRING MATCHING ALGORITHM

String matching is a fundamental operation in computer science, and its performance has great impact on many applications including database query, text processing, DNA and protein sequence analysis. In this paper, we propose a fast string matching algorithm, PAMA (PAttern MAtching). The shift rule used by PAMA not only subsumes both the bad character rule and the good suffix rule employed by the well-known Boyer-Moore algorithm, but also employs an additional key observation to enable faster shifting during the string matching process. Theoretically, we prove that from the same alignment, the next shift of PAMA will be at least as much as that of the Boyer-Moore algorithm. Experimentally, we show that PAMA indeed significantly outperforms the original Boyer-Moore algorithm in almost all cases, and outperforms other Boyer-Moore variants such as Tuned-BM, Turbo-BM and Horspool for long patterns (length ≥ 128) or for small alphabets (size < 8).

Faster Algorithms for Computing Longest Common Increasing Subsequences

We present algorithms for finding a longest common increasing subsequence of two or more input sequences. For two sequences of lengths m and n , where m >= n, we present an algorithm with an output-dependent expected running time of O((m + n l) log log sigma + Sort) and O(m) space, where l is the length of an LCIS, sigma is the size of the alphabet, and 'Sort' is the time to sort each input sequence.<br /> <br />For k >= 3 length-n sequences we present an algorithm running time O(min{k r^2, r log^{k-1} r} + Sort), which improves the previous best bound by more than a factor k for many inputs. In both cases, our algorithms are conceptually quite simple but rely on existing sophisticated data structures.<br /> <br />Finally, we introduce the problem of longest common weakly-increasing (or non-decreasing) subsequences (LCWIS), for which we present an O(m + n log n) time algorithm for the 3-letter alphabet case. For the extensively studied Longest Common Subsequence problem, comparable speedups have not been achieved for small alphabets.

A Space-Efficient Construction of the Burrows–Wheeler Transform for Genomic Data

Algorithms for exact string matching have substantial application in computational biology. Time-efficient data structures which support a variety of exact string matching queries, such as the suffix tree and the suffix array, have been applied to such problems. As sequence databases grow, more space-efficient approaches to exact matching are becoming more important. One such data structure, the compressed suffix array (CSA), based on the Burrows-Wheeler transform, has been shown to require memory which is nearly equal to the memory requirements of the original database, while supporting common sorts of query problems time efficiently. However, building a CSA from a sequence in efficient space and time is challenging. In 2002, the first space-efficient CSA construction algorithm was presented. That implementation used (1+2 log2 |summation|)(1+epsilon) bits per character (where epsilon is a small fraction). The construction algorithm ran in as much as twice that space, in O(| summation|n log(n)) time. We have created an implementation which can also achieve these asymptotic bounds, but for small alphabets, and only uses 1/2 (1+|summation|)(1+epsilon) bits per character, a factor of 2 less space for nucleotide alphabets. We present time and space results for the CSA construction and querying of our implementation on publicly available genome data which demonstrate the practicality of this approach.

Tight approximability results for test set problems in bioinformatics

In this paper, we investigate the test set problem and its variations that appear in a variety of applications. In general, we are given a universe of objects to be “distinguished” by a family of “tests”, and we want to find the smallest sufficient collection of tests. In the simplest version, a test is a subset of the universe and two objects are distinguished by our collection if one test contains exactly one of them. Variations allow tests to be multi-valued functions or unions of “basic” tests, and different notions of the term distinguished. An important version of this problem that has applications in DNA sequence analysis has the universe consisting of strings over a small alphabet and tests that are detecting presence (or absence) of a substring. For most versions of the problem, including the latter, we establish matching lower and upper bounds on approximation ratio. When tests can be formed as unions of basic tests, we show that the problem is as hard as the graph coloring problem. We conclude by reporting preliminary computational results on the implementations of our algorithmic approaches for the minimum cost probe set problems on a data set used by Borneman et al.

On average sequence complexity

In this paper we study the average behavior of the number of distinct substrings in a text of size n over an alphabet of cardinality k. This quantity is called the complexity index and it captures the “richness of the language” used in a sequence. For example, sequences with low complexity index contain a large number of repeated substrings and they eventually become periodic (e.g., tandem repeats in a DNA sequence). In order to identify unusually low- or high-complexity strings one needs to determine how far are the complexities of the strings under study from the average or maximum string complexity. While the maximum string complexity was studied quite extensively in the past, to the best of our knowledge there are no results concerning the average complexity. We first prove that for a sequence generated by a mixing model (which includes Markov sources) the average complexity is asymptotically equal to n 2 / 2 which coincides with the maximum string complexity. However, for a memoryless source we establish a more precise result, namely the average string complexity is n 2 / 2 - n log k n + 1 + ( 1 - γ ) / ln k + φ k ( log k n ) + o ( 1 ) n where γ ≈ 0.577 and φ k ( x ) is a periodic function with a small amplitude for small alphabet size.

Modified Chu sequences with smaller alphabet size

A set of modified Chu sequences is proposed with similar correlation properties to those of normal Chu sequences except for a special case. Compared with Chu sequences, the modified Chu sequences have a smaller alphabet size.

Fast prefix matching of bounded strings

Longest Prefix Matching (LPM) is the problem of finding which string from a given set is the longest prefix of another, given string. LPM is a core problem in many applications, including IP routing, network data clustering, and telephone network management. These applications typically require very fast matching of bounded strings, i.e., strings that are short and based on small alphabets. We note a simple correspondence between bounded strings and natural numbers that maps prefixes to nested intervals so that computing the longest prefix matching a string is equivalent to finding the shortest interval containing its corresponding integer value. We then present retries , a fast and compact data structure for LPM on general alphabets. Performance results show that retries often outperform previously published data structures for IP look-up. By extending LPM to general alphabets, retries admit new applications that could not exploit prior LPM solutions designed for IP look-ups.

Automated identification of RNA conformational motifs: theory and application to the HM LSU 23S rRNA.

We develop novel methods for recognizing and cataloging conformational states of RNA, and for discovering statistical rules governing those states. We focus on the conformation of the large ribosomal subunit from Haloarcula marismortui. The two approaches described here involve torsion matching and binning. Torsion matching is a pattern-recognition code which finds structural repetitions. Binning is a classification technique based on distributional models of the data. In comparing the results of the two methods we have tested the hypothesis that the conformation of a very large complex RNA molecule can be described accurately by a limited number of discrete conformational states. We identify and eliminate extraneous and redundant information without losing accuracy. We conclude, as expected, that four of the torsion angles contain the overwhelming bulk of the structural information. That information is not significantly compromised by binning the continuous torsional information into a limited number of discrete values. The correspondence between torsion matching and binning is 99% (per residue). Binning, however, does have several advantages. In particular, we demonstrate that the conformation of a large complex RNA molecule can be represented by a small alphabet. In addition, the binning method lends itself to a natural graphical representation using trees.

Shift-or string matching with super-alphabets

Given a text T[1… n] and a pattern P[1… m] over some alphabet Σ of size σ, we want to find all the (exact) occurrences of P in T. The well-known shift-or algorithm solves this problem in time O( n⌈ m/ w⌉), where w is the number of bits in machine word, using bit-parallelism. We show how to extend the bit-parallelism in another direction, using super-alphabets. This gives a speed-up by a factor s, where s is the number of characters processed simultaneously. The algorithm is implemented, and we show that it works well in practice too. The result is the fastest known algorithm for exact string matching for short patterns and small alphabets.

High precision simulations of the longest common subsequence problem

The longest common subsequence problem is a long studied prototype of pattern matching problems. In spite of the effort dedicated to it, the numerical value of its central quantity, the Chvatal-Sankoff constant, is not yet known. Numerical estimations of this constant are very difficult due to finite size effects. We propose a numerical method to estimate the Chvatal-Sankoff constant which combines the advantages of an analytically known functional form of the finite size effects with an efficient multi-spin coding scheme. This method yields very high precision estimates of the Chvatal-Sankoff constant. Our results correct earlier estimates for small alphabet size while they are consistent with (albeit more precise than) earlier results for larger alphabet size.

(Coarse coding of shape fragments) + (retinotopy) approximately = representation of structure.

The ability to deal with object structure--to determine what is where in a given object, rather than merely to categorize or identify it--has been hitherto considered the prerogative of 'structural description' approaches, which represent shapes as categorical compositions of generic parts taken from a small alphabet. In this note, we propose a simple extension to a theoretically motivated and extensively tested appearance-based model of recognition and categorization, which should make it capable of representing object structure. We describe a pilot implementation of the extended model, survey independent evidence supporting its modus operandi, and outline a research program focused on achieving a range of object processing capabilities, including reasoning about structure, within a unified appearance-based framework.

Linearly representable codes over chain rings

In this paper, we consider linear codes over finite chain rings. We present a general mapping which produces codes over smaller alphabets. Under special conditions, these codes are linear over a finite field. We introduce the notion of a linearly representable code and prove that certain MacDonald codes are linearly representable. Finally, we give examples for good linear codes over finite fields obtained from special multisets in projective Hjelmslev planes.

On the complexity measures of genetic sequences.

It is well known that the regulatory regions of genomes are highly repetitive. They are rich in direct, symmetric and complemented repeats, and there is no doubt about the functional significance of these repeats. Among known measures of complexity, the Ziv-Lempel complexity measure reflects most adequately repeats occurring in the text. But this measure does not take into account isomorphic repeats. By isomorphic repeats we mean fragments that are identical (or symmetric) modulo some permutation of the alphabet letters. In this paper, two complexity measures of symbolic sequences are proposed that generalize the Ziv-Lempel complexity measure by taking into account any isomorphic repeats in the text (rather than just direct repeats as in Ziv-Lempel). The first of them, the complexity vector, is designed for small alphabets such as the alphabet of nucleotides. The second is based on a search for the longest isomorphic fragment in the history of sequence synthesis and can be used for alphabets of arbitrary cardinality. These measures have been used for recognition of structural regularities in DNA sequences. Some interesting structures related to the regulatory region of the human growth hormone are reported.

Polyphase Barker sequences up to length 45 with small alphabets

Polyphase Barker sequences are finite-length, uniform complex sequences; the magnitudes of their aperiodic autocorrelation sidelobes are bounded by 1. Polyphase Barker sequences with lengths up to 45 and alphabet sizes up to 120 are presented, which are found using a stochastic optimisation procedure.

Algorithms for the longest common subsequence problem for multiple strings based on geometric maxima

Given two or more strings (for example, DNA and amino acid sequences), the longest common subsequence (LCS for short) problem is to determine the longest common subsequence obtained by deleting zero or more symbols from each string. The algorithms for computing an LCS between two strings were given by many papers, but there are few efficient algorithms for computing an LCS between more than two strings. This paper proposes a method for computing efficiently the LCS between three and more strings of small alphabet size, evaluates its theoretical time complexity, and estimates the computing time by computational experiments. Using this method, the LCS problem for eight strings of more than 120 length can be solved in about 40min on a slow workstation.

Almost-perfect polyphase sequences with small phase alphabet

New classes of polyphase sequences with small phase alphabets of size k are presented. These almost-perfect sequences of length N have a vanishing periodic autocorrelation function /spl Theta/(/spl tau/) for all /spl tau//spl ne/0 mod (N/k). The sequences are derived from q-ary m-sequences by a k-valued character operation.

Random structures and evolution of biopolymers: A computational case study on RNA secondary structures

We compute large ensembles of RNA secondary structures by means of various computer algorithms. Their secondary structures are sorted and ranked. The number of neutral neighbours (i.e., one error mutants of a sequence that have the same structure with that sequence) are computed as well as the length of neutral paths (i.e., adjacent neutral neighbours). We calculate and compare correlation lengths which are a reasonable measure for the ruggedness of the sequence to structure mapping and describe the hardness of an optimisation problem. We investigate the influence of size of the spaces as well as frequency and accessibility of structures on the degree of neutrality. We find large neutral nets, percolating through sequence space even for rare structures and small alphabets. We conclude that the characteristic features of the mapping, namely the distribution of structures and the amount of neutrality are robust with respect to the chosen algorithm. This implies a new perspective on the efficiency of evolution by point mutations: from any random structure it is not far to any target and yet many mutations will preserve the structure.

A subquadratic algorithm for approximate limited expression matching

In this paper we present an efficient subquadratic-time algorithm for matching strings and limited expressions in large texts. Limited expressions are a subset of regular expressions that appear often in practice. The generalization from simple strings to limited expressions has a negligible affect on the speed of our algorithm, yet allows much more flexibility. Our algorithm is similar in spirit to that of Masek and Paterson [MP], but it is much faster in practice. Our experiments show a factor of four to five speedup against the algorithms of Sellers [Se] and Ukkonen [Uk1] independent of the sizes of the input strings. Experiments also reveal our algorithm to be faster, in most cases, than a recent improvement by Chang and Lampe [CL2], especially for small alphabet sizes for which it is two to three times faster.