On-line construction of suffix trees

The presence of repeated sequences is a fundamental feature of genomes. Tandemly repeated DNA appears in both eukaryotic and prokaryotic genomes, it is associated with various regulatory mechanisms and plays an important role in genomic fingerprinting. In this paper, we describe mreps, a powerful software tool for a fast identification of tandemly repeated structures in DNA sequences. mreps is able to identify all types of tandem repeats within a single run on a whole genomic sequence. It has a resolution parameter that allows the program to identify 'fuzzy' repeats. We introduce main algorithmic solutions behind mreps, describe its usage, give some execution time benchmarks and present several case studies to illustrate its capabilities. The mreps web interface is accessible through http://www.loria.fr/mreps/.

/pdf/mreps-efficient-and-flexible-detection-of-tandem-repeats-in-3jwx33f8ti.pdf

mreps: efficient and flexible detection of tandem repeats in DNA

This paper considers various flavors of the following online problem: preprocess a text or collection of strings, so that given a query string p, all matches of p with the text can be reported quickly. In this paper we consider matches in which a bounded number of mismatches are allowed, or in which a bounded number of "don't care" characters are allowed. The specific problems we look at are: indexing, in which there is a single text t, and we seek locations where p matches a substring of t; dictionary queries, in which a collection of strings is given upfront, and we seek those strings which match p in their entirety; and dictionary matching, in which a collection of strings is given upfront, and we seek those substrings of a (long) p which match an original string in its entirety. These are all instances of an all-to-all matching problem, for which we provide a single solution.The performance bounds all have a similar character. For example, for the indexing problem with n=|t| and m=|p|, the query time for k substitutions is O(m + (c1 log n)k⁄k! + # matches), with a data structure of size O(n (c2 log n)k⁄k!) and a preprocessing time of O(n (c2 log n)k⁄k!), where c1,c2 > 1 are constants. The deterministic preprocessing assumes a weakly nonuniform RAM model; this assumption is not needed if randomization is used in the preprocessing.

/pdf/dictionary-matching-and-indexing-with-errors-and-don-t-cares-371530tjab.pdf

Dictionary matching and indexing with errors and don't cares

: This project was concerned with the development of correct and reusable software through the use of higher order abstractions (function, control, assignment, process) and reflection. A semantic framework for these notions will be the basis of an experimental system for manipulating and reasoning about programs. The goals of this project were the development of logical formalisms for reasoning about programs that use abstractions and reflection, and the application of these theoretical results to selected software problems. Example applications include (1) clarification of existing programming paradigms, (2) analysis of existing and proposed languages used in the DARPA community for specifying, writing, and transforming programs, and (3) development and implementation of tools for computer aided reasoning about and operating on programs. The accomplishments of this project fit into four categories: (1) logics for reasoning about function and control abstractions; (2) logics for reasoning about data mutation; (3) logics for reasoning about function and control abstractions in the presence of mutable data; and (4) applying methodology for reasoning about programs to the mechanical verification of hardware.

Mathematical Theory of Computation

Linear time algorithms for finding and representing all the tandem repeats in a string

A perfect single tandem repeat is defined as a nonempty string that can be divided into two identical substrings, e.g., abcabc. An approximate single tandem repeat is one in which the substrings are similar, but not identical, e.g., abcdaacd. In this paper we consider two criterions of similarity: the Hamming distance (k mismatches) and the edit distance (k differences). For a string S of length n and an integer k our algorithm reports all locally optimal approximate repeats, r = umacro u, for which the Hamming distance of umacro and u is at most k, in O(nk log (n/k)) time, or all those for which the edit distance of umacro and u is at most k, in O(nk log k log (n/k)) time. This paper concentrates on a more general type of repeat called multiple tandem repeats. A multiple tandem repeat in a sequence S is a (periodic) substring r of S of the form r = u(a)u', where u is a prefix of r and u' is a prefix of u. An approximate multiple tandem repeat is a multiple repeat with errors; the repeated subsequences are similar but not identical. We precisely define approximate multiple repeats, and present an algorithm that finds all repeats that concur with our definition. The time complexity of the algorithm, when searching for repeats with up to k errors in a string S of length n, is O(nka log (n/k)) where a is the maximum number of periods in any reported repeat. We present some experimental results concerning the performance and sensitivity of our algorithm. The problem of finding repeats within a string is a computational problem with important applications in the field of molecular biology. Both exact and inexact repeats occur frequently in the genome, and certain repeats occurring in the genome are known to be related to diseases in the human.

An algorithm for approximate tandem repeats.

In this article, we address a new version of dynamic pattern matching. The dynamic text and static pattern matching problem is the problem of finding a static pattern in a text that is continuously being updated. The goal is to report all new occurrences of the pattern in the text after each text update. We present an algorithm for solving the problem where the text update operation is changing the symbol value of a text location. Given a text of length n and a pattern of length m, our algorithm preprocesses the text in time O(n log log m), and the pattern in time O(m log m). The extra space used is O(n p m log m). Following each text update, the algorithm deletes all prior occurrences of the pattern that no longer match, and reports all new occurrences of the pattern in the text in O(log log m) time. We note that the complexity is not proportional to the number of pattern occurrences, since all new occurrences can be reported in a succinct form.

/pdf/dynamic-text-and-static-pattern-matching-l4g4e7go3a.pdf

Dynamic text and static pattern matching

Motivation: A tandem repeat in DNA is a sequence of two or more contiguous, approximate copies of a pattern of nucleotides. Tandem repeats occur in the genomes of both eukaryotic and prokaryotic organisms. They are important in numerous fields including disease diagnosis, mapping studies, human identity testing (DNA fingerprinting), sequence homology and population studies. Although tandem repeats have been used by biologists for many years, there are few tools available for performing an exhaustive search for all tandem repeats in a given sequence.

Results: In this paper we describe an efficient algorithm for finding all tandem repeats within a sequence, under the edit distance measure. The contributions of this paper are two-fold: theoretical and practical. We present a precise definition for tandem repeats over the edit distance and an efficient, deterministic algorithm for finding these repeats.

Availability: The algorithm has been implemented in C++, and the software is available upon request and can be used at http://www.sci.brooklyn.cuny.edu/~sokol/trepeats. The use of this tool will assist biologists in discovering new ways that tandem repeats affect both the structure and function of DNA and protein molecules.

Contact: sokol@sci.brooklyn.cuny.edu

/pdf/tandem-repeats-over-the-edit-distance-3xlzcw539d.pdf

Tandem repeats over the edit distance

Two equal length strings s and s′, over alphabets Σs and Σs′, parameterize match if there exists a bijection π : Σs r Σs′ such that π (s) = s′, where π (s) is the renaming of each character of s via π. Parameterized matching is the problem of finding all parameterized matches of a pattern string p in a text t, and approximate parameterized matching is the problem of finding at each location a bijection π that maximizes the number of characters that are mapped from p to the appropriate vpv-length substring of t.Parameterized matching was introduced as a model for software duplication detection in software maintenance systems and also has applications in image processing and computational biology. For example, approximate parameterized matching models image searching with variable color maps in the presence of errors.We consider the problem for which an error threshold, k, is given, and the goal is to find all locations in t for which there exists a bijection π which maps p into the appropriate vpv-length substring of t with at most k mismatched mapped elements. Our main result is an algorithm for this problem with O(nk1.5 p mk log m) time complexity, where m = vpv and n=vtv. We also show that when vpv = vtv = m, the problem is equivalent to the maximum matching problem on graphs, yielding a O(m p k1.5) solution.

/pdf/approximate-parameterized-matching-nplr7vzc4h.pdf

Approximate parameterized matching

Two equal length strings s and s′, over alphabets Σ s and Σ s′, parameterize match if there exists a bijection π:Σ s → Σ s′, such that π (s)=s′, where π (s) is the renaming of each character of s via π. Approximate parameterized matching is the problem of finding for a pattern p, at each location of a text string t, a bijection π that maximizes the number of characters that are mapped from p to the appropriate |p|-length substring of t.

Dina Sokol

Papers

An algorithm for approximate tandem repeats.

Dynamic text and static pattern matching

Tandem repeats over the edit distance

Approximate parameterized matching

Approximate Parameterized Matching