Approximate Matching in Weighted Sequences

• Published in: Combinatorial Pattern Matching pp. 365 - 376
• Main Authors: Amir, Amihood, Iliopoulos, Costas, Kapah, Oren, Porat, Ely
• Format: Book Chapter Conference Proceeding
• Language: English
• Published: Berlin, Heidelberg Springer Berlin Heidelberg 2006; Springer
• Series: Lecture Notes in Computer Science
• Subjects: Applied sciences; Artificial intelligence; Computer science; control theory; systems; Data processing. List processing. Character string processing; Edit Distance; Exact sciences and technology; Memory organisation. Data processing; Pattern recognition. Digital image processing. Computational geometry; Software; Text Element; Text Location; Weighted Match; Zero Probability; Data analysis; Statistical analysis; Hamming distance; Probabilistic approach; Similarity; Image processing; Combinatorial problem; String matching; Text; Pattern recognition; Infinite alphabet; Character string; Image sequence; Localization; Pattern matching
• ISBN: 3540354557; 9783540354550
• ISSN: 0302-9743; 1611-3349
• DOI: 10.1007/11780441_33

Weighted sequences have been recently introduced as a tool to handle a set of sequences that are not identical but have many local similarities. The weighted sequence is a “statistical image” of this set, where the probability of every symbol’s occurrence at every text location is given. We address the problem of approximately matching a pattern in such a weighted sequence. The pattern is a given string and we seek all locations in the set where the pattern occurs with a high enough probability. We define the notion of Hamming distance and edit distance in weighted sequences and give efficient algorithms for computing them. We compute two versions of the Hamming distance in time \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$O(n \sqrt{m\log m})$\end{document}, where n is the length of the weighted text and m is the pattern length. The edit distance is computed in time O(nm) and O(nm2), depending on the edit distance definition used. Unfortunately, due to space considerations, the edit distance details are left to the journal version. We also define the notion of weighted matching in infinite alphabets and show that exact weighted matching can be computed in time O(slog2s), where s is the number of text symbols having non-zero probability. The weighted Hamming distance over infinite alphabets can be computed in time \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\min(O(kn\sqrt{s}+s^{3/2}\log^2s), O(s^{4/3}m^{1/3}\log s))$\end{document}.