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Hidden pattern statistics. (English) Zbl 0986.68136

Orejas, Fernando (ed.) et al., Automata, languages and programming. 28th international colloquium, ICALP 2001, Crete, Greece, July 8-12, 2001. Proceedings. Berlin: Springer. Lect. Notes Comput. Sci. 2076, 152-165 (2001).
Summary: We consider the sequence comparison problem, also known as “hidden pattern” problem, where one searches for a given subsequence in a text (rather than a string understood as a sequence of consecutive symbols). A characteristic parameter is the number of occurrences of a given pattern \(w\) of length \(m\) as a subsequence in a random text of length \(n\) generated by a memoryless source. Spacings between letters of the pattern may either be constrained or not in order to define valid occurrences. We determine the mean and the variance of the number of occurrences, and establish a Gaussian limit law. These results are obtained via combinatorics on words, formal language techniques, and methods of analytic combinatorics based on generating functions and convergence of moments. The motivation to study this problem comes from an attempt at finding a reliable threshold for intrusion detections, from textual data processing applications, and from molecular biology.
For the entire collection see [Zbl 0967.00069].

MSC:

68T20 Problem solving in the context of artificial intelligence (heuristics, search strategies, etc.)
68R15 Combinatorics on words
68Q45 Formal languages and automata
60F05 Central limit and other weak theorems
62P99 Applications of statistics
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