Algorithmica

, Volume 22, Issue 4, pp 631-649

First online:

On Pattern Frequency Occurrences in a Markovian Sequence

  • M. RégnierAffiliated withINRIA, Rocquencourt, 78153 Le Chesnay Cedex, France. Mireille.Regnier@inria.fr.
  • , W. SzpankowskiAffiliated withDepartment of Computer Science, Purdue University, West Lafayette, IN 47907, USA. spa@cs.purdue.edu.

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Abstract.

Consider a given pattern H and a random text T generated by a Markovian source. We study the frequency of pattern occurrences in a random text when overlapping copies of the pattern are counted separately. We present exact and asymptotic formulae for moments (including the variance), and probability of r pattern occurrences for three different regions of r , namely: (i) r=O(1) , (ii) central limit regime, and (iii) large deviations regime. In order to derive these results, we first construct certain language expressions that characterize pattern occurrences which are later translated into generating functions. We then use analytical methods to extract asymptotic behaviors of the pattern frequency from the generating functions. These findings are of particular interest to molecular biology problems (e.g., finding patterns with unexpectedly high or low frequencies, and gene recognition), information theory (e.g., second-order properties of the relative frequency), and pattern matching algorithms (e.g., q -gram algorithms).

Key words. Frequency of pattern occurrences, Markov source, Autocorrelation polynomials, Languages, Generating functions, Asymptotic analysis, Large deviations.