When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.AofA.2020.17
URN: urn:nbn:de:0030-drops-120476
URL: https://drops.dagstuhl.de/opus/volltexte/2020/12047/
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### Hidden Words Statistics for Large Patterns

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### Abstract

We study here the so called subsequence pattern matching also known as hidden pattern matching in which one searches for a given pattern w of length m as a subsequence in a random text of length n. The quantity of interest is the number of occurrences of w as a subsequence (i.e., occurring in not necessarily consecutive text locations). This problem finds many applications from intrusion detection, to trace reconstruction, to deletion channel, and to DNA-based storage systems. In all of these applications, the pattern w is of variable length. To the best of our knowledge this problem was only tackled for a fixed length m=O(1) [P. Flajolet et al., 2006]. In our main result Theorem 5 we prove that for m=o(n^{1/3}) the number of subsequence occurrences is normally distributed. In addition, in Theorem 6 we show that under some constraints on the structure of w the asymptotic normality can be extended to m=o(√n). For a special pattern w consisting of the same symbol, we indicate that for m=o(n) the distribution of number of subsequences is either asymptotically normal or asymptotically log normal. We conjecture that this dichotomy is true for all patterns. We use Hoeffding’s projection method for U-statistics to prove our findings.

### BibTeX - Entry

```@InProceedings{janson_et_al:LIPIcs:2020:12047,
author =	{Svante Janson and Wojciech Szpankowski},
title =	{{Hidden Words Statistics for Large Patterns}},
booktitle =	{31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2020)},
pages =	{17:1--17:15},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-147-4},
ISSN =	{1868-8969},
year =	{2020},
volume =	{159},
editor =	{Michael Drmota and Clemens Heuberger},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},