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On the Distribution of the Number of Missing Words in Random Texts

Published online by Cambridge University Press:  28 January 2003

SVEN RAHMANN  and
ERIC RIVALS
SVEN RAHMANN
Affiliation:
Department of Computational Molecular Biology, Max-Planck-Institut für Molekulare Genetik, Ihnestraße 63-73, D-14195 Berlin, [email protected]
ERIC RIVALS
Affiliation:
L.I.R.M.M., CNRS U.M.R. 5506, 161 rue Ada, F-34392 Montpellier Cedex 5, [email protected]
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Abstract

Determining the distribution of the number of empty urns after a number of balls have been thrown randomly into the urns is a classical and well understood problem. We study a generalization: Given a finite alphabet of size σ and a word length q, what is the distribution of the number X of words (of length q) that do not occur in a random text of length n+q−1 over the given alphabet? For q=1, X is the number Y of empty urns with σ urns and n balls. For q[ges ]2, X is related to the number Y of empty urns with σq urns and n balls, but the law of X is more complicated because successive words in the text overlap. We show that, perhaps surprisingly, the laws of X and Y are not as different as one might expect, but some problems remain currently open.

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 12 , Issue 1 , January 2003 , pp. 73 - 87
Copyright
2003 Cambridge University Press

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