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The weighted words collector

Jérémie Du Boisberranger 1 Danièle Gardy 1 Yann Ponty 2, 3, * 
* Corresponding author
3 AMIB - Algorithms and Models for Integrative Biology
LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau], LRI - Laboratoire de Recherche en Informatique, UP11 - Université Paris-Sud - Paris 11, Inria Saclay - Ile de France
Abstract : We consider the word collector problem, i.e. the expected number of calls to a random weighted generator before all the words of a given length in a language are generated. The originality of this instance of the non-uniform coupon collector lies in the, potentially large, multiplicity of the words/coupons of a given probability/composition. We obtain a general theorem that gives an asymptotic equivalent for the expected waiting time of a general version of the Coupon Collector. This theorem is especially well-suited for classes of coupons featuring high multiplicities. Its application to a given language essentially necessitates knowledge on the number of words of a given composition/probability. We illustrate the application of our theorem, in a step-by-step fashion, on four exemplary languages, whose analyses reveal a large diversity of asymptotic waiting times, generally expressible as $\kappa \cdot m^p \cdot (\log{m})^q \cdot (\log \log{m})^r$, for $m$ the number of words, and $p, q, r$ some positive real numbers.
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Submitted on : Monday, April 16, 2012 - 11:16:52 PM
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Jérémie Du Boisberranger, Danièle Gardy, Yann Ponty. The weighted words collector. AOFA - 23rd International Meeting on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms - 2012, Nicolas, Broutin (INRIA, France) and Luc, Devroye (McGill, Canada), Jun 2012, Montreal, Canada. pp.243--264, ⟨10.46298/dmtcs.2998⟩. ⟨hal-00666399v2⟩



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