加权词收集器
The weighted words collector
摘要：
出于生物信息学中的应用，我们考虑的字收集器的问题，即预期数量的调用到一个随机加权生成器的长度$n$之前，获得完整的集合。非均匀优惠券收集器的该实例的独创性在于给定概率/组成的单词/优惠券的潜在大的多样性。我们得到一个一般定理，给出了一个渐近等价的预期等待时间的一般版本的优惠券收集器。这个定理特别适合于具有高多重性的优惠券类。它的应用程序到一个给定的语言基本上需要一些知识的单词数量的一个给定的组成/概率。我们说明了我们的定理的应用，在一步一步的方式，在三个示例性的语言，揭示渐近制度$Θ(μ(n)\cdot n)$和$Θ(μ(n)\cdot \log n)$，其中$μ(n)$是长度为$n$的单词的权重之和。
英文摘要：
Motivated by applications in bioinformatics, we consider the word collector problem, i.e. the expected number of calls to a random weighted generator of words of length $n$ before the full collection is obtained. 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 some 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 three exemplary languages, revealing asymptotic regimes in $Θ(μ(n)\cdot n)$ and $Θ(μ(n)\cdot \log n)$, where $μ(n)$ is the sum of weights over words of length $n$.
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