General asymptotic estimates for the Coupon Collector Problem

Boneh, Shahar; Papanicolaou, Vassilis G.

doi:10.1016/0377-0427(95)00020-8

Cited by 21 publications

(32 citation statements)

References 7 publications

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“…Note that this formula holds regardless of whether ∑ D k= p k = holds (here, it does not). This function has been studied extensively in the mathematics literature and not only [21,6,1,12]. In this study, we want to prove that it is positive and decreasing in p. As shown in Lemma 4, maximizing p amounts to maximizing the degrees of the carrier nodes.…”

Section: Expected *-Ast Timementioning

confidence: 98%

Minimum Expected *-Cast Time in DTNs

Picu

Spyropoulos

2010

Lecture Notes of the Institute for Computer Sciences, Social Informatics and Telecommunications Engineering

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Abstract. Delay Tolerant Networks (DTNs) are wireless networks in which end-to-end connectivity is sporadic. Routing in DTNs uses past connectivity information to predict future node meeting opportunities. Recent research efforts consider the use of social network analysis (i.e., node communities, centralities etc.) for this forecast. However, most of these works focus on unicast. We believe that group communication is the natural basis of most applications envisioned for DTNs. To this end, we study constrained *-cast (broad-, multi-and anycast) in DTNs. The constraint is on the number of copies of a message and the goal is to find the best relay nodes for those copies, that will provide a small delivery delay and a good coverage. After defining a solid probabilistic model for DTNs collecting social information, we prove a near-optimal policy for our constrained *-cast problems that minimizes the expected delivery delay. We verify it through simulation on both real and synthetic mobility traces.

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Section: Expected *-Ast Timementioning

confidence: 98%

Minimum Expected *-Cast Time in DTNs

Picu

Spyropoulos

2010

Lecture Notes of the Institute for Computer Sciences, Social Informatics and Telecommunications Engineering

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“…In this problem, we consider how many trials are needed for collecting all coupons whose occurrence probabilities follow some stable distribution. According to a well-known result about power law distributions (Boneh and Papanicolaou, 1996), we need a corpus of size kW k 1−π k ln W when π k < 1, and W ln 2 W when π k = 1 for collecting all of the k-grams, the number of which is W k . Using results in (Atsonios et al, 2011), we can easily obtain a lower and upper bound of the actual vocabulary sizeW k of k-grams from the corpus size N and vocabulary size W as…”

Section: Lemma 4 For Any Corpus W Following Zipf's Law the Maximum mentioning

confidence: 99%

Perplexity on Reduced Corpora

Kobayashi

2014

Proceedings of the 52nd Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers)

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This paper studies the idea of removing low-frequency words from a corpus, which is a common practice to reduce computational costs, from a theoretical standpoint. Based on the assumption that a corpus follows Zipf's law, we derive tradeoff formulae of the perplexity of k-gram models and topic models with respect to the size of the reduced vocabulary. In addition, we show an approximate behavior of each formula under certain conditions. We verify the correctness of our theory on synthetic corpora and examine the gap between theory and practice on real corpora.

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“…As noted in [4] for E[T N ], the problem of estimating E[T N (T N + 1)] as N → ∞ can be treated as two separate problems, namely estimating A 2 N (i.e. A N ) and estimating Q N (α).…”

Section: Large N Asymptoticsmentioning

confidence: 99%

“…In order to proceed, we assume that f (x) possesses three derivatives satisfying the following conditions as x → ∞: [4] the conditions on f (x) were slightly weaker). These conditions are satisfied by a variety of commonly used functions.…”

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confidence: 99%

The Coupon Collector's Problem Revisited: Asymptotics of the Variance

Doumas

Papanicolaou

2012

Advances in Applied Probability

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We develop techniques for computing the asymptotics of the first and second moments of the number TN of coupons that a collector has to buy in order to find all N existing different coupons as N → ∞. The probabilities (occurring frequencies) of the coupons can be quite arbitrary. From these asymptotics we obtain the leading behavior of the variance V[TN] of TN (see Theorems 3.1 and 4.4). Then, we combine our results with the general limit theorems of Neal in order to derive the limit distribution of TN (appropriately normalized), which, for a large class of probabilities, turns out to be the standard Gumbel distribution. We also give various illustrative examples.

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General asymptotic estimates for the Coupon Collector Problem

Cited by 21 publications

References 7 publications

Minimum Expected *-Cast Time in DTNs

Minimum Expected *-Cast Time in DTNs

Perplexity on Reduced Corpora

The Coupon Collector's Problem Revisited: Asymptotics of the Variance

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