收集优惠券是更快的朋友
Collecting Coupons is Faster with Friends
摘要：
在这篇文章中，我们介绍了一个分布式扭曲的经典优惠券收集器问题：一组$m$收集器希望每个人都获得一组$n$优惠券，为此，他们可以每个样品优惠券均匀随机，但也可以满足成对的互动，在此期间，他们可以交换优惠券。通过这样做，他们希望减少每个收藏家为了获得全套而必须抽样的优惠券数量。当考虑到息票收集器现象的现实表现时，这种扩展是自然的，并且已经被评论和经验研究[Hayes and Hannigan 2006，Ahmad et al. 2014，Delmarcelle 2019]。我们为这种情况提供了第一个理论分析。我们发现，“优惠券收集与朋友”确实可以显着减少优惠券的数量，每个收藏家必须采样，并提出了有趣的连接到更传统的变种的问题。虽然我们的分析是在大多数情况下渐近紧，有几个开放的问题，关于细粒度的分析“优惠券收集与朋友”，和一个长期研究的原始问题的变种，其中一个收藏家需要多个全套的优惠券。
英文摘要：
In this note, we introduce a distributed twist on the classic coupon collector problem: a set of $m$ collectors wish to each obtain a set of $n$ coupons; for this, they can each sample coupons uniformly at random, but can also meet in pairwise interactions, during which they can exchange coupons. By doing so, they hope to reduce the number of coupons that must be sampled by each collector in order to obtain a full set. This extension is natural when considering real-world manifestations of the coupon collector phenomenon, and has been remarked upon and studied empirically [Hayes and Hannigan 2006, Ahmad et al. 2014, Delmarcelle 2019]. We provide the first theoretical analysis for such a scenario. We find that "coupon collecting with friends" can indeed significantly reduce the number of coupons each collector must sample, and raises interesting connections to the more traditional variants of the problem. While our analysis is in most cases asymptotically tight, there are several open questions raised, regarding finer-grained analysis of both "coupon collecting with friends", and of a long-studied variant of the original problem in which a collector requires multiple full sets of coupons.
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