The Careless Coupon Collector's Problem

Abstract

We initiate the study of the Careless Coupon Collector's Problem (CCCP), a novel variation of the classical coupon collector, that we envision as a model for information systems such as web crawlers, dynamic caches, and fault-resilient networks. In CCCP, a collector attempts to gather $n$ distinct coupon types by obtaining one coupon type uniformly at random in each discrete round, however the collector is \textit{careless}: at the end of each round, each collected coupon type is independently lost with probability $p$. We analyze the number of rounds required to complete the collection as a function of $n$ and $p$. In particular, we show that it transitions from $\Theta(n \ln n)$ when $p = o\big(\frac{\ln n}{n^2}\big)$ up to $\Theta\big((\frac{np}{1-p})^n\big)$ when $p=\omega\big(\frac{1}{n}\big)$ in multiple distinct phases. Interestingly, when $p=\frac{c}{n}$, the process remains in a metastable phase, where the fraction of collected coupon types is concentrated around $\frac{1}{1+c}$ with probability $1-o(1)$, for a time window of length $e^{\Theta(n)}$. Finally, we give an algorithm that computes the expected completion time of CCCP in $O(n^2)$ time.