The Careless Coupon Collector's Problem

TL;DR

This paper introduces the Careless Coupon Collector's Problem (CCCP), analyzing how the collection time varies with loss probability and providing algorithms for expected completion time, relevant for information systems like web crawlers and caches.

Contribution

It defines the CCCP model, analyzes its phase transitions, and develops an efficient algorithm to compute expected collection times, extending classical coupon collector insights.

Findings

Collection time transitions from Θ(n log n) to exponential in n depending on p.

Metastable phase where collected fraction concentrates around 1/(1+c).

Algorithm computes expected completion time in O(n^2) time.

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.