Analyzing Collection Strategies: A Computational Perspective on the Coupon Collector Problem

TL;DR

This paper introduces three algorithms based on Markov models for efficiently computing moments of the Coupon Collector Problem under arbitrary drawing probabilities, addressing computational challenges in practical applications.

Contribution

It presents novel algorithms that enable exact computation of expectation and variance for the CCP with any drawing distribution, improving computational efficiency and generality.

Findings

Algorithms compute expectation, variance, and second moments.

Polynomial time algorithm for uniform distribution case.

Extension to arbitrary drawing distributions.

Abstract

The Coupon Collector Problem (CCP) is a well-known combinatorial problem that seeks to estimate the number of random draws required to complete a collection of $n$ distinct coupon types. Various generalizations of this problem have been applied in numerous engineering domains. However, practical applications are often hindered by the computational challenges associated with deriving numerical results for moments and distributions. In this work, we present three algorithms for solving the most general form of the CCP, where coupons are collected under any arbitrary drawing probability, with the objective of obtaining $t$ copies of a subset of $k$ coupons from a total of $n$. The First algorithm provides the base model to compute the expectation, variance, and the second moment of the collection process. The second algorithm utilizes the construction of the base model and computes the same values in polynomial time with respect to $n$ under the uniform drawing distribution, and the third algorithm extends to any general drawing distribution. All algorithms leverage Markov models specifically designed to address computational challenges, ensuring exact computation of the expectation and variance of the collection process. Their implementation uses a dynamic programming approach that follows from the Markov models framework, and their time complexity is analyzed accordingly.