The Weighted Coupon Collector’s Problem and Applications

Abstract

In the classical coupon collector’s problem n coupons are given. In every step one of the n coupons is drawn uniformly at random (with replacement) and the goal is to obtain a copy of all the coupons. It is a well-known fact that in expectation $n \sum_{k=1}^n 1/k \approx n \ln n$ steps are needed to obtain all coupons.

In this paper we show two results. First we revisit the weighted coupon collector case where in each step every coupon i is drawn with probability p _i . Let p = (p ₁,..., p _n ). In this setting exact but complicated bounds are known for ${\mathbf{E}}[{ \mathcal{C} ({\mathbf{p}})] }$, which is the expected time to obtain all n coupons. Here we suggest the following rather simple way to approximate ${\mathbf{E}}[{ \mathcal{C} ({\mathbf{p}})] }$. Assume p ₁ ≤ p ₂ ≤ ⋯ ≤ p _n and take $\sum_{i=1}^n 1/(i p_i)$ as an approximation. We prove that, rather unexpectedly, this expression approximates $\operatorname{\mathbf{E}}\left[ \mathcal{C} ({\mathbf{p}}) \right]$ by a factor of Θ(loglogn). We also present an extension that achieves an approximation factor of ${\mathcal{O}}(\log \log \log n)$.

In the second part of the paper we derive some combinatorial properties of the coupon collecting processes. We apply these properties to show results for the following simple randomized broadcast algorithm. A graph G is given and one node is initially informed. In each round, every informed node chooses a random neighbor and informs it. We restrict G to the class of trees and we show that the expected broadcast time is maximized if and only if G is the star graph. Besides being the first rigorous extremal result, our finding nicely contrasts with a previous result by Brightwell and Winkler [2] showing that for the star graph the cover time of a random walk is minimized among all trees.