Epidemic algorithms for replicated database maintenance
PODC '87 Proceedings of the sixth annual ACM Symposium on Principles of distributed computing
Extermal cover times for random walks on trees
Journal of Graph Theory
Birthday paradox, coupon collectors, caching algorithms and self-organizing search
Discrete Applied Mathematics
A tight upper bound on the cover time for random walks on graphs
Random Structures & Algorithms
A tight lower bound on the cover time for random walks on graphs
Random Structures & Algorithms
FOCS '00 Proceedings of the 41st Annual Symposium on Foundations of Computer Science
SFCS '88 Proceedings of the 29th Annual Symposium on Foundations of Computer Science
On the runtime and robustness of randomized broadcasting
ISAAC'06 Proceedings of the 17th international conference on Algorithms and Computation
The design space of probing algorithms for network-performance measurement
Proceedings of the ACM SIGMETRICS/international conference on Measurement and modeling of computer systems
Hi-index | 0.00 |
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 ***(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.