Document Open Access Logo

The Careless Coupon Collector’s Problem

Authors Emilio Cruciani , Aditi Dudeja

PDF

File

Thumbnail PDF
PDF
LIPIcs.FUN.2026.14.pdf
  • Filesize: 4.18 MB
  • 21 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Mathematics of computing
Keywords
  • Coupon Collector
  • Markov Chains
  • Metastability

Metrics

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 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 Θ(n ln n) when p = o((ln n)/n²) up to Θ(((np)/(1-p))ⁿ) when p = ω(1/n) in multiple distinct phases. Interestingly, when p = c/n, the process remains in a metastable phase, where the fraction of collected coupon types is concentrated around 1/(1+c) with probability 1-o(1), for a time window of length e^{Θ(n)}. Finally, we give an algorithm that computes the expected completion time of CCCP in O(n²) time.

Cite As Get BibTex

Emilio Cruciani and Aditi Dudeja. The Careless Coupon Collector’s Problem. In 13th International Conference on Fun with Algorithms (FUN 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 366, pp. 14:1-14:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.FUN.2026.14

Author Details

Emilio Cruciani
  • European University of Rome, Italy
Aditi Dudeja
  • The School of Data Science, The Chinese University of Hong Kong, Shenzhen, China

Funding

  • Dudeja, Aditi: This work was initiated while the author was affiliated with the University of Salzburg. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 947702).

Acknowledgements

We thank Silvia Cruciani for the CCCP illustration.

References

  1. Ilan Adler, Hyun-Soo Ahn, Richard M Karp, and Sheldon M Ross. A probabilistic model for the survivability of cells. Journal of Applied Probability, 42(4):919-931, 2005. URL: https://www.jstor.org/stable/30040818.
  2. Ilan Adler, Shmuel Oren, and Sheldon M Ross. The coupon-collector’s problem revisited. Journal of Applied Probability, 40(2):513-518, 2003. URL: https://www.jstor.org/stable/3215807.
  3. Dan Alistarh and Peter Davies. Collecting coupons is faster with friends. In International Colloquium on Structural Information and Communication Complexity, pages 3-12. Springer, 2021. URL: https://doi.org/10.1007/978-3-030-79527-6_1.
  4. Emmanuelle Anceaume, Yann Busnel, Ernst Schulte-Geers, and Bruno Sericola. Optimization results for a generalized coupon collector problem. Journal of Applied Probability, 53(2):622-629, 2016. URL: https://doi.org/10.1017/jpr.2016.27.
  5. Emmanuelle Anceaume, Yann Busnel, and Bruno Sericola. New results on a generalized coupon collector problem using markov chains. Journal of Applied Probability, 52(2):405-418, 2015. URL: https://doi.org/10.1239/jap/1437658606.
  6. Srinivasan Arunachalam, Aleksandrs Belovs, Andrew M Childs, Robin Kothari, Ansis Rosmanis, and Ronald de Wolf. Quantum coupon collector. In 15th Conference on the Theory of Quantum Computation, Communication and Cryptography, 2020. URL: https://doi.org/10.4230/LIPIcs.TQC.2020.10.
  7. Siva Athreya, Satyaki Mukherjee, and Soumendu Sundar Mukherjee. On a generalisation of the coupon collector problem. Journal of Theoretical Probability, 38(3):54, 2025. URL: https://doi.org/10.1007/s10959-025-01417-w.
  8. Petra Berenbrink and Thomas Sauerwald. The weighted coupon collector’s problem and applications. In International Computing and Combinatorics Conference, pages 449-458. Springer, 2009. URL: https://doi.org/10.1007/978-3-642-02882-3_45.
  9. Shoham Shimon Berrebi, Eitan Yaakobi, Zohar Yakhini, and Daniella Bar-Lev. The labeled coupon collector problem with random sample sizes and partial recovery. arXiv preprint arXiv:2502.02968, 2025. URL: https://doi.org/10.48550/arXiv.2502.02968.
  10. Arnon Boneh and Micha Hofri. The coupon-collector problem revisited—a survey of engineering problems and computational methods. Stochastic Models, 13(1):39-66, 1997. URL: https://doi.org/10.1080/15326349708807412.
  11. Peter J Brockwell. The extinction time of a birth, death and catastrophe process and of a related diffusion model. Advances in Applied Probability, 17(1):42-52, 1985. URL: https://doi.org/10.2307/1427051.
  12. Gregory Dobson and Tolga Tezcan. Optimal sampling strategies in the coupon collector’s problem with unknown population size. Annals of Operations Research, 233(1):77-99, 2015. URL: https://doi.org/10.1007/s10479-014-1563-0.
  13. Devdatt P. Dubhashi and Alessandro Panconesi. Concentration of measure for the analysis of randomized algorithms. Cambridge University Press, 2009. Google Scholar
  14. Pál Erdős and Alfréd Rényi. On a classical problem of probability theory. A Magyar Tudományos Akadémia Matematikai Kutató Intézetének Közleményei, 6(1-2):215-220, 1961. URL: https://real.mtak.hu/id/eprint/201134.
  15. William Feller et al. An introduction to probability theory and its applications, volume 963. Wiley New York, 1971. Google Scholar
  16. Marco Ferrante and Monica Saltalamacchia. The coupon collector’s problem. Materials matematics, pages 0001-35, 2014. Google Scholar
  17. Philippe Flajolet, Danièle Gardy, and Loÿs Thimonier. Birthday paradox, coupon collectors, caching algorithms and self-organizing search. Discrete Applied Mathematics, 39(3):207-229, 1992. URL: https://doi.org/10.1016/0166-218X(92)90177-C.
  18. Gene H Golub and Charles F Van Loan. Matrix computations. JHU press, 2013. Google Scholar
  19. Lars Holst. On birthday, collectors', occupancy and other classical urn problems. International Statistical Review/Revue Internationale de Statistique, pages 15-27, 1986. URL: https://doi.org/10.2307/1403255.
  20. Jelena Jocković and Bojana Todić. Coupon collector problem with reset button. Mathematics, 12(2):239, 2024. URL: https://doi.org/10.3390/math12020239.
  21. Tipaluck Krityakierne and Thotsaporn Aek Thanatipanonda. The slowest coupon collector’s problem. Journal of Systems Science and Complexity, pages 1-13, 2025. URL: https://doi.org/10.1007/s11424-025-4223-3.
  22. David A Levin and Yuval Peres. Markov chains and mixing times, volume 107. American Mathematical Soc., 2017. Google Scholar
  23. Stephen N Luko. The “coupon collector’s problem” and quality control. Quality Engineering, 21(2):168-181, 2009. URL: https://doi.org/10.1080/08982110802642555.
  24. Michael Mitzenmacher and Eli Upfal. Probability and computing: Randomization and probabilistic techniques in algorithms and data analysis. Cambridge university press, 2017. Google Scholar
  25. Donald J Newman. The double dixie cup problem. The American Mathematical Monthly, 67(1):58-61, 1960. URL: http://www.jstor.org/stable/2308930.
  26. Prabhakar Raghavan and Rajeev Motwani. Randomized algorithms. Cambridge University Press Cambridge, 1995. Google Scholar
  27. Ran Raz and Wei Zhan. The Random-Query Model and the Memory-Bounded Coupon Collector. In Thomas Vidick, editor, 11th Innovations in Theoretical Computer Science Conference (ITCS 2020), volume 151 of Leibniz International Proceedings in Informatics (LIPIcs), pages 20:1-20:11, Dagstuhl, Germany, 2020. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.ITCS.2020.20.
  28. Bojana Todić. Coupon collector problem with penalty coupon. Matematički vesnik, 2023. URL: https://doi.org/10.57016/MV-BGON6192.
  29. Hermann Von Schelling. Coupon collecting for unequal probabilities. The American Mathematical Monthly, 61(5):306-311, 1954. URL: https://doi.org/10.1080/00029890.1954.11988466.
  30. Weiyu Xu and A Kevin Tang. A generalized coupon collector problem. Journal of Applied Probability, 48(4):1081-1094, 2011. URL: https://doi.org/10.1239/jap/1324046020.
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2026 Schloss Dagstuhl – LZI GmbH Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact