TL;DR
The paper introduces the Bottleneck Birthday Problem, a variant of the classic birthday problem, analyzing the maximum group size for a probability threshold that no day exceeds r birthdays.
Contribution
It surveys existing techniques for occupancy problems, introduces a novel recurrence using restricted Stirling numbers, and provides complexity analysis and numerical results.
Findings
Derived recurrence relations for exact probability computation.
Introduced a new recurrence using restricted Stirling numbers.
Provided complexity comparisons and numerical results.
Abstract
We introduce a fun problem that can be considered as a variant of the classic birthday problem, the Bottleneck Birthday Problem (BBP). It is stated as: what is the maximum number of people we have to choose so that no day of the year has more than r >= 1 birthdays incident on it with probability at least 1/2? We provide a survey of techniques used in the literature on occupancy and load balancing problems to derive recurrence relations for exact computation of the probability, and the number of people keeping probability fixed at a threshold. Further, we show that restricted Stirling numbers of the second kind can be used to derive an additional recurrence, in a novel way. We provide complexity comparisons and numerical results from an implementation of the recurrences.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.