Chijul B. Tripathy

Chijul B. Tripathy

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.

Chijul B. Tripathy

We revisit the Strong Birthday Problem (SBP) introduced by DasGupta'05, which asks for the minimum population n required such that, with a probability of at least 1/2, every individual in the group shares a birthday with at least one other person. Formally, we develop and analyze computational frameworks to determine the probability that in a group of n people with birthdays distributed over m days, each day either has two or more birthdays or is birthday-free. We derive both counting-based and probability-based recurrence relations to solve this problem and establish a novel connection to associated Stirling numbers of the second kind. This relationship is exploited to derive new, more efficient recurrences. Finally, we implement these recurrences using dynamic programming, provide analysis of their asymptotic complexities, and present numerical evaluations that demonstrate the practical efficiency and scalability of our proposed approaches.