The Bottleneck Birthday Problem

TL;DR

The paper studies the Bottleneck Birthday Problem, seeking the largest group size n that keeps the maximum daily count of birthdays at or below r with probability at least 1/2. It develops and analyzes multiple exact recurrences for computing P(m,n,r), including one-day-at-a-time, counting-based, restricted-Stirling-number-based, and direct probability recurrences, each with distinct space-time tradeoffs. The authors implement these recurrences with arbitrary-precision arithmetic and compare their performance, highlighting the direct probability recurrence as the most efficient for probability evaluation and the counting approaches for structural insight. The results have practical relevance to hashing, load balancing, and resource allocation, and the work outlines directions for extending BBP to non-uniform distributions and higher-dimensional settings.

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 $r \geq 1$ 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 numerical results from an implementation of the recurrences.