Ode on a random urn (functional pearl)

We present the urn, a simple tree-based data structure that supports sampling from and updating discrete probability distributions in logarithmic time. We avoid the usual complexity of traditional self-balancing binary search trees by not keeping values in a specific order. Instead, we keep the tree maximally balanced at all times using a single machine word of overhead: its size.

Urns provide an alternative interface for the frequency combinator from the QuickCheck library that allows for asymptotically more efficient sampling from dynamically-updated distributions. They also facilitate backtracking in property-based random testing, and can be applied to such complex examples from the literature as generating well-typed lambda terms or information flow machine states, demonstrating significant speedups.