Not all keys can be hashed in constant time.

The authors present a simple new model that captures many natural (sequential and parallel) hashing algorithms. In a game against nature, the algorithm and coin-tosses cause the evolution of a random tree, whose size corresponds to space (hash table size), and two notions of depth correspond respectively to the largest probe sequences for insertion (parallel insertion time) and search of a key. It was observed, that although average insertion time per element is constant, parallel application of this (and other) algorithms cannot work in constant time. The reason is that while the average is constant, some elements will have to be hashed nonconstant number of times. The main results exhibit tight trade-offs between space and parallel time, in the basic model and three variants, which capture standard hashing practice.