 Universal hash family

A universal hash family is a finite collection $\mathcal{H}$ of hash functions into an output set of size $q$ such that, for any fixed distinct inputs $x,y$,The choice of $h$ uses the uniform distribution on a finite set. The keys are fixed before that choice: the randomness belongs to the hash function, so the guarantee does not assume naturally random keys. Choose one member and keep it while using the table; changing it requires redistributing the stored records. The bound holds separately for every distinct pair, without requiring uniform individual outputs or independent random variables. A strongly universal hash family supplies those stronger properties.

The practical payoff follows from linearity of expectation. If $n$ stored keys are distinct from a fixed query, let $C$ count how many share its bucket. Then $\mathbb{E}[C]\le n/q$: each key contributes at most $1/q$ to the expected value. With $100$ stored keys and $100$ buckets, an unsuccessful search encounters at most one record on average over the random choice of hash function. This is an expected value, not a limit on every search; one unlucky choice can still create a long chain.

This calculation needs no independence between the collision events: linearity of expectation adds their individual probabilities. The fixed-query condition matters, since deliberately choosing keys after inspecting the selected hash function can target a crowded bucket.

Carter and Wegman introduced this way of analyzing storage and retrieval in their 1977 paper: average over a chosen class of hash functions while allowing any fixed sequence of inputs. Their 1979 journal version develops the same universal-pair collision condition and constructions. The insight is useful because a randomized algorithm can obtain a performance guarantee through its own choices, even when its inputs have inconvenient patterns.^[ref][ref]