CSE 190, Great ideas in algorithms: Pairwise independent hash functions

The goal of hash functions is to map elements from a large domain to a small one. Typically, to obtain the required guarantees, we would need not just one function, but a family of functions, where we would use randomness to sample a hash function from this family. Let H = {h : U → R} be a family of functions, mapping elements from a (large) universe to a (small) range. Ideally, we would like to obtain certatin “random-like” properties of this family, while keeping its size small. This will be important for applications in data structures and streaming algorithms, where we would need to keep a “seed” which tells us which hash function we chose, as well as in de-randomization, where we can replace true random with pseudo-randomness, obtained via enumerating all the hash functions in a family. We start with describing one of the more basic but very useful properties we can require from hash functions, that of being pairwise-independent.