High Speed Hashing for Integers and Strings

The concept of truly independent hash functions is extremely useful in the design of randomized algorithms. We have a large universe U of keys, e.g., 64-bit numbers, that we wish to map randomly to a range [m] = {0, ...,m− 1} of hash values. A truly random hash function h : U → [m] assigns an independent uniformly random variable h(x) to each key in x. The function h is thus a |U |-dimensional random variable, picked uniformly at random among all functions from U to [m]. When h has been picked, it is no longer random, so h(x) is fixed for all keys x ∈ U . Unfortunately truly random hash functions are idealized objects that cannot be implemented. More precisely, to represent a truly random hash function, we need to store at least |U | log2m bits, and in most applications of hash functions, the whole point in hashing is that the universe is much too large for such a representation (at least not in fast internal memory). The idea is to let hash functions contain only a small element or seed of randomness so that the hash function is sufficiently random for the desired application, yet so that the seed is small enough that we can store it when first it is fixed. In these notes we will discuss some basic forms of random hashing that are very efficient to implement, and yet have sufficient randomness for some very important applications.