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Sensible hashing of variable-length strings is impossible
Consider the problem of hashing an infinite number of keys—such as the set of all strings of any length—to the set of numbers in {1,2,…,b}. Random hashing proceeds by first picking at random a hash function h in a family of H functions.
I will show that there is always an infinite number of keys that are certain to collide over all H functions, that is, they always have the same hash value for all hash functions.
Pick any hash function h: there is at least one hash value y such that the set A={s:h(s)=y} is infinite, that is, you have infinitely many strings colliding. Pick any other hash function h‘ and hash the keys in the set A. There is at least one hash value y‘ such that the set A‘={s is in A: h‘(s)=y‘} is infinite, that is, there are infinitely many strings colliding on two hash functions. You can repeat this argument any number of times. Repeat it H times. Then you have an infinite set of strings where all strings collide over all of your H hash functions. CQFD.
Further reading: Do hash tables work in constant time? (blog post) and Recursive n-gram hashing is pairwise independent, at best (research paper).