Hashing is everywhere. To Start, hash tables are one of the most widely used primitive data structures, with numerous variations (open address, chained, linear probing, multiple-choice, cuckoo, and so on). Hashing is also frequently used for sampling; hash all items and keep only those with certain hash values as the sample. Hashing further plays a major role in a variety of algorithms and data structures for data sketches for both streaming and non-streaming data, such as Bloom filters and approximate counting structures.
For much of the early history of hashing, there was a clear divide between theory and practice. The mathematical analysis of hashing and hashing algorithms was (and often still is) based on perfect randomness. You assume that for each input x, the hash value h(x) is uniformly distributed over all possible values it could take on, and that each value h(x) is independent of all other hash values h(y) for y≠x. Such perfect hash functions make mathematical analysis much simpler, as every new hash value looks completely random.
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