A perfect hash function for a set S is a hash function that maps distinct elements in S to distinct integers, with no collisions. A perfect hash function with values in a limited range can be used for efficient lookup operations, by placing keys from S (or other associated values) in a table indexed by the output of the function. A perfect hash function for a specific set S that can be evaluated in constant time, and with values in a small range, can be found by a randomized algorithm in a number of operations that is proportional to the size of S.
The minimal size of the description of a perfect hash function depends on the range of its function values: The smaller the range, the more space is required. Any perfect hash functions suitable for use with a hash table require at least a number of bits that is proportional to the size of S.
Using a perfect hash function is best in situations where there is a frequently queried large set, S, which is seldom updated. Efficient solutions to performing updates are known as dynamic perfect hashing, but these methods are relatively complicated to implement. A simple alternative to perfect hashing, which also allows dynamic updates, is cuckoo hashing. A minimal perfect hash function is a perfect hash function that maps n keys to n consecutive integers—usually [0..n−1] or [1..n].
A more formal way of expressing this is: Let j and k be elements of some set K. F is a minimal perfect hash function iff F(j) =F(k) implies j=k and there exists an integer a such that the range of F is a..a+|K|−1. It has been proved that any minimal perfect hash scheme requires at least 1.44 bits/key. However the smallest currently use around 2.5 bits/key. Some of these space-efficient perfect hash functions have been implemented in cmph library  and Sux4J .
A minimal perfect hash function F is order preserving if keys are given in some order a1, a2, ..., and for any keys aj and ak, j<k implies F(aj)<F(ak). Order-preserving minimal perfect hash functions require necessarily Ω(n log n) bits to be represented. A minimal perfect hash function F is monotone if it preserves the lexicographical order of the keys. Monotone minimal perfect hash functions can be represented in very little space. Several implementations of monotone minimal perfect hash functions are available in Sux4J
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