Perfect hash function
A perfect hash function for a set S is a hash function that maps distinct elements in S to a set of integers, with no collisions. A perfect hash function has many of the same applications as other hash functions, but with the advantage that no collision resolution has to be implemented. In mathematical terms, it is a total injective function.
Properties and uses
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.
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. 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.
Minimal perfect hash function
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 finite set K. F is a minimal perfect hash function iff F(j) =F(k) implies j=k (injectivity) and there exists an integer a such that the range of F is a..a+|K|−1. It has been proved that a general purpose minimal perfect hash scheme requires at least 1.44 bits/key. However the smallest currently use around 2.5 bits/key.
A minimal perfect hash function F is order preserving if keys are given in some order a1, a2, ..., an 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.
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- Minimal Perfect Hashing by Bob Jenkins
- gperf is an Open Source C and C++ perfect hash generator
- cmph is Open Source implementing many perfect hashing methods
- Sux4J is Open Source implementing perfect hashing, including monotone minimal perfect hashing in Java
- MPHSharp is Open Source implementing many perfect hashing methods in C#