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Perfect hash function
The best currently known minimal perfect hashing schemes can be represented using less than 2.1 bits per key if given enough time.
Order preservation
A minimal perfect hash function is order preserving if keys are given in some order and for any keys and , implies . In this case, the function value is just the position of each key in the sorted ordering of all of the keys. A simple implementation of order-preserving minimal perfect hash functions with constant access time is to use an (ordinary) perfect hash function or cuckoo hashing to store a lookup table of the positions of each key. If the keys to be hashed are themselves stored in a sorted array, it is possible to store a small number of additional bits per key in a data structure that can be used to compute hash values quickly. Order-preserving minimal perfect hash functions require necessarily bits to be represented.
Related constructions
A simple alternative to perfect hashing, which also allows dynamic updates, is cuckoo hashing. This scheme maps keys to two or more locations within a range (unlike perfect hashing which maps each key to a single location) but does so in such a way that the keys can be assigned one-to-one to locations to which they have been mapped. Lookups with this scheme are slower, because multiple locations must be checked, but nevertheless take constant worst-case time.
