Perfect hash function: Difference between revisions

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===Minimal perfect hash function===

A minimal perfect hash function is a perfect hash function that maps {{mvar|n}} keys to {{mvar|n}} consecutive integers – usually the numbers from {{math|0}} to {{math|''n'' &minus; 1}} or from {{math|1}} to {{mvar|n}}. A more formal way of expressing this is: Let {{mvar|j}} and {{mvar|k}} be elements of some finite set {{mvar|S}}. Then {{mvar|h}} is a minimal perfect hash function if and only if {{math|1=''h''(''j'') = ''h''(''k'')}} implies {{math|1=''j'' = ''k''}} ([[injectivity]]) and there exists an integer {{mvar|a}} such that the range of {{mvar|h}} is {{math|1=''a''..''a'' + {{!}}''S''{{!}} &minus; 1}}. It has been proven that a general purpose minimal perfect hash scheme requires at least {{<math|lg>\log_2 ''e'' ≈\approx 1.44}}</math> bits/key.<ref name="CHD">{{citation

| last1 = Belazzougui | first1 = Djamal

| last2 = Botelho | first2 = Fabiano C.

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| year = 2009| citeseerx = 10.1.1.568.130

| url = http://cmph.sourceforge.net/papers/esa09.pdf

}}.</ref> Assuming that <math>S</math> is a set of size <math>n</math> containing integers in the range <math>[1, 2^{o(n)}]</math>, it is known how to efficiently construct an explicit minimal perfect hash function from <math>S</math> to <math>\{1, 2, \ldots, n\}</math> that uses space <math>n \log_2 e + o(n)</math>bits and that supports constant evaluation time.<ref>{{Citation |last=Hagerup |first=Torben |title=Efficient Minimal Perfect Hashing in Nearly Minimal Space |date=2001 |url=http://dx.doi.org/10.1007/3-540-44693-1_28 |work=STACS 2001 |pages=317–326 |access-date=2023-11-12 |place=Berlin, Heidelberg |publisher=Springer Berlin Heidelberg |isbn=978-3-540-41695-1 |last2=Tholey |first2=Torsten}}</ref> In practice, there are minimal perfect hashing schemes that use roughly 1.56 bits/key if given enough time.<ref name="RecSplit">{{citation

}}.</ref> Although this space bound has been achieved by theoretical works, in practice, the best known minimal perfect hashing schemes require roughly 1.56 bits/key if given enough time.<ref name="RecSplit">{{citation

| last1 = Esposito | first1 = Emmanuel

| last2 = Mueller Graf | first2 = Thomas