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Perfect hash function

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In <a href="Computer_science" title="Computer science">computer science</a>, a perfect hash function h for a set S is a <a href="Hash_function" title="Hash function">hash function</a> that maps distinct elements in S to a set of m integers, with no <a href="Hash_collision" title="Hash collision">collisions</a>. In mathematical terms, it is an <a href="Injective_function" title="Injective function">injective function</a>.

Perfect hash functions may be used to implement a <a href="Lookup_table" title="Lookup table">lookup table</a> with constant worst-case access time. A perfect hash function can, as any <a href="Hash_function" title="Hash function">hash function</a>, be used to implement <a href="Hash_table" title="Hash table">hash tables</a>, with the advantage that no <a href="Hash_table#Collision_resolution" title="Hash table">collision resolution</a> has to be implemented. In addition, if the keys are not in the data and if it is known that queried keys will be valid, then the keys do not need to be stored in the lookup table, saving space.

Disadvantages of perfect hash functions are that S needs to be known for the construction of the perfect hash function. Non-dynamic perfect hash functions need to be re-constructed if S changes. For frequently changing S <a href="Dynamic_perfect_hashing" title="Dynamic perfect hashing">dynamic perfect hash functions</a> may be used at the cost of additional space.<a href="#cite_note-DynamicPerfectHashing-1">[1]</a> The space requirement to store the perfect hash function is in O(n) where n is the number of keys in the structure.

The important performance parameters for perfect hash functions are the evaluation time, which should be constant, the construction time, and the representation size.

<div class="mw-heading mw-heading2"><h2 id="Application">Application</h2></div>

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 <a href="Lookup_table" title="Lookup table">lookup table</a> indexed by the output of the function. One can then test whether a key is present in S, or look up a value associated with that key, by looking for it at its cell of the table. Each such lookup takes <a href="Constant_time" class="mw-redirect" title="Constant time">constant time</a> in the <a href="Worst-case_complexity" title="Worst-case complexity">worst case</a>.<a href="#cite_note-inventor-2">[2]</a> With perfect hashing, the associated data can be read or written with a single access to the table.<a href="#cite_note-3">[3]</a>

<div class="mw-heading mw-heading2"><h2 id="Performance_of_perfect_hash_functions">Performance of perfect hash functions</h2></div>

The important performance parameters for perfect hashing are the representation size, the evaluation time, the construction time, and additionally the range requirement <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {m}{n}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\frac {m}{n}}}</annotation>

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</math><img src="./d48d87468620ad6c70385ddd0d024577ccb559e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:2.877ex; height:4.676ex;" alt="{\displaystyle {\frac {m}{n}}}" loading="lazy"> (average number of buckets per key in the hash table).<a href="#cite_note-CHD-4">[4]</a> The evaluation time can be as fast as O(1), which is optimal.<a href="#cite_note-inventor-2">[2]</a><a href="#cite_note-CHD-4">[4]</a> The construction time needs to be at least O(n), because each element in S needs to be considered, and S contains n elements. This lower bound can be achieved in practice.<a href="#cite_note-CHD-4">[4]</a>

The lower bound for the representation size depends on m and n. Let m = (1+ε) n and h a perfect hash function. A good approximation for the lower bound is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log e-\varepsilon \log {\frac {1+\varepsilon }{\varepsilon }}}">

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<annotation encoding="application/x-tex">{\displaystyle \log e-\varepsilon \log {\frac {1+\varepsilon }{\varepsilon }}}</annotation>

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</math><img src="./b02a088abd50352f229055b2573a6cb24e70ea05.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.035ex; height:5.176ex;" alt="{\displaystyle \log e-\varepsilon \log {\frac {1+\varepsilon }{\varepsilon }}}" loading="lazy"> Bits per element. For minimal perfect hashing, ε = 0, the lower bound is log e ≈ 1.44 bits per element.<a href="#cite_note-CHD-4">[4]</a>

<div class="mw-heading mw-heading2"><h2 id="Construction">Construction</h2></div>

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 <a href="Randomized_algorithm" title="Randomized algorithm">randomized algorithm</a> in a number of operations that is proportional to the size of S.

The original construction of <a href="#CITEREFFredmanKomlósSzemerédi1984">Fredman, Komlós & Szemerédi (1984)</a> uses a two-level scheme to map a set S of n elements to a range of O(n) indices, and then map each index to a range of hash values. The first level of their construction chooses a large prime p (larger than the size of the <a href="Universe_(mathematics)" title="Universe (mathematics)">universe</a> from which S is drawn), and a parameter k, and maps each element x of S to the index

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<annotation encoding="application/x-tex">{\displaystyle g(x)=(kx{\bmod {p}}){\bmod {n}}.}</annotation>

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</math><img src="./0eed5c339f42fdccc1685d8d3b1af928e2a3be01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.277ex; height:2.843ex;" alt="{\displaystyle g(x)=(kx{\bmod {p}}){\bmod {n}}.}" loading="lazy"></dd></dl>

If k is chosen randomly, this step is likely to have collisions, but the number of elements ni that are simultaneously mapped to the same index i is likely to be small.

The second level of their construction assigns disjoint ranges of O(ni2) integers to each index i. It uses a second set of linear modular functions, one for each index i, to map each member x of S into the range associated with g(x).<a href="#cite_note-inventor-2">[2]</a>

As <a href="#CITEREFFredmanKomlósSzemerédi1984">Fredman, Komlós & Szemerédi (1984)</a> show, there exists a choice of the parameter k such that the sum of the lengths of the ranges for the n different values of g(x) is O(n). Additionally, for each value of g(x), there exists a linear modular function that maps the corresponding subset of S into the range associated with that value. Both k, and the second-level functions for each value of g(x), can be found in <a href="Polynomial_time" class="mw-redirect" title="Polynomial time">polynomial time</a> by choosing values randomly until finding one that works.<a href="#cite_note-inventor-2">[2]</a>

The hash function itself requires storage space O(n) to store k, p, and all of the second-level linear modular functions. Computing the hash value of a given key x may be performed in constant time by computing g(x), looking up the second-level function associated with g(x), and applying this function to x.

A modified version of this two-level scheme with a larger number of values at the top level can be used to construct a perfect hash function that maps S into a smaller range of length n + o(n).<a href="#cite_note-inventor-2">[2]</a>

A more recent method for constructing a perfect hash function is described by <a href="#CITEREFBelazzouguiBotelhoDietzfelbinger2009">Belazzougui, Botelho & Dietzfelbinger (2009)</a> as "hash, displace, and compress". Here a first-level hash function g is also used to map elements onto a range of r integers. An element x ∈ S is stored in the Bucket Bg(x).<a href="#cite_note-CHD-4">[4]</a>

Then, in descending order of size, each bucket's elements are hashed by a hash function of a sequence of independent fully random hash functions (Φ1, Φ2, Φ3, ...), starting with Φ1. If the hash function does not produce any collisions for the bucket, and the resulting values are not yet occupied by other elements from other buckets, the function is chosen for that bucket. If not, the next hash function in the sequence is tested.<a href="#cite_note-CHD-4">[4]</a>

To evaluate the perfect hash function h(x) one only has to save the mapping σ of the bucket index g(x) onto the correct hash function in the sequence, resulting in h(x) = Φσ(g(x)).<a href="#cite_note-CHD-4">[4]</a>

Finally, to reduce the representation size, the (σ(i))0 ≤ i < r are compressed into a form that still allows the evaluation in O(1).<a href="#cite_note-CHD-4">[4]</a>

This approach needs linear time in n for construction, and constant evaluation time. The representation size is in O(n), and depends on the achieved range. For example, with m = 1.23n <a href="#CITEREFBelazzouguiBotelhoDietzfelbinger2009">Belazzougui, Botelho & Dietzfelbinger (2009)</a> achieved a representation size between 3.03 bits/key and 1.40 bits/key for their given example set of 10 million entries, with lower values needing a higher computation time. The space lower bound in this scenario is 0.88 bits/key.<a href="#cite_note-CHD-4">[4]</a>

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<div class="mw-heading mw-heading3"><h3 id="Pseudocode">Pseudocode</h3></div>

<pre>algorithm hash, displace, and compress is

(1) Split S into buckets Bi := g−1({i})∩S,0 ≤ i < r

(2) Sort buckets Bi in falling order according to size |Bi|

(3) Initialize array T[0...m-1] with 0's

(4) for all i ∈[r], in the order from (2), do

(5) for l ← 1,2,...

(6) repeat forming Ki ← {Φl(x)|x ∈ Bi}

(6) until |Ki|=|Bi| and Ki∩{j|T[j]=1}= ∅

(7) let σ(i):= the successful l

(8) for all j ∈ Ki let T[j]:= 1

(9) Transform (σi)0≤i<r into compressed form, retaining O(1) access.

</pre>

<div class="mw-heading mw-heading2"><h2 id="Space_lower_bounds">Space lower bounds</h2></div>

The use of O(n) words of information to store the function of <a href="#CITEREFFredmanKomlósSzemerédi1984">Fredman, Komlós & Szemerédi (1984)</a> is near-optimal: any perfect hash function that can be calculated in constant time

requires at least a number of bits that is proportional to the size of S.<a href="#cite_note-5">[5]</a>

For minimal perfect hash functions the information theoretic space lower bound is

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<annotation encoding="application/x-tex">{\displaystyle \log _{2}e\approx 1.44}</annotation>

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bits/key.<a href="#cite_note-CHD-4">[4]</a>

For perfect hash functions, it is first assumed that the range of h is bounded by n as m = (1+ε) n. With the formula given by <a href="#CITEREFBelazzouguiBotelhoDietzfelbinger2009">Belazzougui, Botelho & Dietzfelbinger (2009)</a> and for a <a href="Universe_(mathematics)" title="Universe (mathematics)">universe</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\supseteq S}">

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<annotation encoding="application/x-tex">{\displaystyle U\supseteq S}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \log _{2}e-\varepsilon \log {\frac {1+\varepsilon }{\varepsilon }}}</annotation>

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</math><img src="./776f6c7e58c34f514e1be99b98abdf9f15e14e2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.089ex; height:5.176ex;" alt="{\displaystyle \log _{2}e-\varepsilon \log {\frac {1+\varepsilon }{\varepsilon }}}" loading="lazy"></dd></dl>

bits/key, minus log(n) bits overall.<a href="#cite_note-CHD-4">[4]</a>

<div class="mw-heading mw-heading2"><h2 id="Extensions">Extensions</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Dynamic_perfect_hashing">Dynamic perfect hashing</h3></div>

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</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Dynamic_perfect_hashing" title="Dynamic perfect hashing">Dynamic perfect hashing</a></div>

Using a perfect hash function is best in situations where there is a frequently queried large set, S, which is seldom updated. This is because any modification of the set S may cause the hash function to no longer be perfect for the modified set. Solutions which update the hash function any time the set is modified are known as <a href="Dynamic_perfect_hashing" title="Dynamic perfect hashing">dynamic perfect hashing</a>,<a href="#cite_note-DynamicPerfectHashing-1">[1]</a> but these methods are relatively complicated to implement.

<div class="mw-heading mw-heading3"><h3 id="Minimal_perfect_hash_function">Minimal perfect hash function</h3></div>

A minimal perfect hash function is a perfect hash function that maps n keys to n consecutive integers – usually the numbers from 0 to n − 1 or from 1 to n. A more formal way of expressing this is: Let j and k be elements of some finite set S. Then h is a minimal perfect hash function if and only if h(j) = h(k) implies j = k (<a href="Injectivity" class="mw-redirect" title="Injectivity">injectivity</a>) and there exists an integer a such that the range of h is a..a + |S| − 1. It has been proven that a general purpose minimal perfect hash scheme requires at least <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log _{2}e\approx 1.44}">

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<annotation encoding="application/x-tex">{\displaystyle \log _{2}e\approx 1.44}</annotation>

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</math><img src="./0fd72ec3b460d6cf5434f1a0bb5bb5d24b4b2d15.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.729ex; height:2.676ex;" alt="{\displaystyle \log _{2}e\approx 1.44}" loading="lazy"> bits/key.<a href="#cite_note-CHD-4">[4]</a> Assuming that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

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<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

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</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"> is a set of size <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> containing integers in the range <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [1,2^{o(n)}]}">

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</math><img src="./f0ac67d15f0cd602c935e60c92e1d2cf4bb68bff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.948ex; height:3.343ex;" alt="{\displaystyle [1,2^{o(n)}]}" loading="lazy">, it is known how to efficiently construct an explicit minimal perfect hash function from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

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<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

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</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{1,2,\ldots ,n\}}">

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<annotation encoding="application/x-tex">{\displaystyle \{1,2,\ldots ,n\}}</annotation>

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</math><img src="./4ebfec86b3f22a18f086275390917d5aaa2d8c22.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.257ex; height:2.843ex;" alt="{\displaystyle \{1,2,\ldots ,n\}}" loading="lazy"> that uses space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\log _{2}e+o(n)}">

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<annotation encoding="application/x-tex">{\displaystyle n\log _{2}e+o(n)}</annotation>

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</math><img src="./09277f1ba7d62f18c983de0af01dd3deea037948.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.45ex; height:2.843ex;" alt="{\displaystyle n\log _{2}e+o(n)}" loading="lazy">bits and that supports constant evaluation time.<a href="#cite_note-6">[6]</a> In practice, there are minimal perfect hashing schemes that use roughly 1.56 bits/key if given enough time.<a href="#cite_note-RecSplit-7">[7]</a><a href="#cite_note-8">[8]</a>

<div class="mw-heading mw-heading3"><h3 id="k-perfect_hashing">k-perfect hashing</h3></div>

A hash function is k-perfect if at most k elements from S are mapped onto the same value in the range. The "hash, displace, and compress" algorithm can be used to construct k-perfect hash functions by allowing up to k collisions. The changes necessary to accomplish this are minimal, and are underlined in the adapted pseudocode below:

<pre>(4) for all i ∈[r], in the order from (2), do

(5) for l ← 1,2,...

(6) until |Ki|=|Bi| and Ki∩{j|T[j]=k}= ∅

(7) let σ(i):= the successful l

(8) for all j ∈ Ki set T[j]←T[j]+1

</pre>

<div class="mw-heading mw-heading3"><h3 id="Order_preservation">Order preservation</h3></div>

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).<a href="#cite_note-9">[9]</a> 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 to store a lookup table of the positions of each key. This solution uses <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(n\log n)}">

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<annotation encoding="application/x-tex">{\displaystyle O(n\log n)}</annotation>

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</math><img src="./9d2320768fb54880ca4356e61f60eb02a3f9d9f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.118ex; height:2.843ex;" alt="{\displaystyle O(n\log n)}" loading="lazy"> bits, which is optimal in the setting where the comparison function for the keys may be arbitrary.<a href="#cite_note-10">[10]</a> However, if the keys a1, a2, ..., an are integers drawn from a universe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{1,2,\ldots ,U\}}">

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<annotation encoding="application/x-tex">{\displaystyle \{1,2,\ldots ,U\}}</annotation>

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</math><img src="./d2431655ee77a4033df95488cded81c131e07957.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.645ex; height:2.843ex;" alt="{\displaystyle \{1,2,\ldots ,U\}}" loading="lazy">, then it is possible to construct an order-preserving hash function using only <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(n\log \log \log U)}">

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<annotation encoding="application/x-tex">{\displaystyle O(n\log \log \log U)}</annotation>

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</math><img src="./760a3917c91c59e973214c6f49cc11d493cfbd92.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.224ex; height:2.843ex;" alt="{\displaystyle O(n\log \log \log U)}" loading="lazy"> bits of space.<a href="#cite_note-11">[11]</a> Moreover, this bound is known to be optimal.<a href="#cite_note-12">[12]</a>

<div class="mw-heading mw-heading2"><h2 id="Related_constructions">Related constructions</h2></div>

While well-dimensioned hash tables have amortized average O(1) time (amortized average constant time) for lookups, insertions, and deletion, most hash table algorithms suffer from possible worst-case times that take much longer.

A worst-case O(1) time (constant time even in the worst case) would be better for many applications (including <a href="Network_router" class="mw-redirect" title="Network router">network router</a> and <a href="Memory_cache" class="mw-redirect" title="Memory cache">memory caches</a>).<a href="#cite_note-davis-13">[13]</a>: 41

Few hash table algorithms support worst-case O(1) lookup time (constant lookup time even in the worst case). The few that do include: perfect hashing; <a href="Dynamic_perfect_hashing" title="Dynamic perfect hashing">dynamic perfect hashing</a>; <a href="Cuckoo_hashing" title="Cuckoo hashing">cuckoo hashing</a>; <a href="Hopscotch_hashing" title="Hopscotch hashing">hopscotch hashing</a>; and <a href="Extendible_hashing" title="Extendible hashing">extendible hashing</a>.<a href="#cite_note-davis-13">[13]</a>: 42–69

A simple alternative to perfect hashing, which also allows dynamic updates, is <a href="Cuckoo_hashing" title="Cuckoo hashing">cuckoo hashing</a>. 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.<a href="#cite_note-14">[14]</a>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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<li id="cite_note-6"><a href="#cite_ref-6">^</a> <cite id="CITEREFHagerupTholey2001" class="citation cs2">Hagerup, Torben; Tholey, Torsten (2001), <a rel="nofollow" class="external text" href="https://dx.doi.org/10.1007/3-540-44693-1_28">"Efficient Minimal Perfect Hashing in Nearly Minimal Space"</a>, STACS 2001, Berlin, Heidelberg: Springer Berlin Heidelberg, pp. 317–326, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F3-540-44693-1_28">10.1007/3-540-44693-1_28</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-41695-1</bdi>, retrieved 2023-11-12</cite>

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<li id="cite_note-8"><a href="#cite_ref-8">^</a> <a rel="nofollow" class="external text" href="https://github.com/iwiwi/minimal-perfect-hash">minimal-perfect-hash (GitHub)</a>

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<li id="cite_note-9"><a href="#cite_ref-9">^</a> <cite id="CITEREFJenkins2009" class="citation cs2">Jenkins, Bob (14 April 2009), "order-preserving minimal perfect hashing", in Black, Paul E. (ed.), <a rel="nofollow" class="external text" href="https://xlinux.nist.gov/dads/HTML/orderPreservMinPerfectHash.html">Dictionary of Algorithms and Data Structures</a>, U.S. National Institute of Standards and Technology, retrieved 2013-03-05</cite>

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<li id="cite_note-10"><a href="#cite_ref-10">^</a> <cite id="CITEREFFoxChenDaoudHeath1991" class="citation cs2">Fox, Edward A.; Chen, Qi Fan; Daoud, Amjad M.; Heath, Lenwood S. (July 1991), <a rel="nofollow" class="external text" href="http://eprints.cs.vt.edu/archive/00000248/01/TR-91-01.pdf">"Order-preserving minimal perfect hash functions and information retrieval"</a> (PDF), ACM Transactions on Information Systems, 9 (3), New York, NY, USA: ACM: 281–308, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F125187.125200">10.1145/125187.125200</a>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:53239140">53239140</a></cite>.

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<li id="cite_note-11"><a href="#cite_ref-11">^</a> <cite id="CITEREFBelazzouguiBoldiPaghVigna2008" class="citation cs2">Belazzougui, Djamal; Boldi, Paolo; <a href="Rasmus_Pagh" title="Rasmus Pagh">Pagh, Rasmus</a>; Vigna, Sebastiano (November 2008), "Theory and practice of monotone minimal perfect hashing", Journal of Experimental Algorithmics, 16, Art. no. 3.2, 26pp, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F1963190.2025378">10.1145/1963190.2025378</a>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:2367401">2367401</a></cite>.

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<li id="cite_note-12"><a href="#cite_ref-12">^</a> <cite id="CITEREFAssadiFarach-ColtonKuszmaul2023" class="citation cs2">Assadi, Sepehr; Farach-Colton, Martín; Kuszmaul, William (January 2023), <a rel="nofollow" class="external text" href="https://dx.doi.org/10.1137/1.9781611977554.ch20">"Tight Bounds for Monotone Minimal Perfect Hashing"</a>, Proceedings of the 2023 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), Philadelphia, PA: Society for Industrial and Applied Mathematics, pp. 456–476, <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/2207.10556">2207.10556</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1137%2F1.9781611977554.ch20">10.1137/1.9781611977554.ch20</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-61197-755-4</bdi>, retrieved 2023-04-27</cite>

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Timothy A. Davis.

<a rel="nofollow" class="external text" href="https://www.cs.wm.edu/~tadavis/cs303/ch05sm.pdf">"Chapter 5 Hashing"</a>: subsection "Hash Tables with Worst-Case O(1) Access"

</li>

<li id="cite_note-14"><a href="#cite_ref-14">^</a> <cite id="CITEREFPaghRodler2004" class="citation cs2"><a href="Rasmus_Pagh" title="Rasmus Pagh">Pagh, Rasmus</a>; Rodler, Flemming Friche (2004), "Cuckoo hashing", Journal of Algorithms, 51 (2): 122–144, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.jalgor.2003.12.002">10.1016/j.jalgor.2003.12.002</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2050140">2050140</a></cite>.

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</ol></div>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li>Richard J. Cichelli. Minimal Perfect Hash Functions Made Simple, Communications of the ACM, Vol. 23, Number 1, January 1980.</li>

<li><a href="Thomas_H._Cormen" title="Thomas H. Cormen">Thomas H. Cormen</a>, <a href="Charles_E._Leiserson" title="Charles E. Leiserson">Charles E. Leiserson</a>, <a href="Ronald_L._Rivest" class="mw-redirect" title="Ronald L. Rivest">Ronald L. Rivest</a>, and <a href="Clifford_Stein" title="Clifford Stein">Clifford Stein</a>. <a href="Introduction_to_Algorithms" title="Introduction to Algorithms">Introduction to Algorithms</a>, Third Edition. MIT Press, 2009. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0262033848</bdi>. Section 11.5: Perfect hashing, pp. 267, 277–282.</li>

<li>Fabiano C. Botelho, <a href="Rasmus_Pagh" title="Rasmus Pagh">Rasmus Pagh</a> and Nivio Ziviani. <a rel="nofollow" class="external text" href="https://arxiv.org/abs/cs/0702159">"Perfect Hashing for Data Management Applications"</a>.</li>

<li>Fabiano C. Botelho and <a href="Nivio_Ziviani" title="Nivio Ziviani">Nivio Ziviani</a>. <a rel="nofollow" class="external text" href="http://homepages.dcc.ufmg.br/~nivio/papers/cikm07.pdf">"External perfect hashing for very large key sets"</a>. 16th ACM Conference on Information and Knowledge Management (CIKM07), Lisbon, Portugal, November 2007.</li>

<li>Djamal Belazzougui, Paolo Boldi, <a href="Rasmus_Pagh" title="Rasmus Pagh">Rasmus Pagh</a>, and Sebastiano Vigna. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20140125080021/http://vigna.dsi.unimi.it/ftp/papers/MonotoneMinimalPerfectHashing.pdf">"Monotone minimal perfect hashing: Searching a sorted table with O(1) accesses"</a>. In Proceedings of the 20th Annual ACM-SIAM Symposium On Discrete Mathematics (SODA), New York, 2009. ACM Press.</li>

<li>Marshall D. Brain and Alan L. Tharp. "Near-perfect Hashing of Large Word Sets". Software—Practice and Experience, vol. 19(10), 967-078, October 1989. John Wiley & Sons.</li>

<li>Douglas C. Schmidt, <a rel="nofollow" class="external text" href="http://www.dre.vanderbilt.edu/~schmidt/PDF/gperf.pdf">GPERF: A Perfect Hash Function Generator</a>, C++ Report, SIGS, Vol. 10, No. 10, November/December, 1998.</li>

<li>Hans-Peter Lehmann, Thomas Mueller, Rasmus Pagh, Giulio Ermanno Pibiri, Peter Sanders, Sebastiano Vigna, Stefan Walzer, "Modern Minimal Perfect Hashing: A Survey", <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/2506.06536">2506.06536</a>, June 2025. Discusses post-1997 developments in the field.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="https://www.gnu.org/software/gperf/">gperf</a> is an <a href="Open_source" title="Open source">open source</a> C and C++ perfect hash generator (very fast, but only works for small sets)</li>

<li><a rel="nofollow" class="external text" href="http://burtleburtle.net/bob/hash/perfect.html">Minimal Perfect Hashing (bob algorithm)</a> by Bob Jenkins</li>

<li><a rel="nofollow" class="external text" href="https://cmph.sourceforge.net/index.html">cmph</a>: C Minimal Perfect Hashing Library, open source implementations for many (minimal) perfect hashes (works for big sets)</li>

<li><a rel="nofollow" class="external text" href="http://sux.di.unimi.it/">Sux4J</a>: open source monotone minimal perfect hashing in Java</li>

<li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20130729211948/http://www.dupuis.me/node/9">MPHSharp</a>: perfect hashing methods in C#</li>

<li><a rel="nofollow" class="external text" href="https://github.com/rizkg/BBHash">BBHash</a>: minimal perfect hash function in header-only C++</li>

<li><a rel="nofollow" class="external text" href="https://github.com/rurban/Perfect-Hash">Perfect::Hash</a>, perfect hash generator in Perl that makes C code. Has a "prior art" section worth looking at.</li></ul></div><div><div class="zim-footer">

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