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| last1 = Lu | first1 = Yi | author1-link = Yi Lu (computer scientist)
| last2 = Prabhakar | first2 = Balaji | author2-link = Balaji Prabhakar
| last3 = Bonomi | first3 = Flavio | title = 2006 IEEE International Symposium on Information Theory | chapter = Perfect Hashing for Network Applications | author3-link = Flavio Bonomi
| doi = 10.1109/ISIT.2006.261567
| pages = 2774–2778
| year = 2006| isbn = 1-4244-0505-X | s2cid = 1494710 }}</ref>
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===Order preservation===
A minimal perfect hash function {{mvar|F}} is ''order preserving'' if keys are given in some order {{math|''a''<sub>1</sub>, ''a''<sub>2</sub>, ..., ''a''<sub>''n''</sub>}} and for any keys {{math|''a''<sub>''j''</sub>}} and {{math|''a''<sub>''k''</sub>}}, {{math|''j'' < ''k''}} implies {{math|''F''(''a''<sub>''j''</sub>) < F(''a''<sub>''k''</sub>)}}.<ref>{{Citation |first=Bob |last=Jenkins |contribution=order-preserving minimal perfect hashing |title=Dictionary of Algorithms and Data Structures |editor-first=Paul E. |editor-last=Black |publisher=U.S. National Institute of Standards and Technology |date=14 April 2009 |accessdate=2013-03-05 |url=https://xlinux.nist.gov/dads/HTML/orderPreservMinPerfectHash.html}}</ref> 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>O(n \log n)</math> bits, which is optimal in the setting where the comparison function for the keys may be arbitrary.<ref>{{citation |last1=Fox |first1=Edward A. |title=Order-preserving minimal perfect hash functions and information retrieval |date=July 1991 |url=http://eprints.cs.vt.edu/archive/00000248/01/TR-91-01.pdf |journal=ACM Transactions on Information Systems |volume=9 |issue=3 |pages=281–308 |___location=New York, NY, USA |publisher=ACM |doi=10.1145/125187.125200 |s2cid=53239140 |last2=Chen |first2=Qi Fan |last3=Daoud |first3=Amjad M. |last4=Heath |first4=Lenwood S.}}.</ref> However, if the keys {{math|''a''<sub>1</sub>, ''a''<sub>2</sub>, ..., ''a''<sub>''n''</sub>}} are integers drawn from a universe <math>\{1, 2, \ldots, U\}</math>, then it is possible to construct an order-preserving hash function using only <math>O(n \log \log \log U)</math> bits of space.<ref>{{citation |last1=Belazzougui |first1=Djamal |title=Theory and practice of monotone minimal perfect hashing |date=November 2008 |journal=Journal of Experimental Algorithmics |volume=16 |at=Art. no. 3.2, 26pp |doi=10.1145/1963190.2025378 |s2cid=2367401 |last2=Boldi |first2=Paolo |last3=Pagh |first3=Rasmus |last4=Vigna |first4=Sebastiano |author3-link=Rasmus Pagh}}.</ref> Moreover, this bound is known to be optimal.<ref>{{Citation |
==Related constructions==
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