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	Hi there~~, I am Andrew Berryhill~~. ~~Office supervising~~ is ~~exactly where her main earnings arrives from but she~~'s ~~already utilized~~ for an ~~additional one~~. ~~Mississippi~~ is ~~where her house~~ is but ~~her spouse wants them~~ to ~~move~~. It's ~~not~~ a ~~typical factor but what she likes performing~~ is ~~to perform domino but she doesn~~'~~t have~~ the ~~time recently~~.<br><br>~~my site~~ ... ~~online psychic reading ([http~~://test.~~jeka-nn~~.ru/~~node~~/~~129 click this link here now~~])		In [[computational complexity theory]], the '''element distinctness problem''' or '''element uniqueness problem''' is the problem of determining whether all the elements of a list are distinct.

			It is a well studied problem in many different models of computation. The problem may be solved by [[sorting]] the list and then checking if there are any consecutive equal elements; it may also be solved in linear expected time by a [[randomized algorithm]] that inserts each item into a [[hash table]] and compares only those elements that are placed in the same hash table cell.<ref>{{citation
			\| last1 = Gil \| first1 = J.
			\| last2 = Meyer auf der Heide \| first2 = F.
			\| last3 = Wigderson \| first3 = A. \| author3-link = Avi Wigderson
			\| contribution = Not all keys can be hashed in constant time
			\| doi = 10.1145/100216.100247
			\| pages = 244–253
			\| title = [[Symposium on Theory of Computing\|Proc. 22nd ACM Symposium on Theory of Computing]]
			\| year = 1990}}.</ref>

			It is known that, for lists of numbers, the problem's [[time complexity]] is [[Big O notation\|Θ]](''n'' log ''n''), i.e., both the upper and lower bounds on its time complexity are of order of the [[linearithmic function]] in the [[algebraic decision tree]] [[model of computation]],<ref>{{citation\|first=Michael\|last=Ben-Or\|contribution=Lower bounds for algebraic computation trees\|title=[[Symposium on Theory of Computing\|Proc. 15th ACM Symposium on Theory of Computing]]\|year=1983\|pages=80–86\|doi=10.1145/800061.808735}}.</ref> a model of computation in which the elements may not be used to index the computer's memory (as in the hash table solution) but may only be accessed by computing and comparing simple algebraic functions of their values. In other words, an [[asymptotically optimal]] algorithm of linearithmic time complexity is known for this model. The algebraic computation tree model basically means that the allowable algorithms are only the ones that can perform polynomial operations of bounded degree on the input data and comparisons of the results of these computations.

			The same lower bound was later proved for the [[randomized complexity\|randomized]] [[algebraic decision tree]] model.<ref>{{citation\|doi=10.1007/BF01270387\|title=A lower bound for randomized algebraic decision trees\|year=1996\|last=Grigoriev\|first=Dima\|authorlink=Dima Grigoriev\|journal=Computational Complexity\|volume=6\|pages=357\|last2=Karpinski\|first2=Marek\|author2-link=Marek Karpinski\|last3=Heide\|first3=Friedhelm Meyer\|last4=Smolensky\|first4=Roman\|issue=4}}.</ref><ref>{{citation\|doi=10.1007/s000370050002\|title=Complexity lower bounds for randomized computation trees over zero characteristic fields\|year=1999\|last=Grigoriev\|first=Dima\|authorlink=Dima Grigoriev\|journal=Computational Complexity\|volume=8\|pages=316\|issue=4}}.</ref>

			It is also known that [[quantum algorithm]]s can solve this problem faster in <math>\Theta\left(n^{2/3}\right)</math> queries. The optimal algorithm is by [[Andris Ambainis]].<ref>{{citation \| last1=Ambainis \| first1=Andris \| author1-link=Andris Ambainis \| title=Quantum walk algorithm for element distinctness \| doi = 10.1137/S0097539705447311 \| journal = [[SIAM Journal on Computing]] \| volume = 37 \| issue = 1 \| year = 2007 \| pages=210–239 \| arxiv=quant-ph/0311001 }}</ref> [[Scott Aaronson]] and Yaoyun Shi first proved a tight lower bound for a large but restricted class of functions.<ref>
			{{Cite journal
			\| last1=Aaronson \| first1=S.
			\| last2=Shi \| first2=Y.
			\| year=2004
			\| title=Quantum lower bounds for the collision and the element distinctness problems
			\| journal=[[Journal of the ACM]]
			\| volume=51 \| issue=4 \| pages=595–605
			\| arxiv=
			\| bibcode=
			\| doi=10.1145/1008731.1008735
			}}</ref> Ambainis<ref>
			{{cite journal
			\|last=Ambainis \|first=A.
			\|year=2005
			\|title=Polynomial Degree and Lower Bounds in Quantum Complexity: Collision and Element Distinctness with Small Range
			\|journal=[[Theory of Computing]]
			\|volume=1 \|issue=1 \|pages=37–46
			\|arxiv=
			\|bibcode=
			\|doi=10.4086/toc.2005.v001a003
			}}</ref> and Kutin<ref>
			{{cite journal
			\|last1=Kutin \|first1=S.
			\|year=2005
			\|title=Quantum Lower Bound for the Collision Problem with Small Range
			\|journal=[[Theory of Computing]]
			\|volume=1 \|issue=1 \|pages=29–36
			\|arxiv=
			\|bibcode=
			\|doi=10.4086/toc.2005.v001a002
			}}</ref> independently (and via different proofs) extended their work to obtain the lower bound for all functions.

			Several lower bounds in computational complexity are proved by reducing the element distinctness problem to the problem in question, i.e., by demonstrating that the solution of the element uniqueness problem may be quickly found after solving the problem in question.

			==Restrictions==
			[[Decision tree model]]s are inapplicable for determining lower bounds for algorithmic problems for objects that have some ''a priori'' properties which can be exploited in construction of algorithms. For example, if it is known that the ''n'' objects are [[integer number]]s from the range [1..''n''], then the element uniqueness problem may be solved in [[Big O notation\|O]](''n'') time by a modification of [[bucket sort]].

			==Generalization: Finding repeated elements==
			Elements that occur more than ''n''/''k'' times in a multiset of size n may be found in time O(''n'' log ''k''). The element distinctness problem is a special case of ''k''=''n''. This algorithm is optimal under the [[decision tree model]] of computation.<ref>{{citation\|first1=J.\|last1=Misra\|first2=D.\|last2=Gries\|author2-link=David Gries\|title=Finding repeated elements\|journal=Science of Computer Programming\|volume=2\|year=1982\|pages=143–152}}.</ref>

			The algorithm is a generalization of the one for a special case of ''k''=2, which had a rather convoluted history of publication.<ref name=bm>{{citation\|first1=R. S.\|last1=Boyer\|author1-link=Robert S. Boyer\|first2=J S.\|last2=Moore\|author2-link=J Strother Moore\|contribution=MJRTY - A Fast Majority Vote Algorithm\|editor-first=R. S.\|editor-last=Boyer\|title=Automated Reasoning: Essays in Honor of Woody Bledsoe\|series=Automated Reasoning Series\|publisher=Kluwer Academic Publishers\|location=Dordrecht, The Netherlands\|year=1991\|pages=105–117}}.</ref>

			The above algorithms rely only on the test of identity of the elements. If sorting is allowed, previously known [[order statistics]] finding algorithms may be exploited. For example, for ''k''=2, a [[median]] may be found first in [[linear time]], and then it may be easily tested whether there are more than ''n''/2 median elements. However the above algorithms require fewer comparisons than the order statistics algorithms.<ref name=bm/>

			==See also==
			*[[Collision problem]]

			==References==
			{{reflist}}

			[[Category:Articles with inconsistent citation formats]]
			[[Category:Polynomial-time problems]]

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