Element distinctness problem: Difference between revisions

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It is known that the problem's [[time complexity]] is [[Big O notation|&Theta;]](''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>[[Michael Ben-Or]], "Lower bounds for algebraic computation trees", Proceedings of the 15th Annual ACM Symposium on Theory of Computing, 1983, pp. 80-86</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>{{cite journal|doi=10.1007/BF01270387|title=A lower bound for randomized algebraic decision trees|year=1996|author=Grigoriev, Dima|journal=Computational Complexity|volume=6|pages=357|last2=Karpinski|first2=Marek|last3=Heide|first3=Friedhelm Meyer|last4=Smolensky|first4=Roman|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>{{Cite journal | last1=Ambainis | first1=Andris | author1-link=Andris Ambainis | title=Quantum walk algorithm for element distinctness | id = {{arxiv|quant-ph/0311001}} | doi = 10.1137/S0097539705447311 | journal = [[SIAM Journal on Computing]] | volume = 37 | issue = 1 | year = 2007 | pages=210–239 [[Category:Articles with inconsistent citation formats]] | arxiv=quant-ph/0311001 }}.</ref> and the lower bound is due to [[Scott Aaronson]] and Yaoyun Shi.<ref>{{Cite journal | last1=Aaronson | first1=S. | last2=Shi | first2=Y. | title=Quantum lower bounds for the collision and the element distinctness problems | publisher=Association for Computing Machinery, Inc, One Astor Plaza, 1515 Broadway, New York, NY, 10036-5701, USA, | year=2004 | journal=[[Journal of the ACM]] | issn=0004-5411 | volume=51 | issue=4 | pages=595–605 | doi=10.1145/1008731.1008735}}</ref>

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.