Element distinctness problem: Difference between revisions

Revision as of 16:09, 18 October 2010

Revision as of 09:45, 21 April 2011

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| year = 1990}}.</ref>

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 athe [[modelalgebraic ofdecision computationtree]] known as the [[algebraicmodel of computation tree]] model,<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}}</ref>