Element distinctness problem: Difference between revisions

Revision as of 18:58, 26 May 2020

Revision as of 21:03, 26 May 2020

Line 14:

==Decision tree complexity==

It is known that, for lists of numbers, the problem's [[time complexity]] is [[Big O notation|Θ&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>{{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=316–329|issue=4}}.</ref>