Element distinctness problem: Difference between revisions

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| title = [[Symposium on Theory of Computing|Proc. 22nd ACM Symposium on Theory of Computing]]

| year = 1990}}.</ref>| s2cid = 11943779

| doi-access = free

}}.</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.

It is known that, for lists of numbers, 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>{{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.

==Decision tree complexity==

The number of comparisons needed to solve the problem of size <math>n</math>, in a comparison-based model of computation such as a [[decision tree]] or [[algebraic decision tree]], is <math>\Theta(n\log n)</math>. Here, <math>\Theta</math> invokes [[Big O notation|big theta notation]], meaning that the problem can be solved in a number of comparisons proportional to <math>n\log n</math> (a [[linearithmic function]]) and that all solutions require this many comparisons.<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|doi-access=free}}.</ref> In these models of computation, the input numbers 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. For these models, an algorithm based on [[comparison sort]] solves the problem within a constant factor of the best possible number of comparisons. The same lower bound applies as well to the [[expected value|expected number]] of comparisons in 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|last1=Grigoriev|first1=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|s2cid=1462184 }}.</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|s2cid=10641238 }}.</ref>

==Real RAM Complexity==

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>

If the elements in the problem are [[real number|real numbers]], the decision-tree lower bound extends to the [[real RAM|real random-access machine]] model with an instruction set that includes addition, subtraction and multiplication of real numbers, as well as comparison and either division or remaindering ("floor").<ref>{{citation|doi=10.1137/S0097539797329397|title=Topological Lower Bounds on Algebraic Random Access Machines|year=2001|last1=Ben-Amram|first1=Amir M.|journal=SIAM Journal on Computing|volume=31|issue=3|pages=722–761|last2=Galil|first2=Zvi|author2-link=Zvi Galil}}.</ref> It follows that the problem's complexity in this model is also <math>\Theta(n\log n)</math>. This RAM model covers more algorithms than the algebraic decision-tree model, as it encompasses algorithms that use indexing into tables. However, in this model all program steps are counted, not just decisions.

==Turing Machine complexity==

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>

A single-tape deterministic [[Turing machine]] can solve the problem, for ''n'' elements of {{math|''m'' &geq; log ''n''}} bits each, in time {{math|''O''(''n''²''m''(''m''+2–log ''n''))}},

{{Cite journal

while on a nondeterministic machine the time complexity is {{math|''O''(''nm''(''n'' + log ''m''))}}.<ref>{{citation|doi=10.1007/s00236-003-0125-8|title=Element distinctness on one-tape Turing machines: a complete solution.|year=2003|last1=Ben-Amram|first1=Amir M.|journal=Acta Informatica|volume=40|issue=2|pages=81–94|last2=Berkman|first2=Omer|last3=Petersen|first3=Holger|s2cid=24821585 }}</ref>

| last1=Aaronson | first1=S.

| last2=Shi | first2=Y.

==Quantum complexity==

| year=2004

It is also known that [[quantumQuantum algorithm]]s can solve this problem faster, in <math display=inline>\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> [[ScottYaoyun AaronsonShi]] and Yaoyun Shi first proved a tight lower bound forwhen athe largesize butof restrictedthe classrange ofis functionssufficiently large.<ref>

{{cite conference

| last1=AaronsonShi | first1=SY.

| year=20042002

| title=Quantum lower bounds for the collision and the element distinctness problems

| conference = Proceedings of the 43rd [[Symposium on Foundations of Computer Science]]

| journal=[[Journal of the ACM]]

| volume=51 | issue=4 | pages=595–605513–519

| arxiv = quant-ph/0112086

| doi=10.1109/SFCS.2002.1181975

| bibcode=

| doi=10.1145/1008731.1008735

}}</ref> Ambainis<ref>

{{cite journal

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|bibcode=

|doi=10.4086/toc.2005.v001a003

|doi-access=free

}}</ref> and Kutin<ref>

{{cite journal

|last1=Kutin |first1=S.

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|bibcode=

|doi=10.4086/toc.2005.v001a002

|doi-access=free

}}</ref> independently (and via different proofs) extended theirhis 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==

{{main|Misra–Gries heavy hitters algorithm}}

Elements that occur more than ''<math>n''/''k''</math> times in a multiset of size <math>n</math> may be found by a comparison-based algorithm, the [[Misra–Gries heavy hitters algorithm]], in time <math>O(''n'' \log ''k'')</math>. The element distinctness problem is a special case of ''this problem where <math>k''=''n''</math>. This algorithmtime 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|issue=2|pages=143–152|doi=10.1016/0167-6423(82)90012-0|hdl=1813/6345|hdl-access=free}}.</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==

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{{reflist}}

[[Category:Articles with inconsistent citation formats]]

[[Category:Polynomial-time problems]]