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Element distinctness problem

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr">In <a href="Computational_complexity_theory" title="Computational complexity theory">computational complexity theory</a>, 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 <a href="Sorting" title="Sorting">sorting</a> the list and then checking if there are any consecutive equal elements; it may also be solved in linear expected time by a <a href="Randomized_algorithm" title="Randomized algorithm">randomized algorithm</a> that inserts each item into a <a href="Hash_table" title="Hash table">hash table</a> and compares only those elements that are placed in the same hash table cell.<a href="#cite_note-1">[1]</a>

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.

<div class="mw-heading mw-heading2"><h2 id="Decision_tree_complexity">Decision tree complexity</h2></div>

The number of comparisons needed to solve the problem of size <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">, in a comparison-based model of computation such as a <a href="Decision_tree" title="Decision tree">decision tree</a> or <a href="Algebraic_decision_tree" class="mw-redirect" title="Algebraic decision tree">algebraic decision tree</a>, is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Theta (n\log n)}">

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<annotation encoding="application/x-tex">{\displaystyle \Theta (n\log n)}</annotation>

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</math><img src="./1b8781cea4259c3bd43204e02d08b9b9ce8fe0ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.153ex; height:2.843ex;" alt="{\displaystyle \Theta (n\log n)}" loading="lazy">. Here, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Theta }">

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<annotation encoding="application/x-tex">{\displaystyle \Theta }</annotation>

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</math><img src="./bc927b19f46d005b4720db7a0f96cd5b6f1a0d9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \Theta }" loading="lazy"> invokes <a href="Big_O_notation" title="Big O notation">big theta notation</a>, meaning that the problem can be solved in a number of comparisons proportional to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\log n}">

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<annotation encoding="application/x-tex">{\displaystyle n\log n}</annotation>

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</math><img src="./560dfdce0353a330e03e4b3e0b7ca6e484bb40fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.535ex; height:2.509ex;" alt="{\displaystyle n\log n}" loading="lazy"> (a <a href="Linearithmic_function" class="mw-redirect" title="Linearithmic function">linearithmic function</a>) and that all solutions require this many comparisons.<a href="#cite_note-2">[2]</a> 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 <a href="Comparison_sort" title="Comparison sort">comparison sort</a> solves the problem within a constant factor of the best possible number of comparisons. The same lower bound applies as well to the <a href="Expected_value" title="Expected value">expected number</a> of comparisons in the <a href="Randomized_complexity" class="mw-redirect" title="Randomized complexity">randomized</a> <a href="Algebraic_decision_tree" class="mw-redirect" title="Algebraic decision tree">algebraic decision tree</a> model.<a href="#cite_note-3">[3]</a><a href="#cite_note-4">[4]</a>

<div class="mw-heading mw-heading2"><h2 id="Real_RAM_Complexity">Real RAM Complexity</h2></div>

If the elements in the problem are <a href="Real_number" title="Real number">real numbers</a>, the decision-tree lower bound extends to the <a href="Real_RAM" title="Real RAM">real random-access machine</a> model with an instruction set that includes addition, subtraction and multiplication of real numbers, as well as comparison and either division or remaindering ("floor").<a href="#cite_note-5">[5]</a> It follows that the problem's complexity in this model is also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Theta (n\log n)}">

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<annotation encoding="application/x-tex">{\displaystyle \Theta (n\log n)}</annotation>

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</math><img src="./1b8781cea4259c3bd43204e02d08b9b9ce8fe0ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.153ex; height:2.843ex;" alt="{\displaystyle \Theta (n\log n)}" loading="lazy">. 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.

<div class="mw-heading mw-heading2"><h2 id="Turing_Machine_complexity">Turing Machine complexity</h2></div>

A single-tape deterministic <a href="Turing_machine" title="Turing machine">Turing machine</a> can solve the problem, for n elements of m ≥ log n bits each, in time O(n2m(m+2–log n)),

while on a nondeterministic machine the time complexity is O(nm(n + log m)).<a href="#cite_note-6">[6]</a>

<div class="mw-heading mw-heading2"><h2 id="Quantum_complexity">Quantum complexity</h2></div>

<a href="Quantum_algorithm" title="Quantum algorithm">Quantum algorithms</a> can solve this problem faster, in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \Theta (n^{2/3})}">

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<annotation encoding="application/x-tex">{\textstyle \Theta (n^{2/3})}</annotation>

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</math><img src="./6e72fd4d9c8f1c1f6a23c756dc72682d0c071f4c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.71ex; height:3.176ex;" alt="{\textstyle \Theta (n^{2/3})}" loading="lazy"> queries. The optimal algorithm is by <a href="Andris_Ambainis" title="Andris Ambainis">Andris Ambainis</a>.<a href="#cite_note-7">[7]</a> Yaoyun Shi first proved a tight lower bound when the size of the range is sufficiently large.<a href="#cite_note-8">[8]</a> Ambainis<a href="#cite_note-9">[9]</a> and Kutin<a href="#cite_note-10">[10]</a> independently (and via different proofs) extended his work to obtain the lower bound for all functions.

<div class="mw-heading mw-heading2"><h2 id="Generalization:_Finding_repeated_elements">Generalization: Finding repeated elements</h2></div>

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</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Misra%E2%80%93Gries_heavy_hitters_algorithm" title="Misra–Gries heavy hitters algorithm">Misra–Gries heavy hitters algorithm</a></div>

Elements that occur more than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n/k}">

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<annotation encoding="application/x-tex">{\displaystyle n/k}</annotation>

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</math><img src="./761557c05efc5c1109da15c578dc9df4a7a32790.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.768ex; height:2.843ex;" alt="{\displaystyle n/k}" loading="lazy"> times in a multiset of size <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> may be found by a comparison-based algorithm, the <a href="Misra%E2%80%93Gries_heavy_hitters_algorithm" title="Misra–Gries heavy hitters algorithm">Misra–Gries heavy hitters algorithm</a>, in time <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(n\log k)}">

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<annotation encoding="application/x-tex">{\displaystyle O(n\log k)}</annotation>

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</math><img src="./99e71cfbbb1dbd170050e63d7b342b07b4b014e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.934ex; height:2.843ex;" alt="{\displaystyle O(n\log k)}" loading="lazy">. The element distinctness problem is a special case of this problem where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=n}">

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<annotation encoding="application/x-tex">{\displaystyle k=n}</annotation>

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</math><img src="./c4b8cc6fcba0c0b24656b0fb33414d2e6cffb83c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.704ex; height:2.176ex;" alt="{\displaystyle k=n}" loading="lazy">. This time is optimal under the <a href="Decision_tree_model" title="Decision tree model">decision tree model</a> of computation.<a href="#cite_note-11">[11]</a>

<ul><li><a href="Collision_problem" title="Collision problem">Collision problem</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFGilMeyer_auf_der_HeideWigderson1990" class="citation cs2">Gil, J.; Meyer auf der Heide, F.; <a href="Avi_Wigderson" title="Avi Wigderson">Wigderson, A.</a> (1990), "Not all keys can be hashed in constant time", <a href="Symposium_on_Theory_of_Computing" title="Symposium on Theory of Computing">Proc. 22nd ACM Symposium on Theory of Computing</a>, pp. 244–253, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F100216.100247">10.1145/100216.100247</a>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:11943779">11943779</a></cite>.

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<li id="cite_note-2"><a href="#cite_ref-2">^</a> <cite id="CITEREFBen-Or1983" class="citation cs2">Ben-Or, Michael (1983), "Lower bounds for algebraic computation trees", <a href="Symposium_on_Theory_of_Computing" title="Symposium on Theory of Computing">Proc. 15th ACM Symposium on Theory of Computing</a>, pp. 80–86, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F800061.808735">10.1145/800061.808735</a></cite>.

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<li id="cite_note-3"><a href="#cite_ref-3">^</a> <cite id="CITEREFGrigorievKarpinskiHeideSmolensky1996" class="citation cs2"><a href="Dima_Grigoriev" title="Dima Grigoriev">Grigoriev, Dima</a>; <a href="Marek_Karpinski" title="Marek Karpinski">Karpinski, Marek</a>; Heide, Friedhelm Meyer; Smolensky, Roman (1996), "A lower bound for randomized algebraic decision trees", Computational Complexity, 6 (4): 357, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01270387">10.1007/BF01270387</a>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:1462184">1462184</a></cite>.

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<li id="cite_note-4"><a href="#cite_ref-4">^</a> <cite id="CITEREFGrigoriev1999" class="citation cs2"><a href="Dima_Grigoriev" title="Dima Grigoriev">Grigoriev, Dima</a> (1999), "Complexity lower bounds for randomized computation trees over zero characteristic fields", Computational Complexity, 8 (4): 316–329, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs000370050002">10.1007/s000370050002</a>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:10641238">10641238</a></cite>.

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<li id="cite_note-5"><a href="#cite_ref-5">^</a> <cite id="CITEREFBen-AmramGalil2001" class="citation cs2">Ben-Amram, Amir M.; <a href="Zvi_Galil" title="Zvi Galil">Galil, Zvi</a> (2001), "Topological Lower Bounds on Algebraic Random Access Machines", SIAM Journal on Computing, 31 (3): 722–761, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1137%2FS0097539797329397">10.1137/S0097539797329397</a></cite>.

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<li id="cite_note-6"><a href="#cite_ref-6">^</a> <cite id="CITEREFBen-AmramBerkmanPetersen2003" class="citation cs2">Ben-Amram, Amir M.; Berkman, Omer; Petersen, Holger (2003), "Element distinctness on one-tape Turing machines: a complete solution.", Acta Informatica, 40 (2): 81–94, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs00236-003-0125-8">10.1007/s00236-003-0125-8</a>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:24821585">24821585</a></cite>

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<li id="cite_note-7"><a href="#cite_ref-7">^</a> <cite id="CITEREFAmbainis2007" class="citation cs2"><a href="Andris_Ambainis" title="Andris Ambainis">Ambainis, Andris</a> (2007), "Quantum walk algorithm for element distinctness", <a href="SIAM_Journal_on_Computing" title="SIAM Journal on Computing">SIAM Journal on Computing</a>, 37 (1): 210–239, <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/quant-ph/0311001">quant-ph/0311001</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1137%2FS0097539705447311">10.1137/S0097539705447311</a></cite>

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<cite id="CITEREFShi2002" class="citation conference cs1">Shi, Y. (2002). Quantum lower bounds for the collision and the element distinctness problems. Proceedings of the 43rd <a href="Symposium_on_Foundations_of_Computer_Science" title="Symposium on Foundations of Computer Science">Symposium on Foundations of Computer Science</a>. pp. 513–519. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/quant-ph/0112086">quant-ph/0112086</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FSFCS.2002.1181975">10.1109/SFCS.2002.1181975</a>.</cite>

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<cite id="CITEREFAmbainis2005" class="citation journal cs1">Ambainis, A. (2005). <a rel="nofollow" class="external text" href="https://doi.org/10.4086%2Ftoc.2005.v001a003">"Polynomial Degree and Lower Bounds in Quantum Complexity: Collision and Element Distinctness with Small Range"</a>. <a href="Theory_of_Computing" title="Theory of Computing">Theory of Computing</a>. 1 (1): 37–46. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4086%2Ftoc.2005.v001a003">10.4086/toc.2005.v001a003</a>.</cite>

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<cite id="CITEREFKutin2005" class="citation journal cs1">Kutin, S. (2005). <a rel="nofollow" class="external text" href="https://doi.org/10.4086%2Ftoc.2005.v001a002">"Quantum Lower Bound for the Collision Problem with Small Range"</a>. <a href="Theory_of_Computing" title="Theory of Computing">Theory of Computing</a>. 1 (1): 29–36. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4086%2Ftoc.2005.v001a002">10.4086/toc.2005.v001a002</a>.</cite>

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<li id="cite_note-11"><a href="#cite_ref-11">^</a> <cite id="CITEREFMisraGries1982" class="citation cs2">Misra, J.; <a href="David_Gries" title="David Gries">Gries, D.</a> (1982), "Finding repeated elements", Science of Computer Programming, 2 (2): 143–152, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0167-6423%2882%2990012-0">10.1016/0167-6423(82)90012-0</a>, <a href="Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/1813%2F6345">1813/6345</a></cite>.

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