189:) and that all solutions require this many comparisons. 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
57:
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.
220:
model with an instruction set that includes addition, subtraction and multiplication of real numbers, as well as comparison and either division or remaindering ("floor"). It follows that the problem's complexity in this model is also
259:. 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.
389:
first proved a tight lower bound when the size of the range is sufficiently large. Ambainis and Kutin independently (and via different proofs) extended his work to obtain the lower bound for all functions.
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130:
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183:
150:
518:
430:
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84:
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17:
730:
Ben-Amram, Amir M.; Berkman, Omer; Petersen, Holger (2003), "Element distinctness on one-tape Turing machines: a complete solution.",
193:
solves the problem within a constant factor of the best possible number of comparisons. The same lower bound applies as well to the
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399:
46:
the list and then checking if there are any consecutive equal elements; it may also be solved in linear expected time by a
959:
594:
559:
28:
844:"Polynomial Degree and Lower Bounds in Quantum Complexity: Collision and Element Distinctness with Small Range"
626:; Heide, Friedhelm Meyer; Smolensky, Roman (1996), "A lower bound for randomized algebraic decision trees",
224:
97:
769:
664:(1999), "Complexity lower bounds for randomized computation trees over zero characteristic fields",
42:
It is a well studied problem in many different models of computation. The problem may be solved by
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8:
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69:
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156:, meaning that the problem can be solved in a number of comparisons proportional to
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is the problem of determining whether all the elements of a list are distinct.
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54:
and compares only those elements that are placed in the same hash table cell.
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554:
87:
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Quantum lower bounds for the collision and the element distinctness problems
494:. The element distinctness problem is a special case of this problem where
677:
604:
569:
913:
819:
783:
213:
703:(2001), "Topological Lower Bounds on Algebraic Random Access Machines",
592:
Ben-Or, Michael (1983), "Lower bounds for algebraic computation trees",
639:
51:
938:
700:
217:
43:
879:"Quantum Lower Bound for the Collision Problem with Small Range"
618:
552:
66:
The number of comparisons needed to solve the problem of size
307:, while on a nondeterministic machine the time complexity is
393:
767:(2007), "Quantum walk algorithm for element distinctness",
729:
557:(1990), "Not all keys can be hashed in constant time",
86:, in a comparison-based model of computation such as a
343:
500:
462:
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227:
162:
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100:
72:
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452:may be found by a comparison-based algorithm, the
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144:
124:
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595:Proc. 15th ACM Symposium on Theory of Computing
560:Proc. 22nd ACM Symposium on Theory of Computing
216:, the decision-tree lower bound extends to the
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811:Symposium on Foundations of Computer Science
262:
61:
911:
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861:
818:
782:
660:
603:
568:
394:Generalization: Finding repeated elements
841:
763:
14:
952:
591:
207:
916:(1982), "Finding repeated elements",
876:
381:queries. The optimal algorithm is by
329:
804:
454:Misra–Gries heavy hitters algorithm
400:Misra–Gries heavy hitters algorithm
212:If the elements in the problem are
24:
553:Gil, J.; Meyer auf der Heide, F.;
520:. This time is optimal under the
344:
337:can solve this problem faster, in
228:
139:
101:
25:
971:
252:{\displaystyle \Theta (n\log n)}
125:{\displaystyle \Theta (n\log n)}
918:Science of Computer Programming
905:
870:
29:computational complexity theory
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404:Elements that occur more than
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50:that inserts each item into a
13:
1:
539:
374:{\textstyle \Theta (n^{2/3})}
930:10.1016/0167-6423(82)90012-0
432:times in a multiset of size
267:A single-tape deterministic
33:element distinctness problem
7:
527:
271:can solve the problem, for
10:
976:
809:. Proceedings of the 43rd
487:{\displaystyle O(n\log k)}
397:
218:real random-access machine
37:element uniqueness problem
18:Element uniqueness problem
898:10.4086/toc.2005.v001a002
863:10.4086/toc.2005.v001a003
829:10.1109/SFCS.2002.1181975
793:10.1137/S0097539705447311
770:SIAM Journal on Computing
744:10.1007/s00236-003-0125-8
717:10.1137/S0097539797329397
705:SIAM Journal on Computing
263:Turing Machine complexity
960:Polynomial-time problems
666:Computational Complexity
628:Computational Complexity
62:Decision tree complexity
202:algebraic decision tree
178:{\displaystyle n\log n}
145:{\displaystyle \Theta }
92:algebraic decision tree
514:
488:
446:
426:
375:
253:
197:of comparisons in the
179:
146:
126:
80:
842:Ambainis, A. (2005).
678:10.1007/s000370050002
605:10.1145/800061.808735
570:10.1145/100216.100247
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489:
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187:linearithmic function
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813:. pp. 513–519.
699:Ben-Amram, Amir M.;
563:, pp. 244–253,
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341:
225:
160:
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48:randomized algorithm
884:Theory of Computing
849:Theory of Computing
522:decision tree model
513:{\displaystyle k=n}
425:{\displaystyle n/k}
285:bits each, in time
208:Real RAM Complexity
877:Kutin, S. (2005).
640:10.1007/BF01270387
598:, pp. 80–86,
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484:
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371:
335:Quantum algorithms
330:Quantum complexity
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154:big theta notation
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534:Collision problem
445:{\displaystyle n}
79:{\displaystyle n}
16:(Redirected from
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805:Shi, Y. (2002).
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765:Ambainis, Andris
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924:(2): 143–152,
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797:
777:(1): 210–239,
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711:(3): 722–761,
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672:(4): 316–329,
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555:Wigderson, A.
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88:decision tree
73:
59:
55:
53:
49:
45:
40:
38:
34:
30:
19:
921:
917:
907:
891:(1): 29–36.
888:
882:
872:
856:(1): 37–46.
853:
847:
837:
806:
800:
774:
768:
759:
738:(2): 81–94,
735:
731:
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298:
294:
291:
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281:
280:≥ log
277:
275:elements of
272:
266:
214:real numbers
211:
65:
56:
41:
36:
32:
26:
912:Misra, J.;
701:Galil, Zvi
634:(4): 357,
540:References
456:, in time
387:Yaoyun Shi
199:randomized
52:hash table
939:1813/6345
914:Gries, D.
476:
345:Θ
241:
229:Θ
170:
140:Θ
114:
102:Θ
954:Category
752:24821585
686:10641238
579:11943779
528:See also
152:invokes
132:. Here,
648:1462184
301:+2–log
204:model.
44:sorting
750:
684:
646:
577:
320:+ log
31:, the
815:arXiv
779:arXiv
748:S2CID
682:S2CID
644:S2CID
575:S2CID
94:, is
934:hdl
926:doi
893:doi
858:doi
825:doi
789:doi
740:doi
713:doi
674:doi
636:doi
600:doi
565:doi
473:log
238:log
185:(a
167:log
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90:or
35:or
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