Difference between revisions of "Template:Heap Running Times"

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imported>Arthur MILCHIOR
(Undid revision 740157890 by 2001:388:E000:BA00:4D0B:E82B:C093:F632 (talk) O and Theta have different meaning, as explained above the tabl)
(Fixed dead link, moved wrong Fredman paper to Fibonacci heap column where it belongs, insert right Fredman paper in pairing heap bounds footnote, other reference maintenance.)
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! [[Binary heap|Binary]]<ref name="CLRS"/>
 
! [[Binary heap|Binary]]<ref name="CLRS"/>
 
! [[Binomial heap|Binomial]]<ref name="CLRS"/>
 
! [[Binomial heap|Binomial]]<ref name="CLRS"/>
! [[Fibonacci heap|Fibonacci]]<ref name="CLRS"/>
+
! [[Fibonacci heap|Fibonacci]]<ref name="CLRS"/><ref name="Fredman And Tarjan">{{cite journal
 +
|first1=Michael Lawrence |last1=Fredman |authorlink1=Michael Fredman
 +
|first2=Robert E. |last2=Tarjan |authorlink2=Robert Tarjan
 +
|title=Fibonacci heaps and their uses in improved network optimization algorithms
 +
|url=http://bioinfo.ict.ac.cn/~dbu/AlgorithmCourses/Lectures/Fibonacci-Heap-Tarjan.pdf
 +
|journal=[[Journal of the Association for Computing Machinery]]
 +
|volume=34 |issue=3 |date=July 1987 |pages=596&ndash;615
 +
|ref=harv |doi=10.1145/28869.28874
 +
}}<!--See also 1994 paper by the same title at https://www.computer.org/csdl/proceedings/focs/1984/0591/00/0715934.pdf doi=10.1109/SFCS.1984.715934 --></ref>
 
! [[Pairing heap|Pairing]]<ref name="Iacono">{{citation
 
! [[Pairing heap|Pairing]]<ref name="Iacono">{{citation
 
  | last = Iacono | first = John
 
  | last = Iacono | first = John
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  | series = Lecture Notes in Computer Science
 
  | series = Lecture Notes in Computer Science
 
  | title = Proc. 7th Scandinavian Workshop on Algorithm Theory
 
  | title = Proc. 7th Scandinavian Workshop on Algorithm Theory
 +
| isbn = 3-540-67690-2
 
  | volume = 1851
 
  | volume = 1851
  | year = 2000}}</ref>
+
  | year = 2000
! [[Brodal queue|Brodal]]<ref>{{citation | last=Brodal | first=Gerth S. | url=http://www.cs.au.dk/~gerth/papers/soda96.pdf | contribution=Worst-Case Efficient Priority Queues | title=Proc. 7th Annual ACM-SIAM Symposium on Discrete Algorithms |pages=52–58 | year=1996}}</ref>{{efn|name=brodal|Brodal and Okasaki later describe a [[Persistent_data_structure|persistent]] variant with the same bounds except for decrease-key, which is not supported.
+
| arxiv = 1110.4428}}</ref>
Heaps with ''n'' elements can be constructed bottom-up in ''O''(''n'').<ref>{{cite book|title=Data Structures and Algorithms in Java|first1=Michael T.|last1=Goodrich|authorlink1=Michael T. Goodrich|first2=Roberto|last2=Tamassia|authorlink2=Roberto Tamassia|edition=3rd|year=2004|chapter=7.3.6. Bottom-Up Heap Construction|pages=338&ndash;341}}</ref>}}
+
! [[Brodal queue|Brodal]]<ref>{{citation | last=Brodal | first=Gerth S. | contribution-url=http://www.cs.au.dk/~gerth/papers/soda96.pdf | contribution=Worst-Case Efficient Priority Queues | title=Proc. 7th Annual ACM-SIAM Symposium on Discrete Algorithms |pages=52–58 | year=1996}}</ref>{{efn|name=brodal|Brodal and Okasaki later describe a [[Persistent_data_structure|persistent]] variant with the same bounds except for decrease-key, which is not supported.
 +
Heaps with ''n'' elements can be constructed bottom-up in ''O''(''n'').<ref>{{cite book|title=Data Structures and Algorithms in Java|first1=Michael T.|last1=Goodrich|authorlink1=Michael T. Goodrich|first2=Roberto|last2=Tamassia|authorlink2=Roberto Tamassia|edition=3rd|year=2004|chapter=7.3.6. Bottom-Up Heap Construction|pages=338&ndash;341|isbn=0-471-46983-1}}</ref>}}
 
! [[Rank-pairing heap|Rank-pairing]]<ref>{{cite journal
 
! [[Rank-pairing heap|Rank-pairing]]<ref>{{cite journal
 
  | last1 = Haeupler | first1 = Bernhard
 
  | last1 = Haeupler | first1 = Bernhard
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|style="background:#ffffdd"| ''Θ''(log ''n'')
 
|style="background:#ffffdd"| ''Θ''(log ''n'')
 
|style="background:#ddffdd"| ''Θ''(1){{efn|name=amortized}}
 
|style="background:#ddffdd"| ''Θ''(1){{efn|name=amortized}}
|style="background:#ffffdd"| ''o''(log ''n''){{efn|name=amortized}}{{efn|name=pairingdecreasekey|Bounded by <math>\Omega(\log\log n), O(2^{2\sqrt{\log\log n}})</math><ref name="Fredman And Tarjan">{{cite journal|first1=Michael Lawrence|last1=Fredman|authorlink1=Michael Fredman|first2=Robert E.|last2=Tarjan|authorlink2=Robert Tarjan |title=Fibonacci heaps and their uses in improved network optimization algorithms| url = http://www.cl.cam.ac.uk/~sos22/supervise/dsaa/fib_heaps.pdf | format = PDF |journal=[[Journal of the Association for Computing Machinery]]|volume=34|year=1987|pages=596&ndash;615|ref=harv|doi=10.1145/28869.28874|issue=3}}</ref><ref>{{cite journal|last=Pettie|first=Seth|title=Towards a Final Analysis of Pairing Heaps|journal=Max Planck Institut f&uuml;r Informatik|year=2005|url=http://web.eecs.umich.edu/~pettie/papers/focs05.pdf}}</ref>}}
+
|style="background:#ffffdd"| ''o''(log ''n''){{efn|name=amortized}}{{efn|name=pairingdecreasekey|Lower bound of <math>\Omega(\log\log n),</math><ref name="Fredman1999">{{cite journal |first=Michael Lawrence |last=Fredman |authorlink=Michael Fredman |title=On the Efficiency of Pairing Heaps and Related Data Structures |url=http://www.uqac.ca/azinflou/Fichiers840/EfficiencyPairingHeap.pdf |journal=[[Journal of the Association for Computing Machinery]] |volume=46 |issue=4 |pages=473&ndash;501 |date=July 1999 |doi=10.1145/320211.320214}}</ref> upper bound of <math>O(2^{2\sqrt{\log\log n}}).</math><ref>{{cite conference |last=Pettie |first=Seth |title=Towards a Final Analysis of Pairing Heaps |conference=FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science |pages=174&ndash;183 |isbn=0-7695-2468-0 |doi=10.1109/SFCS.2005.75 |citeseerx=10.1.1.549.471 |year=2005 |url=http://web.eecs.umich.edu/~pettie/papers/focs05.pdf}}</ref>}}
 
|style="background:#ddffdd"| ''Θ''(1)
 
|style="background:#ddffdd"| ''Θ''(1)
 
|style="background:#ddffdd"| ''Θ''(1){{efn|name=amortized}}
 
|style="background:#ddffdd"| ''Θ''(1){{efn|name=amortized}}

Revision as of 09:01, 29 December 2016

In the following time complexities[1] O(f) is an asymptotic upper bound and Θ(f) is an asymptotically tight bound (see Big O notation). Function names assume a min-heap.

Operation Binary[1] Binomial[1] Fibonacci[1][2] Pairing[3] Brodal[4][lower-alpha 1] Rank-pairing[6] Strict Fibonacci[7]
find-min Θ(1) Θ(log n) Θ(1) Θ(1) Θ(1) Θ(1) Θ(1)
delete-min Θ(log n) Θ(log n) O(log n)[lower-alpha 2] O(log n)[lower-alpha 2] O(log n) O(log n)[lower-alpha 2] O(log n)
insert O(log n) Θ(1)[lower-alpha 2] Θ(1) Θ(1) Θ(1) Θ(1) Θ(1)
decrease-key Θ(log n) Θ(log n) Θ(1)[lower-alpha 2] o(log n)[lower-alpha 2][lower-alpha 3]).</math>[9]}} Θ(1) Θ(1)[lower-alpha 2] Θ(1)
merge Θ(n) O(log n)[lower-alpha 4] Θ(1) Θ(1) Θ(1) Θ(1) Θ(1)
  1. Brodal and Okasaki later describe a persistent variant with the same bounds except for decrease-key, which is not supported. Heaps with n elements can be constructed bottom-up in O(n).[5]
  2. Jump up to: 2.0 2.1 2.2 2.3 2.4 2.5 2.6 Amortized time.
  3. Lower bound of <math>\Omega(\log\log n),</math>[8] upper bound of <math>O(2^{2\sqrt{\log\log n
  4. n is the size of the larger heap.
  1. Jump up to: 1.0 1.1 1.2 1.3 Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L. (1990). Introduction to Algorithms (1st ed.). MIT Press and McGraw-Hill. ISBN 0-262-03141-8.
  2. Fredman, Michael Lawrence; Tarjan, Robert E. (July 1987). "Fibonacci heaps and their uses in improved network optimization algorithms" (PDF). Journal of the Association for Computing Machinery. 34 (3): 596–615. doi:10.1145/28869.28874.CS1 maint: ref=harv (link)
  3. Iacono, John (2000), "Improved upper bounds for pairing heaps", Proc. 7th Scandinavian Workshop on Algorithm Theory, Lecture Notes in Computer Science, 1851, Springer-Verlag, pp. 63–77, arXiv:1110.4428, doi:10.1007/3-540-44985-X_5, ISBN 3-540-67690-2
  4. Brodal, Gerth S. (1996), "Worst-Case Efficient Priority Queues" (PDF), Proc. 7th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 52–58
  5. Goodrich, Michael T.; Tamassia, Roberto (2004). "7.3.6. Bottom-Up Heap Construction". Data Structures and Algorithms in Java (3rd ed.). pp. 338–341. ISBN 0-471-46983-1.
  6. Haeupler, Bernhard; Sen, Siddhartha; Tarjan, Robert E. (2009). "Rank-pairing heaps" (PDF). SIAM J. Computing: 1463–1485.
  7. Brodal, G. S. L.; Lagogiannis, G.; Tarjan, R. E. (2012). Strict Fibonacci heaps (PDF). Proceedings of the 44th symposium on Theory of Computing - STOC '12. p. 1177. doi:10.1145/2213977.2214082. ISBN 9781450312455.
  8. Fredman, Michael Lawrence (July 1999). "On the Efficiency of Pairing Heaps and Related Data Structures" (PDF). Journal of the Association for Computing Machinery. 46 (4): 473–501. doi:10.1145/320211.320214.
  9. Pettie, Seth (2005). Towards a Final Analysis of Pairing Heaps (PDF). FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science. pp. 174–183. CiteSeerX 10.1.1.549.471. doi:10.1109/SFCS.2005.75. ISBN 0-7695-2468-0.