Difference between revisions of "Template:Heap Running Times"

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(Transpose table to avoid excessive width. Now 6 columns, 10 rows, and additional heap types add rows rather than width.)
(Improve existing refs with citeseerx links, authorlinks, etc.)
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  |journal=[[Journal of the Association for Computing Machinery]]
 
  |journal=[[Journal of the Association for Computing Machinery]]
 
  |volume=34 |issue=3 |date=July 1987 |pages=596-615
 
  |volume=34 |issue=3 |date=July 1987 |pages=596-615
  |ref=harv |doi=10.1145/28869.28874
+
  |ref=harv |doi=10.1145/28869.28874 |citeseerx=10.1.1.309.8927
}}<!--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>
+
}}<!-- An earlier version of this paper appeared in 1984 {{doi|10.1109/SFCS.1984.715934}}--></ref>
 
|style="background:#ddffdd"| ''Θ''(1)
 
|style="background:#ddffdd"| ''Θ''(1)
 
|style="background:#ffffdd"| ''O''(log&nbsp;''n''){{efn|name=amortized}}
 
|style="background:#ffffdd"| ''O''(log&nbsp;''n''){{efn|name=amortized}}
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|-
 
|-
 
! [[Pairing heap|Pairing]]<ref name="Iacono">{{citation
 
! [[Pairing heap|Pairing]]<ref name="Iacono">{{citation
  | last = Iacono | first = John
+
  | last = Iacono | first = John | authorlink = John Iacono
 
  | contribution = Improved upper bounds for pairing heaps
 
  | contribution = Improved upper bounds for pairing heaps
 
  | url = http://john2.poly.edu/papers/swat00/paper.pdf
 
  | url = http://john2.poly.edu/papers/swat00/paper.pdf
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  | volume = 1851
 
  | volume = 1851
 
  | year = 2000
 
  | year = 2000
  | arxiv = 1110.4428}}</ref>
+
  | arxiv = 1110.4428
 +
| citeseerx = 10.1.1.748.7812}}</ref>
 
|style="background:#ddffdd"| ''Θ''(1)
 
|style="background:#ddffdd"| ''Θ''(1)
 
|style="background:#ffffdd"| ''O''(log ''n''){{efn|name=amortized}}
 
|style="background:#ffffdd"| ''O''(log ''n''){{efn|name=amortized}}
 
|style="background:#ddffdd"| ''Θ''(1)
 
|style="background:#ddffdd"| ''Θ''(1)
|style="background:#ffffdd"| ''o''(log&nbsp;''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:#ffffdd"| ''o''(log&nbsp;''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)
 
|-
 
|-
! [[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.
+
! [[Brodal queue|Brodal]]<ref>{{citation
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=338-341|isbn=0-471-46983-1}}</ref>}}
+
| last=Brodal | first=Gerth S. | authorlink=Gerth Stølting Brodal
 +
| 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=338-341
 +
|isbn=0-471-46983-1}}</ref>}}
 
|style="background:#ddffdd"| ''Θ''(1)
 
|style="background:#ddffdd"| ''Θ''(1)
 
|style="background:#ffffdd"| ''O''(log&nbsp;''n'')
 
|style="background:#ffffdd"| ''O''(log&nbsp;''n'')
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  | last1 = Haeupler | first1 = Bernhard
 
  | last1 = Haeupler | first1 = Bernhard
 
  | last2 = Sen | first2 = Siddhartha
 
  | last2 = Sen | first2 = Siddhartha
  | last3 = Tarjan | first3 = Robert E.
+
  | last3 = Tarjan | first3 = Robert E. |authorlink3 = Robert Tarjan
 
  | title = Rank-pairing heaps
 
  | title = Rank-pairing heaps
 
  | journal = SIAM J. Computing
 
  | journal = SIAM J. Computing
  | pages = 1463–1485
+
  | volume = 40 | issue = 6 | pages = 1463–1485
 
  | date = November 2011
 
  | date = November 2011
 
  | doi = 10.1137/100785351
 
  | doi = 10.1137/100785351
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|style="background:#ddffdd"| ''Θ''(1)
 
|style="background:#ddffdd"| ''Θ''(1)
 
|-
 
|-
! [[Fibonacci heap|Strict Fibonacci]]<ref>{{Cite conference| doi = 10.1145/2213977.2214082| title = Strict Fibonacci heaps| conference = Proceedings of the 44th symposium on Theory of Computing - STOC '12| pages = 1177| year = 2012| last1 = Brodal | first1 = G. S. L. | last2 = Lagogiannis | first2 = G. | last3 = Tarjan | first3 = R. E. | isbn = 9781450312455| url = http://www.cs.au.dk/~gerth/papers/stoc12.pdf}}</ref>
+
! [[Fibonacci heap|Strict Fibonacci]]<ref>{{Cite conference
 +
| doi = 10.1145/2213977.2214082
 +
| title = Strict Fibonacci heaps
 +
| conference = Proceedings of the 44th symposium on Theory of Computing - STOC '12
 +
| pages = 1177–1184 | year = 2012
 +
| last1 = Brodal | first1 = Gerth Stølting | authorlink1 = Gerth Stølting Brodal
 +
| last2 = Lagogiannis | first2 = George
 +
| last3 = Tarjan | first3 = Robert E.| authorlink3 = Robert Tarjan
 +
| isbn = 978-1-4503-1245-5
 +
| citeseerx = 10.1.1.233.1740
 +
| url = http://www.cs.au.dk/~gerth/papers/stoc12.pdf}}</ref>
 
|style="background:#ddffdd"| ''Θ''(1)
 
|style="background:#ddffdd"| ''Θ''(1)
 
|style="background:#ffffdd"| ''O''(log ''n'')
 
|style="background:#ffffdd"| ''O''(log ''n'')

Revision as of 03:38, 10 March 2019

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 find-min delete-min insert decrease-key merge
Binary[1] Θ(1) Θ(log n) O(log 'n) O(log n) Θ(n)
Leftist Θ(1) Θ(log n) Θ(log n) Θ(n) Θ(log n)
Binomial[1] Θ(log n) Θ(log n) Θ(1)[lower-alpha 1] Θ(log n) O(log n)[lower-alpha 2]
Fibonacci[1][2] Θ(1) O(log n)[lower-alpha 1] Θ(1) Θ(1)[lower-alpha 1] Θ(1)
Pairing[3] Θ(1) O(log n)[lower-alpha 1] Θ(1) o(log n)[lower-alpha 1][lower-alpha 3]).</math>[5]}} Θ(1)
Brodal[6][lower-alpha 4] Θ(1) O(log n) Θ(1) Θ(1) Θ(1)
Rank-pairing[8] Θ(1) O(log n)[lower-alpha 1] Θ(1) Θ(1)[lower-alpha 1] Θ(1)
Strict Fibonacci[9] Θ(1) O(log n) Θ(1) Θ(1) Θ(1)
2-3 heap ? O(log n)[lower-alpha 1] O(log n)[lower-alpha 1] Θ(1) ?
  1. 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 Amortized time.
  2. n is the size of the larger heap.
  3. Lower bound of <math>\Omega(\log\log n),</math>[4] upper bound of <math>O(2^{2\sqrt{\log\log n
  4. 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).[7]
  1. 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. CiteSeerX 10.1.1.309.8927. 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 (PDF), Lecture Notes in Computer Science, 1851, Springer-Verlag, pp. 63–77, arXiv:1110.4428, CiteSeerX 10.1.1.748.7812, doi:10.1007/3-540-44985-X_5, ISBN 3-540-67690-2
  4. 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.
  5. 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.
  6. Brodal, Gerth S. (1996), "Worst-Case Efficient Priority Queues" (PDF), Proc. 7th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 52–58
  7. 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.
  8. Haeupler, Bernhard; Sen, Siddhartha; Tarjan, Robert E. (November 2011). "Rank-pairing heaps" (PDF). SIAM J. Computing. 40 (6): 1463–1485. doi:10.1137/100785351.
  9. Brodal, Gerth Stølting; Lagogiannis, George; Tarjan, Robert E. (2012). Strict Fibonacci heaps (PDF). Proceedings of the 44th symposium on Theory of Computing - STOC '12. pp. 1177–1184. CiteSeerX 10.1.1.233.1740. doi:10.1145/2213977.2214082. ISBN 978-1-4503-1245-5.