Difference between revisions of "Template:Heap Running Times"

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(Use dash rather than hyphen in 2–3 heap)
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|-
 
|-
 
! [[Fibonacci heap|Fibonacci]]<ref name="CLRS"/><ref name="Fredman And Tarjan">{{cite journal
 
! [[Fibonacci heap|Fibonacci]]<ref name="CLRS"/><ref name="Fredman And Tarjan">{{cite journal
  |first1=Michael Lawrence |last1=Fredman |authorlink1=Michael Fredman
+
  |first1=Michael Lawrence |last1=Fredman |author-link1=Michael Fredman
  |first2=Robert E. |last2=Tarjan |authorlink2=Robert Tarjan
+
  |first2=Robert E. |last2=Tarjan |author-link2=Robert Tarjan
 
  |title=Fibonacci heaps and their uses in improved network optimization algorithms
 
  |title=Fibonacci heaps and their uses in improved network optimization algorithms
 
  |url=http://bioinfo.ict.ac.cn/~dbu/AlgorithmCourses/Lectures/Fibonacci-Heap-Tarjan.pdf
 
  |url=http://bioinfo.ict.ac.cn/~dbu/AlgorithmCourses/Lectures/Fibonacci-Heap-Tarjan.pdf
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|-
 
|-
 
! [[Pairing heap|Pairing]]<ref name="Iacono">{{citation
 
! [[Pairing heap|Pairing]]<ref name="Iacono">{{citation
  | last = Iacono | first = John | authorlink = John Iacono
+
  | last = Iacono | first = John | author-link = 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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|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
 
|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
+
  |first=Michael Lawrence |last=Fredman |author-link=Michael Fredman
 
  |title=On the Efficiency of Pairing Heaps and Related Data Structures
 
  |title=On the Efficiency of Pairing Heaps and Related Data Structures
 
  |url=http://www.uqac.ca/azinflou/Fichiers840/EfficiencyPairingHeap.pdf
 
  |url=http://www.uqac.ca/azinflou/Fichiers840/EfficiencyPairingHeap.pdf
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|-
 
|-
 
! [[Brodal queue|Brodal]]<ref>{{citation
 
! [[Brodal queue|Brodal]]<ref>{{citation
  | last=Brodal | first=Gerth S. | authorlink=Gerth Stølting Brodal
+
  | last=Brodal | first=Gerth S. | author-link=Gerth Stølting Brodal
 
  | contribution-url=http://www.cs.au.dk/~gerth/papers/soda96.pdf
 
  | contribution-url=http://www.cs.au.dk/~gerth/papers/soda96.pdf
 
  | contribution=Worst-Case Efficient Priority Queues
 
  | contribution=Worst-Case Efficient Priority Queues
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Heaps with ''n'' elements can be constructed bottom-up in ''O''(''n'').<ref>{{cite book
 
Heaps with ''n'' elements can be constructed bottom-up in ''O''(''n'').<ref>{{cite book
 
  |title=Data Structures and Algorithms in Java
 
  |title=Data Structures and Algorithms in Java
  |first1= Michael T. |last1=Goodrich |authorlink1=Michael T. Goodrich
+
  |first1= Michael T. |last1=Goodrich |author-link1=Michael T. Goodrich
  |first2=Roberto |last2=Tamassia |authorlink2=Roberto Tamassia
+
  |first2=Roberto |last2=Tamassia |author-link2=Roberto Tamassia
 
  |edition=3rd |year=2004 |chapter=7.3.6. Bottom-Up Heap Construction |pages=338-341
 
  |edition=3rd |year=2004 |chapter=7.3.6. Bottom-Up Heap Construction |pages=338-341
 
  |isbn=0-471-46983-1}}</ref>}}
 
  |isbn=0-471-46983-1}}</ref>}}
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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. |authorlink3 = Robert Tarjan
+
  | last3 = Tarjan | first3 = Robert E. |author-link3 = Robert Tarjan
 
  | title = Rank-pairing heaps
 
  | title = Rank-pairing heaps
 
  | journal = SIAM J. Computing
 
  | journal = SIAM J. Computing
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  | conference = Proceedings of the 44th symposium on Theory of Computing - STOC '12
 
  | conference = Proceedings of the 44th symposium on Theory of Computing - STOC '12
 
  | pages = 1177–1184 | year = 2012
 
  | pages = 1177–1184 | year = 2012
  | last1 = Brodal | first1 = Gerth Stølting | authorlink1 = Gerth Stølting Brodal  
+
  | last1 = Brodal | first1 = Gerth Stølting | author-link1 = Gerth Stølting Brodal  
 
  | last2 = Lagogiannis | first2 = George
 
  | last2 = Lagogiannis | first2 = George
  | last3 = Tarjan | first3 = Robert E.| authorlink3 = Robert Tarjan
+
  | last3 = Tarjan | first3 = Robert E.| author-link3 = Robert Tarjan
 
  | isbn = 978-1-4503-1245-5
 
  | isbn = 978-1-4503-1245-5
 
  | citeseerx = 10.1.1.233.1740
 
  | citeseerx = 10.1.1.233.1740
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|-
 
|-
 
! [[2–3 heap]]<ref>{{citation
 
! [[2–3 heap]]<ref>{{citation
  | last=Takaoka | first=Tadao | authorlink=Tadao Takaoka
+
  | last=Takaoka | first=Tadao | author-link=Tadao Takaoka
 
  | url=https://ir.canterbury.ac.nz/bitstream/handle/10092/14769/2-3heaps.pdf
 
  | url=https://ir.canterbury.ac.nz/bitstream/handle/10092/14769/2-3heaps.pdf
 
  | title=Theory of 2–3 Heaps
 
  | title=Theory of 2–3 Heaps

Latest revision as of 20:21, 16 December 2020

Template documentation[view] [edit] [history] [purge]

Objective

{{Heap Running Times}} provides time complexity information for operations across different types of heaps.

Usage

{{Heap Running Times |mode = min}}

where

mode 
optional parameter. If present and set to "max", present information for max heap; otherwise present for min heap

Examples

Here are time complexities[1] of various heap data structures. Function names assume a min-heap. For the meaning of "O(f)" and "Θ(f)" see Big O notation.

Operation find-min delete-min insert decrease-key meld
Binary[1] Θ(1) Θ(log n) O(log n) O(log n) Θ(n)
Leftist Θ(1) Θ(log n) Θ(log n) O(log n) Θ(log n)
Binomial[1][2] Θ(1) Θ(log n) Θ(1)[lower-alpha 1] Θ(log n) O(log n)[lower-alpha 2]
Fibonacci[1][3] Θ(1) O(log n)[lower-alpha 1] Θ(1) Θ(1)[lower-alpha 1] Θ(1)
Pairing[4] Θ(1) O(log n)[lower-alpha 1] Θ(1) o(log n)[lower-alpha 1][lower-alpha 3]).</math>[6]}} Θ(1)
Brodal[7][lower-alpha 4] Θ(1) O(log n) Θ(1) Θ(1) Θ(1)
Rank-pairing[9] Θ(1) O(log n)[lower-alpha 1] Θ(1) Θ(1)[lower-alpha 1] Θ(1)
Strict Fibonacci[10] Θ(1) O(log n) Θ(1) Θ(1) Θ(1)
2–3 heap[11] O(log n) 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>[5] 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).[8]
  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. "Binomial Heap | Brilliant Math & Science Wiki". brilliant.org. Retrieved 2019-09-30.
  3. 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.
  4. 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
  5. 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.
  6. 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.
  7. Brodal, Gerth S. (1996), "Worst-Case Efficient Priority Queues" (PDF), Proc. 7th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 52–58
  8. 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.
  9. Haeupler, Bernhard; Sen, Siddhartha; Tarjan, Robert E. (November 2011). "Rank-pairing heaps" (PDF). SIAM J. Computing. 40 (6): 1463–1485. doi:10.1137/100785351.
  10. 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.
  11. Takaoka, Tadao (1999), Theory of 2–3 Heaps (PDF), p. 12