Difference between revisions of "Template:Heap Running Times"

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imported>Alex Kindel
(Decrease-key operation may terminate in constant time binary binary heaps and therefor is not lower-bound logn but O(1), making it O(log n) and not Theta(log n))
(Stop stupid line breaks between "log" and "n". Better having to horizontally scroll than read that mess.)
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|style="background:#ffffdd"| ''Θ''(log ''n'')
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|style="background:#ffffdd"| ''Θ''(log ''n'')
|style="background:#ffffdd"| ''Θ''(log ''n'')
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|style="background:#ffffdd"| ''Θ''(log ''n'')
|style="background:#ffffdd"| ''Θ''(log ''n'')
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|style="background:#ffffdd"| ''Θ''(log ''n'')
|style="background:#ffffdd"| ''O''(log ''n''){{efn|name=amortized|Amortized time.}}
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|style="background:#ffffdd"| ''O''(log ''n''){{efn|name=amortized|Amortized time.}}
|style="background:#ffffdd"| ''O''(log ''n''){{efn|name=amortized}}
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|style="background:#ffffdd"| ''O''(log ''n''){{efn|name=amortized}}
|style="background:#ffffdd"| ''O''(log ''n'')
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|style="background:#ffffdd"| ''O''(log ''n'')
|style="background:#ffffdd"| ''O''(log ''n''){{efn|name=amortized}}
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|style="background:#ffffdd"| ''O''(log ''n''){{efn|name=amortized}}
|style="background:#ffffdd"| ''O''(log ''n'')
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|style="background:#ffffdd"| ''O''(log ''n'')
|style="background:#ffffdd"| ''O''(log ''n''){{efn|name=amortized}}
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|style="background:#ffffdd"| ''O''(log ''n''){{efn|name=amortized}}
 
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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|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>}}
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|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)
 
|style="background:#ddffdd"| ''Θ''(1){{efn|name=amortized}}
 
|style="background:#ddffdd"| ''Θ''(1){{efn|name=amortized}}

Revision as of 04:24, 3 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 Binary[1] Leftist Binomial[1] Fibonacci[1][2] Pairing[3] Brodal[4][lower-alpha 1] Rank-pairing[6] Strict Fibonacci[7] 2-3 heap
find-min Θ(1) Θ(1) Θ(log n) Θ(1) Θ(1) Θ(1) Θ(1) Θ(1) ?
delete-min Θ(log n) 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) O(log n)[lower-alpha 2]
insert O(log n) Θ(log n) Θ(1)[lower-alpha 2] Θ(1) Θ(1) Θ(1) Θ(1) Θ(1) O(log n)[lower-alpha 2]
decrease-key O(log n) Θ(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) Θ(1)
merge Θ(n) Θ(log 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. 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 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. 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 (PDF), 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. (November 2011). "Rank-pairing heaps" (PDF). SIAM J. Computing: 1463–1485. doi:10.1137/100785351.
  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.