Difference between revisions of "Template:Heap Running Times"
Jump to navigation
Jump to search
(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.) |
||
| Line 38: | Line 38: | ||
|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 |
| − | }}<!-- | + | }}<!-- 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 ''n''){{efn|name=amortized}} | |style="background:#ffffdd"| ''O''(log ''n''){{efn|name=amortized}} | ||
| Line 47: | Line 47: | ||
|- | |- | ||
! [[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 | ||
| Line 58: | Line 58: | ||
| 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 ''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=473–501 |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=174–183 |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 ''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=473–501 |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=174–183 | ||
| + | |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 ''n'') | |style="background:#ffffdd"| ''O''(log ''n'') | ||
| Line 76: | Line 105: | ||
| 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 | ||
| Line 89: | Line 118: | ||
|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 = | + | ! [[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) | ? |
- ↑ 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 Amortized time.
- ↑ n is the size of the larger heap.
- ↑ Lower bound of <math>\Omega(\log\log n),</math>[4] upper bound of <math>O(2^{2\sqrt{\log\log n
- ↑ 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]
- ↑ 1.0 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.
- ↑ 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)
- ↑ 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
- ↑ 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.
- ↑ 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.
- ↑ Brodal, Gerth S. (1996), "Worst-Case Efficient Priority Queues" (PDF), Proc. 7th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 52–58
- ↑ 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.
- ↑ Haeupler, Bernhard; Sen, Siddhartha; Tarjan, Robert E. (November 2011). "Rank-pairing heaps" (PDF). SIAM J. Computing. 40 (6): 1463–1485. doi:10.1137/100785351.
- ↑ 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.