Difference between revisions of "Template:Heap Running Times"

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(Since n >= m, we have log(n+m)=O(log n) (in fact, log(n) + O(1)).)
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|style="background:#ffdddd"| ''Θ''(''m'' log(''n''+''m'')){{efn|name=merge2|''n'' is the size of the larger heap and ''m'' is the size of the smaller heap.}}
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|style="background:#ffdddd"| ''Θ''(''m'' log ''n''){{efn|name=merge2|''n'' is the size of the larger heap and ''m'' is the size of the smaller heap.}}
 
|style="background:#ffffdd"| ''O''(log ''n''){{efn|name=merge|''n'' is the size of the larger heap.}}
 
|style="background:#ffffdd"| ''O''(log ''n''){{efn|name=merge|''n'' is the size of the larger heap.}}
 
|style="background:#ddffdd"| ''Θ''(1)
 
|style="background:#ddffdd"| ''Θ''(1)

Revision as of 14:14, 11 June 2015

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] Pairing[2] Brodal[3][lower-alpha 1] Rank-pairing[5] Strict Fibonacci[6]
find-min Θ(1) Θ(1) 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 Θ(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>[7][8]}} Θ(1) Θ(1)[lower-alpha 2] Θ(1)
merge Θ(m log n)[lower-alpha 4] O(log n)[lower-alpha 5] 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).[4]
  2. 2.0 2.1 2.2 2.3 2.4 2.5 2.6 Amortized time.
  3. Bounded by <math>\Omega(\log\log n), O(2^{2\sqrt{\log\log n
  4. n is the size of the larger heap and m is the size of the smaller heap.
  5. n is the size of the larger heap.
  1. 1.0 1.1 1.2 1.3 Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest (1990): Introduction to algorithms. MIT Press / McGraw-Hill.
  2. 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, doi:10.1007/3-540-44985-X_5
  3. Proc. 7th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 52-58, 1996
  4. Goodrich, Michael T.; Tamassia, Roberto (2004). "7.3.6. Bottom-Up Heap Construction". Data Structures and Algorithms in Java (3rd ed.). pp. 338–341.
  5. Haeupler, Bernhard; Sen, Siddhartha; Tarjan, Robert E. (2009). "Rank-pairing heaps" (PDF). SIAM J. Computing: 1463–1485.
  6. Attention: This template ({{cite doi}}) is deprecated. To cite the publication identified by doi:10.1145/2213977.2214082, please use {{cite journal}} with |doi=10.1145/2213977.2214082 instead.
  7. Fredman, Michael Lawrence; Tarjan, Robert E. (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)
  8. Pettie, Seth (2005). "Towards a Final Analysis of Pairing Heaps" (PDF). Max Planck Institut für Informatik.