User:IntGrah/Rank-pairing heap
| Rank-pairing heap | |||||||||||||||||||||||||||||||||||||
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| Type | heap | ||||||||||||||||||||||||||||||||||||
| Invented | 2011 | ||||||||||||||||||||||||||||||||||||
| Invented by | Bernhard Haeupler, Siddhartha Sen, and Robert E. Tarjan | ||||||||||||||||||||||||||||||||||||
| Complexities in big O notation | |||||||||||||||||||||||||||||||||||||
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In computer science, a rank-pairing heap is a data structure for priority queue operations. Rank-pairing heaps were designed to match the amortized running times of Fibonacci heaps whilst maintaining the simplicity of pairing heaps. Rank-pairing heaps were invented in 2011 by Bernhard Haeupler, Siddhartha Sen, and Robert E. Tarjan.[1]
Structure
[edit]A rank-pairing heap is a list of heap-ordered trees represented in the left-child right-sibling binary tree format. This means that each node has one pointer to its left-most child, and another pointer to its right sibling. Additionally, each node maintains a pointer to its parent.
Operations
[edit]Merge
[edit]Insert
[edit]Find-min
[edit]We maintain a pointer to the node containing the minimum key. This will always be a root within the list of trees. The minimum key can thus be found at a constant cost, with only a minor overhead in the other operations.
Delete-min
[edit]Decrease-key
[edit]To decrease the key of a node , reduce its key, and update the minimum key pointer, if it is the new minimum. Then, the subtree rooted at is detached; in the left-child right-sibling representation, this is equivalent to replacing with its right child . The detached subtree is added to the list of trees, and the ranks are recalculated: the rank of is set to be one greater than its left child, and the ancestors of have their ranks reduced.
Summary of running times
[edit]Here are time complexities[2] of various heap data structures. The abbreviation am. indicates that the given complexity is amortized, otherwise it is a worst-case complexity. For the meaning of "O(f)" and "Θ(f)" see Big O notation. Names of operations assume a min-heap.
| Operation | find-min | delete-min | decrease-key | insert | meld | make-heap[a] |
|---|---|---|---|---|---|---|
| Binary[2] | Θ(1) | Θ(log n) | Θ(log n) | Θ(log n) | Θ(n) | Θ(n) |
| Skew[3] | Θ(1) | O(log n) am. | O(log n) am. | O(log n) am. | O(log n) am. | Θ(n) am. |
| Leftist[4] | Θ(1) | Θ(log n) | Θ(log n) | Θ(log n) | Θ(log n) | Θ(n) |
| Binomial[2][6] | Θ(1) | Θ(log n) | Θ(log n) | Θ(1) am. | Θ(log n)[b] | Θ(n) |
| Skew binomial[7] | Θ(1) | Θ(log n) | Θ(log n) | Θ(1) | Θ(log n)[b] | Θ(n) |
| 2–3 heap[9] | Θ(1) | O(log n) am. | Θ(1) | Θ(1) am. | O(log n)[b] | Θ(n) |
| Bottom-up skew[3] | Θ(1) | O(log n) am. | O(log n) am. | Θ(1) am. | Θ(1) am. | Θ(n) am. |
| Pairing[10] | Θ(1) | O(log n) am. | o(log n) am.[c] | Θ(1) | Θ(1) | Θ(n) |
| Rank-pairing[13] | Θ(1) | O(log n) am. | Θ(1) am. | Θ(1) | Θ(1) | Θ(n) |
| Fibonacci[2][14] | Θ(1) | O(log n) am. | Θ(1) am. | Θ(1) | Θ(1) | Θ(n) |
| Strict Fibonacci[15][d] | Θ(1) | Θ(log n) | Θ(1) | Θ(1) | Θ(1) | Θ(n) |
| Brodal[16][d] | Θ(1) | Θ(log n) | Θ(1) | Θ(1) | Θ(1) | Θ(n)[17] |
- ^ make-heap is the operation of building a heap from a sequence of n unsorted elements. It can be done in Θ(n) time whenever meld runs in O(log n) time (where both complexities can be amortized).[3][4] Another algorithm achieves Θ(n) for binary heaps.[5]
- ^ a b c For persistent heaps (not supporting decrease-key), a generic transformation reduces the cost of meld to that of insert, while the new cost of delete-min is the sum of the old costs of delete-min and meld.[8] Here, it makes meld run in Θ(1) time (amortized, if the cost of insert is) while delete-min still runs in O(log n). Applied to skew binomial heaps, it yields Brodal-Okasaki queues, persistent heaps with optimal worst-case complexities.[7]
- ^ Lower bound of [11] upper bound of [12]
- ^ a b Brodal queues and strict Fibonacci heaps achieve optimal worst-case complexities for heaps. They were first described as imperative data structures. The Brodal-Okasaki queue is a persistent data structure achieving the same optimum, except that decrease-key is not supported.
References
[edit]- ^ Haeupler, Bernhard; Sen, Siddhartha; Tarjan, Robert E. (Jan 2011). "Rank-Pairing Heaps". SIAM Journal on Computing. 40 (6): 1463–1485. doi:10.1137/100785351. ISSN 0097-5397.
- ^ a b c d 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.
- ^ a b c Sleator, Daniel Dominic; Tarjan, Robert Endre (February 1986). "Self-Adjusting Heaps". SIAM Journal on Computing. 15 (1): 52–69. CiteSeerX 10.1.1.93.6678. doi:10.1137/0215004. ISSN 0097-5397.
- ^ a b Tarjan, Robert (1983). "3.3. Leftist heaps". Data Structures and Network Algorithms. pp. 38–42. doi:10.1137/1.9781611970265. ISBN 978-0-89871-187-5.
- ^ Hayward, Ryan; McDiarmid, Colin (1991). "Average Case Analysis of Heap Building by Repeated Insertion" (PDF). J. Algorithms. 12: 126–153. CiteSeerX 10.1.1.353.7888. doi:10.1016/0196-6774(91)90027-v. Archived from the original (PDF) on 2016-02-05. Retrieved 2016-01-28.
- ^ "Binomial Heap | Brilliant Math & Science Wiki". brilliant.org. Retrieved 2019-09-30.
- ^ a b Brodal, Gerth Stølting; Okasaki, Chris (November 1996), "Optimal purely functional priority queues", Journal of Functional Programming, 6 (6): 839–857, doi:10.1017/s095679680000201x
- ^ Okasaki, Chris (1998). "10.2. Structural Abstraction". Purely Functional Data Structures (1st ed.). pp. 158–162. ISBN 9780521631242.
- ^ Takaoka, Tadao (1999), Theory of 2–3 Heaps (PDF), p. 12
- ^ Iacono, John (2000), "Improved upper bounds for pairing heaps", Proc. 7th Scandinavian Workshop on Algorithm Theory (PDF), Lecture Notes in Computer Science, vol. 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.
- ^ Haeupler, Bernhard; Sen, Siddhartha; Tarjan, Robert E. (November 2011). "Rank-pairing heaps" (PDF). SIAM J. Computing. 40 (6): 1463–1485. doi:10.1137/100785351.
- ^ 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.
- ^ 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.
- ^ 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.