Brodal queue

Brodal queue
Brodal queue
Type	Heap/priority queue
Invented	1996
Invented by	Gerth Stølting Brodal
Operation
Time complexity in big O notation
Operation	Average
Space complexity

In computer science, the Brodal queue is a heap/priority queue structure with very low worst case time bounds: $O(1)$ for insertion, find-minimum, meld (merge two queues) and decrease-key and $O(\mathrm {log} (n))$ for delete-minimum and general deletion. They are the first heap variant to achieve these bounds without resorting to amortization of operational costs. Brodal queues are named after their inventor Gerth Stølting Brodal.^[1]

While having better asymptotic bounds than other priority queue structures, they are, in the words of Brodal himself, "quite complicated" and "[not] applicable in practice."^[1] Brodal and Okasaki describe a persistent (purely functional) version of Brodal queues.^[2]

Summary of running times

Here are time complexities^[3] 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^[3]	Θ(1)	Θ(log n)	Θ(log n)	Θ(log n)	Θ(n)	Θ(n)
Skew^[4]	Θ(1)	O(log n) am.	O(log n) am.	O(log n) am.	O(log n) am.	Θ(n) am.
Leftist^[5]	Θ(1)	Θ(log n)	Θ(log n)	Θ(log n)	Θ(log n)	Θ(n)
Binomial^[3]^[7]	Θ(1)	Θ(log n)	Θ(log n)	Θ(1) am.	Θ(log n)^[b]	Θ(n)
Skew binomial^[8]	Θ(1)	Θ(log n)	Θ(log n)	Θ(1)	Θ(log n)^[b]	Θ(n)
2–3 heap^[10]	Θ(1)	O(log n) am.	Θ(1)	Θ(1) am.	O(log n)^[b]	Θ(n)
Bottom-up skew^[4]	Θ(1)	O(log n) am.	O(log n) am.	Θ(1) am.	Θ(1) am.	Θ(n) am.
Pairing^[11]	Θ(1)	O(log n) am.	o(log n) am.^[c]	Θ(1)	Θ(1)	Θ(n)
Rank-pairing^[14]	Θ(1)	O(log n) am.	Θ(1) am.	Θ(1)	Θ(1)	Θ(n)
Fibonacci^[3]^[15]	Θ(1)	O(log n) am.	Θ(1) am.	Θ(1)	Θ(1)	Θ(n)
Strict Fibonacci^[16]^[d]	Θ(1)	Θ(log n)	Θ(1)	Θ(1)	Θ(1)	Θ(n)
Brodal^[17]^[d]	Θ(1)	Θ(log n)	Θ(1)	Θ(1)	Θ(1)	Θ(n)^[18]

^ 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).^[4]^[5] Another algorithm achieves Θ(n) for binary heaps.^[6]
^ ^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.^[9] 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.^[8]
^ Lower bound of $\Omega (\log \log n),$ ^[12] upper bound of $O(2^{2{\sqrt {\log \log n}}}).$ ^[13]
^ ^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

^ ^a ^b Gerth Stølting Brodal (1996). Worst-case efficient priority queues. Proc. 7th ACM-SIAM Symposium on Discrete Algorithms, pp. 52–58
^ Gerth Stølting Brodal and Chris Okasaki (1996). Optimal purely functional priority queues. Journal of Functional Programming.
^ ^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.

This algorithms or data structures-related article is a stub. You can help Wikipedia by expanding it.

[7] 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).^[4]^[5] Another algorithm achieves Θ(n) for binary heaps.^[6]

[bootstrap-meld-9] 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.^[9] 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.^[8]

[pairingdecreasekey-16] Lower bound of $\Omega (\log \log n),$ ^[12] upper bound of $O(2^{2{\sqrt {\log \log n}}}).$ ^[13]

[optimum-20] 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.

[Brodal-1] Gerth Stølting Brodal (1996). Worst-case efficient priority queues. Proc. 7th ACM-SIAM Symposium on Discrete Algorithms, pp. 52–58

[BrodalOkasaki-2] Gerth Stølting Brodal and Chris Okasaki (1996). Optimal purely functional priority queues. Journal of Functional Programming.

[CLRS-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.

[sleator-tarjan-skew-4] 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.

[tarjan-leftist-5] 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-mcdiarmid-heap-build-6] 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.

[8] "Binomial Heap | Brilliant Math & Science Wiki". brilliant.org. Retrieved 2019-09-30.

[brodal-okasaki-10] Brodal, Gerth Stølting; Okasaki, Chris (November 1996), "Optimal purely functional priority queues", Journal of Functional Programming, 6 (6): 839–857, doi:10.1017/s095679680000201x

[11] Okasaki, Chris (1998). "10.2. Structural Abstraction". Purely Functional Data Structures (1st ed.). pp. 158–162. ISBN 9780521631242.

[12] Takaoka, Tadao (1999), Theory of 2–3 Heaps (PDF), p. 12

[Iacono-13] 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

[Fredman1999-14] 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.

[15] 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.

[17] 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_And_Tarjan-18] 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.

[19] 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.

[21] Brodal, Gerth S. (1996), "Worst-Case Efficient Priority Queues" (PDF), Proc. 7th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 52–58

[22] 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.

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