# Brodal queue

In computer science, the Brodal queue is a heap/priority queue structure with very low worst case time bounds: ${\displaystyle O(1)}$ for insertion, find-minimum, meld (merge two queues) and decrease-key and ${\displaystyle O(\mathrm {log} (n))}$ 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. Function names assume a min-heap. For the meaning of "O(f)" and "Θ(f)" see Big O notation.

Operation find-min delete-min insert decrease-key meld
Binary[3] Θ(1) Θ(log n) O(log n) O(log n) Θ(n)
Leftist Θ(1) Θ(log n) Θ(log n) O(log n) Θ(log n)
Binomial[3][4] Θ(1) Θ(log n) Θ(1)[lower-alpha 1] Θ(log n) O(log n)[lower-alpha 2]
Fibonacci[3][5] Θ(1) O(log n)[lower-alpha 1] Θ(1) Θ(1)[lower-alpha 1] Θ(1)
Pairing[6] Θ(1) O(log n)[lower-alpha 1] Θ(1) o(log n)[lower-alpha 1][lower-alpha 3] Θ(1)
Brodal[9][lower-alpha 4] Θ(1) O(log n) Θ(1) Θ(1) Θ(1)
Rank-pairing[11] Θ(1) O(log n)[lower-alpha 1] Θ(1) Θ(1)[lower-alpha 1] Θ(1)
Strict Fibonacci[12] Θ(1) O(log n) Θ(1) Θ(1) Θ(1)
2-3 heap[13] O(log n) O(log n)[lower-alpha 1] O(log n)[lower-alpha 1] Θ(1) ?
1. Amortized time.
2. n is the size of the larger heap.
3. Lower bound of ${\displaystyle \Omega (\log \log n),}$[7] upper bound of ${\displaystyle O(2^{2{\sqrt {\log \log n}}}).}$[8]
4. 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).[10]

## References

1. Gerth Stølting Brodal (1996). Worst-case efficient priority queues. Proc. 7th ACM-SIAM Symposium on Discrete Algorithms, pp. 52–58
2. Gerth Stølting Brodal and Chris Okasaki (1996). Optimal purely functional priority queues. Journal of Functional Programming.
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.
4. "Binomial Heap | Brilliant Math & Science Wiki". brilliant.org. Retrieved 2019-09-30.
5. 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.
6. 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
7. 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.
8. 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.
9. Brodal, Gerth S. (1996), "Worst-Case Efficient Priority Queues" (PDF), Proc. 7th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 52–58
10. 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.
11. Haeupler, Bernhard; Sen, Siddhartha; Tarjan, Robert E. (November 2011). "Rank-pairing heaps" (PDF). SIAM J. Computing. 40 (6): 1463–1485. doi:10.1137/100785351.
12. 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.
13. Takaoka, Tadao (1999), Theory of 2–3 Heaps (PDF), p. 12