Abstract
Heaps are well-studied fundamental data structures, having myriads of applications, both theoretical and practical.
We consider the problem of designing a heap with an “optimal” extract-min operation. Assuming an arbitrary linear ordering of keys, a heap with n elements typically takes \(O(\log n)\) time to extract the minimum. Extracting all elements faster is impossible as this would violate the \(\varOmega (n \log n)\) bound for comparison-based sorting. It is known, however, that is takes only \(O(n + k \log k)\) time to sort just k smallest elements out of n given, which prompts that there might be a faster heap, whose extract-min performance depends on the number of elements extracted so far.
In this paper we show that is indeed the case. We present a version of heap that performs insert in O(1) time and takes only \(O(\log ^* n + \log k)\) time to carry out the k-th extraction (where \(\log ^*\) denotes the iterated logarithm). All the above bounds are worst-case.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 3rd edn. The MIT Press, Cambridge (2009)
Williams, J.W.J.: Heapsort. Commun. ACM 7, 347–348 (1964)
Fredman, M.L., Tarjan, R.E.: Fibonacci heaps and their uses in improved network optimization algorithms. J. ACM 34(3), 596–615 (1987)
Brodal, G.S.: Worst-case efficient priority queues. In: Proceedings of the Seventh Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 1996), pp. 52–58, Philadelphia, PA, USA. Society for Industrial and Applied Mathematics (1996)
van Emde Boas, P.: Preserving order in a forest in less than logarithmic time. In: Proceedings of the 16th Annual Symposium on Foundations of Computer Science (SFCS 1975), pp. 75–84, Washington, DC, USA. IEEE Computer Society (1975)
Thorup, M.: On ram priority queues. SIAM J. Comput. 30(1), 86–109 (2000)
Blum, M., Floyd, R.W., Pratt, V., Rivest, R.L., Tarjan, R.E.: Time bounds for selection. J. Comput. Syst. Sci. 7(4), 448–461 (1973)
Navarro, G., Paredes, R.: On sorting, heaps, and minimum spanning trees. Algorithmica 57(4), 585–620 (2010)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Babenko, M., Kolesnichenko, I., Smirnov, I. (2017). Cascade Heap: Towards Time-Optimal Extractions. In: Weil, P. (eds) Computer Science – Theory and Applications. CSR 2017. Lecture Notes in Computer Science(), vol 10304. Springer, Cham. https://doi.org/10.1007/978-3-319-58747-9_8
Download citation
DOI: https://doi.org/10.1007/978-3-319-58747-9_8
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-58746-2
Online ISBN: 978-3-319-58747-9
eBook Packages: Computer ScienceComputer Science (R0)