Theory of Computing Systems

, Volume 59, Issue 2, pp 209–230 | Cite as

QuickHeapsort: Modifications and Improved Analysis

  • Volker Diekert
  • Armin WeißEmail author


QuickHeapsort is a combination of Quicksort and Heapsort. We show that the expected number of comparisons for QuickHeapsort is always better than for Quicksort if a usual median-of-constant strategy is used for choosing pivot elements. In order to obtain the result we present a new analysis for QuickHeapsort splitting it into the analysis of the partition-phases and the analysis of the heap-phases. This enables us to consider samples of non-constant size for the pivot selection and leads to better theoretical bounds for the algorithm. Furthermore, we introduce some modifications of QuickHeapsort. We show that for every input the expected number of comparisons is at most \(n\log _{2}n - 0.03n + o(n)\) for the in-place variant. If we allow n extra bits, then we can lower the bound to \( n\log _{2} n -0.997 n+ o (n)\). Thus, spending n extra bits we can save more that 0.96n comparisons if n is large enough. Both estimates improve the previously known results. Moreover, our non-in-place variant does essentially use the same number of comparisons as index based Heapsort variants and Relaxed-Weak-Heapsort which use \( n\log _{2}n -0.9 n+ o (n)\) comparisons in the worst case. However, index based Heapsort variants and Relaxed-Weak-Heapsort require \({\Theta }(n\log n)\) extra bits whereas we need n bits only. Our theoretical results are upper bounds and valid for every input. Our computer experiments show that the gap between our bounds and the actual values on random inputs is small. Moreover, the computer experiments establish QuickHeapsort as competitive with Quicksort in terms of running time.


In-place sorting Heapsort Quicksort Analysis of algorithms 



We thank Martin Dietzfelbinger, Stefan Edelkamp, Jyrki Katajainen and the anonymous referees for various comments which improved the presentation of the paper. We also thank Simon Paridon for helping us with the implementation of the algorithms.

Conflict of interests

The authors declare that they have no conflict of interest.


  1. 1.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Cantone, D., Cincotti, G.: QuickHeapsort, an efficient mix of classical sorting algorithms. Theor. Comput. Sci. 285(1), 25–42 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Carlsson, S., Chen, J., Mattsson, C.: Heaps with Bits. In: D.-Z. Du, X.-S. Zhang (eds.) ISAAC, vol. 834 of LNCS, pp. 288–296. Springer (1994)Google Scholar
  4. 4.
    Chen, J.: A Framework for Constructing Heap-like structures in-place. In: K.-W. Ng, et al. (eds.) ISAAC, vol. 762 of LNCS, pp. 118–127. Springer (1993)Google Scholar
  5. 5.
    Chen, J., Edelkamp, S., Elmasry, A., Katajainen, J.: In-place Heap Construction with Optimized Comparisons, Moves, and Cache Misses. In: B. Rovan, V. Sassone, P. Widmayer (eds.) MFCS, vol. 7464 of LNCS, pp. 259–270. Springer (2012)Google Scholar
  6. 6.
    Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms. The MIT Press, 3rd edn. (2009)Google Scholar
  7. 7.
    Dutton, R.D.: Weak-heap sort. BIT 33(3), 372–381 (1993)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Edelkamp, S., Stiegeler, P.: Implementing HEAPSORT with \(n \lg n - 0.9n\) and QUICKSORT with \(n \lg n + 0.2n\) comparisons. ACM J. of Exp. Alg., 7:5 (2002)Google Scholar
  9. 9.
    Floyd, R.W.: Algorithm 245: Treesort. Commun. ACM 7(12), 701 (1964)CrossRefGoogle Scholar
  10. 10.
    Gonnet, G.H., Munro, J.I.: Heaps on Heaps. SIAM J. Comput. 15(4), 964–971 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Katajainen, J.: The ultimate heapsort. In: X. Lin (ed.) CATS, vol. 20 of Australian Computer Science Communications, pp. 87–96. Springer-Verlag (1998)Google Scholar
  12. 12.
    Knuth, D.E.: The art of computer programming. vol. 3 Addison-Wesley (1998)Google Scholar
  13. 13.
    Martínez, C., Roura, S.: Optimal sampling strategies in Quicksort and Quickselect. SIAM J. Comput. 31(3), 683–705 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    McDiarmid, C., Reed, B.A.: Building Heaps Fast. J. Alg. 10(3), 352–365 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Reinhardt, K.: Sorting in-place with a worst case complexity of \( n \log n -1.3 n + O(\log n)\) comparisons and \(\epsilon n \log n + O(1)\) transports. In: T. Ibaraki, et al. (eds.) ISAAC, vol. 650 of LNCS, pp. 489–498. Springer (1992)Google Scholar
  16. 16.
    Wang, X.-D., Wu, Y.-J.: An improved HEAPSORT Algorithm with \(n\lg n - 0.788928n\) comparisons in the Worst Case. J. Comput. Sci. Techn. 22, 898–903 (2007)CrossRefGoogle Scholar
  17. 17.
    Wegener, I.: The worst case complexity of McDiarmid and Reed’s variant of Bottom-Up-Heap sort is less than \(n \lg n + 1.1n\). In: C. Choffrut, M. Jantzen (eds.) STACS, vol. 480 of LNCS, pp. 137–147. Springer (1991)Google Scholar
  18. 18.
    Wegener, I.: BOTTOM-UP-HEAPSORT, a new variant of HEAPSORT, beating, on an average, QUICKSORT (if n is not very small). Theor. Comp. Sci. 118(1), 81–98 (1993)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.FMI, Universität StuttgartStuttgartGermany

Personalised recommendations