QuickHeapsort: Modifications and Improved Analysis

  • Volker Diekert
  • Armin Weiß
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7913)

Abstract

We present a new analysis for QuickHeapsort. 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, both in-place and using n extra bits. We show that on every input the expected number of comparisons is \(n\lg n - 0.03n + o(n)\) (in-place) respectively \( n\lg n -0.997 n+ o (n)\) (always \(\lg n= \log_2 n\)). Both estimates improve the previously known best results. (It is conjectured [17] that the in-place algorithm Bottom-Up-Heapsort uses at most \(n\lg n + 0.4 n\) on average and for Weak-Heapsort which uses n extra bits the average number of comparisons is at most \(n\lg n -0.42n\) [8].) Moreover, our non-in-place variant can even compete with index based Heapsort variants (e.g. Rank-Heapsort [15]) and Relaxed-Weak-Heapsort (\( n\lg n -0.9 n+ o (n)\) comparisons in the worst case) for which no \(\mathcal{O}(n)\)-bound on the number of extra bits is known.

Keywords

in-place sorting heapsort quicksort analysis of algorithms 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Volker Diekert
    • 1
  • Armin Weiß
    • 1
  1. 1.FMIUniversität StuttgartStuttgartGermany

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