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Smoothed Analysis of Binary Search Trees and Quicksort under Additive Noise

  • Bodo Manthey
  • Till Tantau
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5162)

Abstract

Binary search trees are a fundamental data structure and their height plays a key role in the analysis of divide-and-conquer algorithms like quicksort. We analyze their smoothed height under additive uniform noise: An adversary chooses a sequence of n real numbers in the range [0,1], each number is individually perturbed by adding a value drawn uniformly at random from an interval of size d, and the resulting numbers are inserted into a search tree. An analysis of the smoothed tree height subject to n and d lies at the heart of our paper: We prove that the smoothed height of binary search trees is \(\Theta (\sqrt{n/d} + \log n)\), where d ≥ 1/n may depend on n. Our analysis starts with the simpler problem of determining the smoothed number of left-to-right maxima in a sequence. We establish matching bounds, namely once more \(\Theta (\sqrt{n/d} + \log n)\). We also apply our findings to the performance of the quicksort algorithm and prove that the smoothed number of comparisons made by quicksort is \(\Theta(\frac{n}{d+1} \sqrt{n/d} + n \log n)\).

Keywords

Tree Height Additive Noise Search Tree Binary Search Tree Perturbation Model 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Bodo Manthey
    • 1
  • Till Tantau
    • 2
  1. 1.Computer ScienceSaarland UniversitySaarbrückenGermany
  2. 2.Institut für Theoretische InformatikUniversität zu LübeckLübeckGermany

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