Smoothed Analysis of Binary Search Trees

  • Bodo Manthey
  • Rüdiger Reischuk
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3827)


Binary search trees are one of the most fundamental data structures. While the height of such a tree may be linear in the worst case, the average height with respect to the uniform distribution is only logarithmic. The exact value is one of the best studied problems in average-case complexity.

We investigate what happens in between by analysing the smoothed height of binary search trees: Randomly perturb a given (adversarial) sequence and then take the expected height of the binary search tree generated by the resulting sequence. As perturbation models, we consider partial permutations, partial alterations, and partial deletions.

On the one hand, we prove tight lower and upper bounds of roughly \({\it \Theta}(\sqrt{n})\) for the expected height of binary search trees under partial permutations and partial alterations. This means that worst-case instances are rare and disappear under slight perturbations. On the other hand, we examine how much a perturbation can increase the height of a binary search tree, i.e. how much worse well balanced instances can become.


Tight Bound Partial Deletion Binary Search Tree Perturbation Model Marked Element 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Bodo Manthey
    • 1
  • Rüdiger Reischuk
    • 1
  1. 1.Institut für Theoretische InformatikUniversität zu LübeckLübeckGermany

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