The height of a random binary search tree
Journal of the ACM (JACM)
An analytic approach to the height of binary search trees II
Journal of the ACM (JACM)
Smoothed analysis of algorithms: Why the simplex algorithm usually takes polynomial time
Journal of the ACM (JACM)
The smoothed analysis of algorithms
FCT'05 Proceedings of the 15th international conference on Fundamentals of Computation Theory
Hi-index | 0.01 |
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.