Smoothed Analysis of Binary Search Trees and Quicksort under Additive Noise

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 $\Theta(\frac{n}{d+1} \sqrt{n/d} + n \log n)$.