Smoothed Analysis of Binary Search Trees and Quicksort under Additive Noise

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. Their worst-case height is linear; their average height, whose exact value is one of the best-studied problems in average-case complexity, is logarithmic. We analyze their smoothed height under additive noise: An adversary chooses a sequence of n real numbers in the range [0, 1]; each number is individually perturbed by adding a random value from an interval of size d; and the resulting numbers are inserted into a search tree. The expected height of this tree is called smoothed tree height. If d is very small, namely for d ≤ 1/n, the smoothed tree height is the same as the worst-case height; if d is very large, the smoothed tree height approaches the logarithmic average-case height. An analysis of what happens between these extremes lies at the heart of our paper: We prove that the smoothed height of binary search trees is Θ( √ n/d + logn), where d ≥ 1/n may depend on n. This implies that the logarithmic average-case height becomes manifest only for d ∈ Ω(n/ log n). For the analysis, we first prove that the smoothed number of left-to-right maxima in a sequence is also Θ( √ n/d+logn). We apply these findings to the performance of the quicksort algorithm, which needs Θ(n) comparisons in the worst case and Θ(n logn) on average, and prove that the smoothed number of comparisons made by quicksort is Θ ( n d+1 √