Smoothed analysis of left-to-right maxima with applications

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Abstract

A left-to-right maximum in a sequence of n numbers s(1), ..., s(n) is a number that is strictly larger than all preceding numbers. In this article we present a smoothed analysis of the number of left-to-right maxima in the presence of additive random noise. We show that for every sequence of n numbers s(i) is an element of[0, 1] that are perturbed by uniform noise from the interval [-epsilon, epsilon], the expected number of left-to-right maxima is Theta(root n/epsilon + log n) for epsilon > 1/n. For Gaussian noise with standard deviation sigma we obtain a bound of O((log(3/2) n)/sigma + log n).

We apply our results to the analysis of the smoothed height of binary search trees and the smoothed number of comparisons in the quicksort algorithm and prove bounds of Theta(root n/epsilon + log n) and Theta(n/epsilon+1 root n/epsilon + n log n), respectively, for uniform random noise from the interval [-epsilon, epsilon]. Our results can also be applied to bound the smoothed number of points on a convex hull of points in the two-dimensional plane and to smoothed motion complexity, a concept we describe in this article. We bound how often one needs to update a data structure storing the smallest axis-aligned box enclosing a set of points moving in d-dimensional space.

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