Smoothed analysis of left-to-right maxima with applications

V. Damerow, B. Manthey, F. Meyer auf der Heide, H. Räcke, C. Scheideler, C. Sohler, T. Tantau, Transactions on Algorithms (2012) 30.

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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 ∈ [0,1] that are perturbed by uniform noise from the interval [-ε,ε], the expected number of left-to-right maxima is Θ(&sqrt;n/ε + log n) for ε>1/n. For Gaussian noise with standard deviation σ we obtain a bound of O((log3/2 n)/σ + 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 Θ(&sqrt;n/ε + log n) and Θ(n/ε+1&sqrt;n/ε + n log n), respectively, for uniform random noise from the interval [-ε,ε]. 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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Transactions on Algorithms
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3
Page
30
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Damerow V, Manthey B, Meyer auf der Heide F, et al. Smoothed analysis of left-to-right maxima with applications. Transactions on Algorithms. 2012;(3):30. doi:10.1145/2229163.2229174.
Damerow, V., Manthey, B., Meyer auf der Heide, F., Räcke, H., Scheideler, C., Sohler, C., & Tantau, T. (2012). Smoothed analysis of left-to-right maxima with applications. Transactions on Algorithms, (3), 30. http://doi.org/10.1145/2229163.2229174
@article{Damerow_Manthey_Meyer auf der Heide_Räcke_Scheideler_Sohler_Tantau_2012, title={Smoothed analysis of left-to-right maxima with applications}, DOI={10.1145/2229163.2229174}, number={3}, journal={Transactions on Algorithms}, author={Damerow, Valentina and Manthey, Bodo and Meyer auf der Heide, Friedhelm and Räcke, Harald and Scheideler, Christian and Sohler, Christian and Tantau, Till}, year={2012}, pages={30}}
Damerow, Valentina, Bodo Manthey, Friedhelm Meyer auf der Heide, Harald Räcke, Christian Scheideler, Christian Sohler, and Till Tantau. “Smoothed Analysis of Left-to-Right Maxima with Applications.” Transactions on Algorithms, no. 3 (2012): 30. doi:10.1145/2229163.2229174.
V. Damerow, B. Manthey, F. Meyer auf der Heide, H. Räcke, C. Scheideler, C. Sohler, and T. Tantau, “Smoothed analysis of left-to-right maxima with applications,” Transactions on Algorithms, no. 3, p. 30, 2012.
Damerow, Valentina et al. “Smoothed Analysis of Left-to-Right Maxima with Applications.” Transactions on Algorithms 3 (2012): 30. Web.
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