Smoothed analysis of left-to-right maxima with applications
Damerow, Valentina
Manthey, Bodo
Meyer auf der Heide, Friedhelm
Räcke, Harald
Scheideler, Christian
Sohler, Christian
Tantau, Till
ddc:040
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.
ACM
2012
info:eu-repo/semantics/article
doc-type:article
text
http://purl.org/coar/resource_type/c_6501
https://ris.uni-paderborn.de/record/579
Damerow V, Manthey B, Meyer auf der Heide F, et al. Smoothed analysis of left-to-right maxima with applications. <i>Transactions on Algorithms</i>. 2012;(3):30. doi:<a href="https://doi.org/10.1145/2229163.2229174">10.1145/2229163.2229174</a>
info:eu-repo/semantics/altIdentifier/doi/10.1145/2229163.2229174
info:eu-repo/semantics/closedAccess