Randomized algorithms
Proximity problems on moving points
SCG '97 Proceedings of the thirteenth annual symposium on Computational geometry
Probabilistic analysis for combinatorial functions of moving points
SCG '97 Proceedings of the thirteenth annual symposium on Computational geometry
The art of computer programming, volume 3: (2nd ed.) sorting and searching
The art of computer programming, volume 3: (2nd ed.) sorting and searching
Data structures for mobile data
Journal of Algorithms
Maintaining approximate extent measures of moving points
SODA '01 Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms
Simplified kinetic connectivity for rectangles and hypercubes
SODA '01 Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms
Fundamentals of the Average Case Analysis of Particular Algorithms
Fundamentals of the Average Case Analysis of Particular Algorithms
Journal of Algorithms - Analysis of algorithms
The height of a random binary search tree
Journal of the ACM (JACM)
An analytic approach to the height of binary search trees II
Journal of the ACM (JACM)
Smoothed analysis of algorithms: Why the simplex algorithm usually takes polynomial time
Journal of the ACM (JACM)
Discrete & Computational Geometry
Smoothed analysis of binary search trees
Theoretical Computer Science
Smoothed Analysis of Binary Search Trees and Quicksort under Additive Noise
MFCS '08 Proceedings of the 33rd international symposium on Mathematical Foundations of Computer Science
MFCS'12 Proceedings of the 37th international conference on Mathematical Foundations of Computer Science
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A left-to-right maximum in a sequence of n numbers s1, …, sn 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 si ∈ [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 ε1/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.