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Available from: Friedhelm Meyer auf der Heide, Mar 15, 2014
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    • "This means that the running time of the algorithm is expected to be polynomial in terms of the input size and the variance of the Gaussian perturbation. Since then, smoothed analysis has been applied to a variety of fields [30], for instance several variants of linear programming [8] [12] [32], online and other algorithms [5] [18] [28], discrete optimisation [6] [26], and other topics [4] [10] [11] [31]. "
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    ABSTRACT: Binary search trees are one of the most fundamental data structures. While the height of such a tree may be linear in the worst case, the average height with respect to the uniform distribution is only logarithmic. The exact value is one of the best studied problems in average-case complexity.We investigate what happens in between by analysing the smoothed height of binary search trees: Randomly perturb a given (adversarial) sequence and then take the expected height of the binary search tree generated by the resulting sequence. As perturbation models, we consider partial permutations, partial alterations, and partial deletions.On the one hand, we prove tight lower and upper bounds of roughly for the expected height of binary search trees under partial permutations and partial alterations, where n is the number of elements and p is the smoothing parameter. This means that worst-case instances are rare and disappear under slight perturbations. On the other hand, we examine how much a perturbation can increase the height of a binary search tree, i.e. how much worse well balanced instances can become.
    Theoretical Computer Science 06/2007; 378(3-378):292-315. DOI:10.1016/j.tcs.2007.02.035 · 0.52 Impact Factor
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    ABSTRACT: Binary search trees are one of the most fundamental data structures. While the height of such a tree may be linear in the worst case, the average height with respect to the uniform distribution is only logarithmic. The exact value is one of the best studied problems in average-case complexity. We investigate what happens in between by analysing the smoothed height of binary search trees: Randomly perturb a given (adversarial) sequence and then take the expected height of the binary search tree generated by the resulting sequence. As perturbation models, we consider partial permutations, partial alterations, and partial deletions. On the one hand, we prove tight lower and upper bounds of roughly Q(Ön){\it \Theta}(\sqrt{n}) for the expected height of binary search trees under partial permutations and partial alterations. This means that worst-case instances are rare and disappear under slight perturbations. On the other hand, we examine how much a perturbation can increase the height of a binary search tree, i.e. how much worse well balanced instances can become.
    03/2005: pages 483-492;
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    ABSTRACT: Many algorithms and heuristics work well on real data, de- spite having poor complexity under the standard worst-case measure. Smoothed analysis (36) is a step towards a the- ory that explains the behavior of algorithms in practice. It is based on the assumption that inputs to algorithms are subject to random perturbation and modification in their formation. A concrete example of such a smoothed analysis is a proof that the simplex algorithm for linear programming usually runs in polynomial time, when its input is subject to modeling or measurement noise.
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