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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.
    Full-text · Article · Jun 2007 · Theoretical Computer Science
  • Conference Paper: Labeling Smart Dust
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    ABSTRACT: Given n distinct points p 1,p 2, ..., p n in the plane, the map labeling problem with four squares is to place n axis-parallel equi-sized squares Q 1, ..., Q n of maximum possible size such that p i is a corner of Q i and no two squares overlap. This problem is NP-hard and no algorithm with approximation ratio better than \frac12\frac{1}{2} exists unless P=NP [10]. In this paper, we consider a scenario where we want to visualize the information gathered by smart dust, i.e. by a large set of simple devices, each consisting of a sensor and a sender that can gather sensor data and send it to a central station. Our task is to label (the positions of) these sensors in a way described by the labeling problem above. Since these devices are not positioned accurately (for example, they might be dropped from an airplane), this gives rise to consider the map labeling problem under the assumption, that the positions of the points are not fixed precisely, but perturbed by random noise. In other words, we consider the smoothed complexity of the map labeling problem. We present an algorithm that, under such an assumption and Gaussian random noise with sufficiently large variance, has linear smoothed complexity.
    No preview · Conference Paper · Sep 2004
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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.
    Preview · Chapter · Mar 2005
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