@inproceedings{18778, abstract = {{Given a point set P in the d-dimensional unit hypercube, we give upper bounds on the maximal expected number of extreme points when each point is perturbed by small random noise chosen independently for each point from the same noise distribution Δ. Our results are parametrized by the variance of the noise distribution. For large variance we essentially consider the average case for distribution Δ while for variance 0 we consider the worst case. Hence our results give upper bounds on the number of extreme points where our input distributions range from average case to worst case.
Our main contribution is a rather general lemma that can be used to obtain upper bounds on the expected number of extreme points for a large class of noise distributions. We then apply this lemma to obtain explicit bounds for random noise coming from the Gaussian normal distribution of variance σ² and the uniform distribution in a hypercube of side length &epsilon. For these noise distributions we show upper bounds of O( (1/ σ )^d * log^3/2 * d - 1 n ) and O( ( (n log n) / ε )^d/(d+1) ), respectively. Besides its theoretical motivation our model is also motivated by the observation that in many applications of convex hull algorithms the input data is inherently noisy, e.g. when the data comes from physical measurement or imprecise arithmetic is used.}}, author = {{Damerow, Valentina and Sohler, Christian}}, booktitle = {{Proceedings of the 12th European Symposium on Algorithms (ESA'04)}}, isbn = {{9783540230250}}, issn = {{0302-9743}}, title = {{{Extreme Points Under Random Noise}}}, doi = {{10.1007/978-3-540-30140-0_25}}, year = {{2004}}, } @inproceedings{18263, abstract = {{We generalize univariate multipoint evaluation of polynomials of degree n at sublinear amortized cost per point. More precisely, it is shown how to evaluate a bivariate polynomial p of maximum degree less than n, specified by its n^2 coefficients, simultaneously at n^2 given points using a total of O(n^2.667) arithmetic operations. In terms of the input size N being quadratic in n, this amounts to an amortized cost of O(N^0.334) per point.}}, author = {{Nüsken, Michael and Ziegler, Martin}}, booktitle = {{Proc. 12th Annual Symposium on Algorithms (ESA'04)}}, isbn = {{9783540230250}}, issn = {{0302-9743}}, pages = {{544--555}}, publisher = {{Springer}}, title = {{{Fast Multipoint Evaluation of Bivariate Polynomials}}}, doi = {{10.1007/978-3-540-30140-0_49}}, volume = {{3221}}, year = {{2004}}, } @inproceedings{16474, abstract = {{Given n distinct points p1, p2, ... , pn in the plane, the map labeling problem with four squares is to place n axis-parallel equi-sized squares Q1, ... ,Qn of maximum possible size such that pi is a corner of Qi and no two squares overlap. This problem is NP-hard and no algorithm with approximation ratio better than 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.}}, author = {{Bansal, Vikas and Meyer auf der Heide, Friedhelm and Sohler, Christian}}, booktitle = {{12th Annual European Symposium on Algorithms (ESA 2004)}}, isbn = {{9783540230250}}, issn = {{0302-9743}}, title = {{{Labeling Smart Dust}}}, doi = {{10.1007/978-3-540-30140-0_9}}, volume = {{3221}}, year = {{2004}}, }