[{"department":[{"_id":"63"}],"user_id":"15415","_id":"18778","language":[{"iso":"eng"}],"publication":"Proceedings of the 12th European Symposium on Algorithms (ESA'04)","type":"conference","status":"public","abstract":[{"text":"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 &#916;. Our results are parametrized by the variance of the noise distribution. For large variance we essentially consider the average case for distribution &#916; 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.<br>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 &#963;² and the uniform distribution in a hypercube of side length &epsilon. For these noise distributions we show upper bounds of O( (1/ &#963; )^d * log^3/2 * d - 1 n ) and O( ( (n log n) / &#949; )^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.","lang":"eng"}],"date_created":"2020-09-01T12:26:47Z","author":[{"last_name":"Damerow","full_name":"Damerow, Valentina","first_name":"Valentina"},{"full_name":"Sohler, Christian","last_name":"Sohler","first_name":"Christian"}],"date_updated":"2022-01-06T06:53:52Z","doi":"10.1007/978-3-540-30140-0_25","title":"Extreme Points Under Random Noise","publication_identifier":{"issn":["0302-9743","1611-3349"],"isbn":["9783540230250","9783540301400"]},"publication_status":"published","citation":{"mla":"Damerow, Valentina, and Christian Sohler. “Extreme Points Under Random Noise.” <i>Proceedings of the 12th European Symposium on Algorithms (ESA’04)</i>, 2004, doi:<a href=\"https://doi.org/10.1007/978-3-540-30140-0_25\">10.1007/978-3-540-30140-0_25</a>.","short":"V. Damerow, C. Sohler, in: Proceedings of the 12th European Symposium on Algorithms (ESA’04), Berlin, Heidelberg, 2004.","bibtex":"@inproceedings{Damerow_Sohler_2004, place={Berlin, Heidelberg}, title={Extreme Points Under Random Noise}, DOI={<a href=\"https://doi.org/10.1007/978-3-540-30140-0_25\">10.1007/978-3-540-30140-0_25</a>}, booktitle={Proceedings of the 12th European Symposium on Algorithms (ESA’04)}, author={Damerow, Valentina and Sohler, Christian}, year={2004} }","apa":"Damerow, V., &#38; Sohler, C. (2004). Extreme Points Under Random Noise. In <i>Proceedings of the 12th European Symposium on Algorithms (ESA’04)</i>. Berlin, Heidelberg. <a href=\"https://doi.org/10.1007/978-3-540-30140-0_25\">https://doi.org/10.1007/978-3-540-30140-0_25</a>","chicago":"Damerow, Valentina, and Christian Sohler. “Extreme Points Under Random Noise.” In <i>Proceedings of the 12th European Symposium on Algorithms (ESA’04)</i>. Berlin, Heidelberg, 2004. <a href=\"https://doi.org/10.1007/978-3-540-30140-0_25\">https://doi.org/10.1007/978-3-540-30140-0_25</a>.","ieee":"V. Damerow and C. Sohler, “Extreme Points Under Random Noise,” in <i>Proceedings of the 12th European Symposium on Algorithms (ESA’04)</i>, 2004.","ama":"Damerow V, Sohler C. Extreme Points Under Random Noise. In: <i>Proceedings of the 12th European Symposium on Algorithms (ESA’04)</i>. Berlin, Heidelberg; 2004. doi:<a href=\"https://doi.org/10.1007/978-3-540-30140-0_25\">10.1007/978-3-540-30140-0_25</a>"},"year":"2004","place":"Berlin, Heidelberg"},{"title":"Fast Multipoint Evaluation of Bivariate Polynomials","date_created":"2020-08-25T11:25:11Z","publisher":"Springer","year":"2004","language":[{"iso":"eng"}],"abstract":[{"text":"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.","lang":"eng"}],"publication":"Proc. 12th Annual Symposium on Algorithms (ESA'04)","doi":"10.1007/978-3-540-30140-0_49","volume":3221,"author":[{"full_name":"Nüsken, Michael","last_name":"Nüsken","first_name":"Michael"},{"first_name":"Martin","last_name":"Ziegler","full_name":"Ziegler, Martin"}],"date_updated":"2022-01-06T06:53:28Z","page":"544-555","intvolume":"      3221","citation":{"chicago":"Nüsken, Michael, and Martin Ziegler. “Fast Multipoint Evaluation of Bivariate Polynomials.” In <i>Proc. 12th Annual Symposium on Algorithms (ESA’04)</i>, 3221:544–55. Berlin, Heidelberg: Springer, 2004. <a href=\"https://doi.org/10.1007/978-3-540-30140-0_49\">https://doi.org/10.1007/978-3-540-30140-0_49</a>.","ieee":"M. Nüsken and M. Ziegler, “Fast Multipoint Evaluation of Bivariate Polynomials,” in <i>Proc. 12th Annual Symposium on Algorithms (ESA’04)</i>, 2004, vol. 3221, pp. 544–555.","ama":"Nüsken M, Ziegler M. Fast Multipoint Evaluation of Bivariate Polynomials. In: <i>Proc. 12th Annual Symposium on Algorithms (ESA’04)</i>. Vol 3221. Berlin, Heidelberg: Springer; 2004:544-555. doi:<a href=\"https://doi.org/10.1007/978-3-540-30140-0_49\">10.1007/978-3-540-30140-0_49</a>","apa":"Nüsken, M., &#38; Ziegler, M. (2004). Fast Multipoint Evaluation of Bivariate Polynomials. In <i>Proc. 12th Annual Symposium on Algorithms (ESA’04)</i> (Vol. 3221, pp. 544–555). Berlin, Heidelberg: Springer. <a href=\"https://doi.org/10.1007/978-3-540-30140-0_49\">https://doi.org/10.1007/978-3-540-30140-0_49</a>","mla":"Nüsken, Michael, and Martin Ziegler. “Fast Multipoint Evaluation of Bivariate Polynomials.” <i>Proc. 12th Annual Symposium on Algorithms (ESA’04)</i>, vol. 3221, Springer, 2004, pp. 544–55, doi:<a href=\"https://doi.org/10.1007/978-3-540-30140-0_49\">10.1007/978-3-540-30140-0_49</a>.","bibtex":"@inproceedings{Nüsken_Ziegler_2004, place={Berlin, Heidelberg}, title={Fast Multipoint Evaluation of Bivariate Polynomials}, volume={3221}, DOI={<a href=\"https://doi.org/10.1007/978-3-540-30140-0_49\">10.1007/978-3-540-30140-0_49</a>}, booktitle={Proc. 12th Annual Symposium on Algorithms (ESA’04)}, publisher={Springer}, author={Nüsken, Michael and Ziegler, Martin}, year={2004}, pages={544–555} }","short":"M. Nüsken, M. Ziegler, in: Proc. 12th Annual Symposium on Algorithms (ESA’04), Springer, Berlin, Heidelberg, 2004, pp. 544–555."},"place":"Berlin, Heidelberg","publication_identifier":{"issn":["0302-9743","1611-3349"],"isbn":["9783540230250","9783540301400"]},"publication_status":"published","department":[{"_id":"63"}],"user_id":"15415","_id":"18263","status":"public","type":"conference"},{"publication_status":"published","publication_identifier":{"issn":["0302-9743","1611-3349"],"isbn":["9783540230250","9783540301400"]},"citation":{"apa":"Bansal, V., Meyer auf der Heide, F., &#38; Sohler, C. (2004). Labeling Smart Dust. In <i>12th Annual European Symposium on Algorithms (ESA 2004)</i> (Vol. 3221). Berlin, Heidelberg. <a href=\"https://doi.org/10.1007/978-3-540-30140-0_9\">https://doi.org/10.1007/978-3-540-30140-0_9</a>","mla":"Bansal, Vikas, et al. “Labeling Smart Dust.” <i>12th Annual European Symposium on Algorithms (ESA 2004)</i>, vol. 3221, 2004, doi:<a href=\"https://doi.org/10.1007/978-3-540-30140-0_9\">10.1007/978-3-540-30140-0_9</a>.","bibtex":"@inproceedings{Bansal_Meyer auf der Heide_Sohler_2004, place={Berlin, Heidelberg}, series={Lecture Notes in Computer Science}, title={Labeling Smart Dust}, volume={3221}, DOI={<a href=\"https://doi.org/10.1007/978-3-540-30140-0_9\">10.1007/978-3-540-30140-0_9</a>}, booktitle={12th Annual European Symposium on Algorithms (ESA 2004)}, author={Bansal, Vikas and Meyer auf der Heide, Friedhelm and Sohler, Christian}, year={2004}, collection={Lecture Notes in Computer Science} }","short":"V. Bansal, F. Meyer auf der Heide, C. Sohler, in: 12th Annual European Symposium on Algorithms (ESA 2004), Berlin, Heidelberg, 2004.","chicago":"Bansal, Vikas, Friedhelm Meyer auf der Heide, and Christian Sohler. “Labeling Smart Dust.” In <i>12th Annual European Symposium on Algorithms (ESA 2004)</i>, Vol. 3221. Lecture Notes in Computer Science. Berlin, Heidelberg, 2004. <a href=\"https://doi.org/10.1007/978-3-540-30140-0_9\">https://doi.org/10.1007/978-3-540-30140-0_9</a>.","ieee":"V. Bansal, F. Meyer auf der Heide, and C. Sohler, “Labeling Smart Dust,” in <i>12th Annual European Symposium on Algorithms (ESA 2004)</i>, 2004, vol. 3221.","ama":"Bansal V, Meyer auf der Heide F, Sohler C. Labeling Smart Dust. In: <i>12th Annual European Symposium on Algorithms (ESA 2004)</i>. Vol 3221. Lecture Notes in Computer Science. Berlin, Heidelberg; 2004. doi:<a href=\"https://doi.org/10.1007/978-3-540-30140-0_9\">10.1007/978-3-540-30140-0_9</a>"},"intvolume":"      3221","place":"Berlin, Heidelberg","year":"2004","date_created":"2020-04-08T11:07:57Z","author":[{"first_name":"Vikas","last_name":"Bansal","full_name":"Bansal, Vikas"},{"first_name":"Friedhelm","full_name":"Meyer auf der Heide, Friedhelm","id":"15523","last_name":"Meyer auf der Heide"},{"first_name":"Christian","last_name":"Sohler","full_name":"Sohler, Christian"}],"volume":3221,"date_updated":"2022-01-06T06:52:50Z","doi":"10.1007/978-3-540-30140-0_9","title":"Labeling Smart Dust","type":"conference","publication":"12th Annual European Symposium on Algorithms (ESA 2004)","status":"public","abstract":[{"lang":"eng","text":"Given n distinct points p1, p2, ... , pn in the plane, the map labeling\r\nproblem with four squares is to place n axis-parallel equi-sized squares Q1, ... ,Qn\r\nof maximum possible size such that pi is a corner of Qi and no two squares overlap.\r\nThis problem is NP-hard and no algorithm with approximation ratio better\r\nthan 1/2 exists unless P = NP [10].\r\nIn this paper, we consider a scenario where we want to visualize the information\r\ngathered by smart dust, i.e. by a large set of simple devices, each consisting of\r\na sensor and a sender that can gather sensor data and send it to a central station.\r\nOur task is to label (the positions of) these sensors in a way described by the\r\nlabeling problem above. Since these devices are not positioned accurately (for\r\nexample, they might be dropped from an airplane), this gives rise to consider the\r\nmap labeling problem under the assumption, that the positions of the points are\r\nnot fixed precisely, but perturbed by random noise. In other words, we consider\r\nthe smoothed complexity of the map labeling problem. We present an algorithm\r\nthat, under such an assumption and Gaussian random noise with sufficiently large\r\nvariance, has linear smoothed complexity."}],"user_id":"15415","series_title":"Lecture Notes in Computer Science","department":[{"_id":"63"}],"_id":"16474","language":[{"iso":"eng"}]}]