35 research outputs found

    On the Smoothed Complexity of Convex Hulls

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    We establish an upper bound on the smoothed complexity of convex hulls in R^d under uniform Euclidean (L^2) noise. Specifically, let {p_1^*, p_2^*, ..., p_n^*} be an arbitrary set of n points in the unit ball in R^d and let p_i = p_i^* + x_i, where x_1, x_2, ..., x_n are chosen independently from the unit ball of radius r. We show that the expected complexity, measured as the number of faces of all dimensions, of the convex hull of {p_1, p_2, ..., p_n} is O(n^{2-4/(d+1)} (1+1/r)^{d-1}); the magnitude r of the noise may vary with n. For d=2 this bound improves to O(n^{2/3} (1+r^{-2/3})). We also analyze the expected complexity of the convex hull of L^2 and Gaussian perturbations of a nice sample of a sphere, giving a lower-bound for the smoothed complexity. We identify the different regimes in terms of the scale, as a function of n, and show that as the magnitude of the noise increases, that complexity varies monotonically for Gaussian noise but non-monotonically for L^2 noise

    Analyse de complexité d'enveloppes convexes aléatoires

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    In this thesis, we give some new results about the average size of convex hulls made of points chosen in a convex body. This size is known when the points are chosen uniformly (and independently) in a convex polytope or in a "smooth" enough convex body. This average size is also known if the points are independently chosen according to a centered Gaussian distribution. In the first part of this thesis, we introduce a technique that will give new results when the points are chosen arbitrarily in a convex body, and then noised by some random perturbations. This kind of analysis, called smoothed analysis, has been initially developed by Spielman and Teng in their study of the simplex algorithm. For an arbitrary set of point in a ball, we obtain a lower and a upper bound for this smoothed complexity, in the case of uniform perturbation in a ball (in arbitrary dimension) and in the case of Gaussian perturbations in dimension 2. The asymptotic behavior of the expected size of the convex hull of uniformly random points in a convex body is polynomial for a "smooth" body and polylogarithmic for a polytope. In the second part, we construct a convex body so that the expected size of the convex hull of points uniformly chosen in that body oscillates between these two behaviors when the number of points increases. In the last part, we present an algorithm to generate efficiently a random convex hull made of points chosen uniformly and independently in a disk. We also compute its average time and space complexity. This algorithm can generate a random convex hull without explicitly generating all the points. It has been implemented in C++ and integrated in the CGAL library.Dans cette thĂšse, nous donnons de nouveaux rĂ©sultats sur la taille moyenne d’enveloppes convexes de points choisis dans un convexe. Cette taille est connue lorsque les points sont choisis uniformĂ©ment (et indĂ©pendamment) dans un polytope convexe, ou un convexe suffisamment «lisse» ; ou encore lorsque les points sont choisis indĂ©pendamment selon une loi normale centrĂ©e. Dans la premiĂšre partie de cette thĂšse, nous dĂ©veloppons une technique nous permettant de donner de nouveaux rĂ©sultats lorsque les points sont choisis arbitrairement dans un convexe, puis «bruitĂ©s» par une perturbation alĂ©atoire. Ce type d’analyse, appelĂ©e analyse lissĂ©e, a initialement Ă©tĂ© dĂ©veloppĂ©e par Spielman et Teng dans leur Ă©tude de l’algorithme du simplexe. Pour un ensemble de points arbitraires dans une boule, nous obtenons une borne infĂ©rieure et une borne supĂ©rieure de cette complexitĂ© lissĂ©e dans le cas de perturbations uniformes dans une boule en dimension arbitraire, ainsi que dans le cas de perturbations gaussiennes en dimension 2. La taille de l'enveloppe convexe de points choisis uniformĂ©ment dans un convexe, peut avoir un comportement logarithmique si ce convexe est un polytope ou polynomial s’il est lisse. Nous construisons un convexe produisant un comportement oscillant entre ces deux extrĂȘmes. Dans la derniĂšre partie, nous prĂ©sentons un algorithme pour engendrer efficacement une enveloppe convexe alĂ©atoire de points choisis uniformĂ©ment et indĂ©pendamment dans un disque sans avoir Ă  engendrer explicitement tous les points. Il a Ă©tĂ© implĂ©mentĂ© en C++ et intĂ©grĂ© dans la bibliothĂšque CGAL

    Complexity analysis of random geometric structures made simpler

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    Average-case analysis of data-structures or algorithms is commonly used in compu- tational geometry when the, more classical, worst-case analysis is deemed overly pessimistic. Since these analyses are often intricate, the models of random geometric data that can be handled are often simplistic and far from "realistic inputs". We present a new simple scheme for the analy- sis of geometric structures. While this scheme only produces results up to a polylog factor, it is much simpler to apply than the classical techniques and therefore succeeds in analyzing new input distributions related to smoothed complexity analysis. Abstract: We illustrate our method on two classical structures: convex hulls and Delaunay triangulations. Specifically, we give short and elementary proofs of the classical results that n points uniformly distributed in a ball in Rd have a convex hull and a Delaunay triangulation of respective expected complexities Θ~(n^((d+1)/(d-1)) ) and Θ~(n). We then prove that if we start with n points well-spread on a sphere, e.g. an (Δ,Îș)-sample of that sphere, and perturb that sample by moving each point ran- domly and uniformly within distance at most ÎŽ of its initial position, then the expected complexity of the convex hull of the resulting point set is Θ~( sqrt(n)^(1−1/d) ÎŽ^(-(d-1/d)/4)). .L'analyse en moyenne de structure de donnĂ©es et d'algorithmes gĂ©omĂ©triques est frĂ©quemment utilisĂ©e en gĂ©omĂ©trie algorithmique, un domaine ou' l'analyse dans le cas le pire est souvent trĂšs pessimiste. La difficultĂ© de ce type d'analyse fait que les modĂšles probabilistes utilisĂ©s restent simples et relativement Ă©loignĂ©es de donnĂ©es rĂ©alistes. Nous prĂ©sentons une nouvelle approche pour l'analyse des structures gĂ©omĂ©triques. Nos rĂ©sultats sont seulement 'a des facteurs logarithmiques prĂšs, mais notre mĂ©thode est plus simple que les classiques du domaine et nous rĂ©ussissons 'a analyser de nouveau type de distribution liĂ©e Ă  la smooth analysis. Nous illustrons notre mĂ©thode sur deux structures classiques: l'enveloppe convexe et la triangulation de Delaunay. Plus prĂ©cisĂ©ment, nous dĂ©montrons simplement le fait, classique, que n points uniformĂ©ment distribuĂ©s dans une boule de Rd ont une enveloppe convexe et une triangulation de Delaunay dont l'espĂ©rance de la taille est respectivement Θ~(n^((d+1)/(d-1)) ) et Θ~(n). Nous dĂ©montrons ensuite que si on se donne ensemble de n points bien distribu ́es sur une sphĂšre, par exemple un (Δ, Îș)-Ă©chantillon de la sphĂšre, et qu'on le perturbe ensuite en dĂ©plaçant chaque point uniformĂ©ment d'une distance ÎŽ Ă  partir de sa position initiale, alors l'espĂ©rance de la taille de l'enveloppe convexe de ces points est Θ~( sqrt(n)^(1−1/d) ÎŽ^(-(d-1/d)/4)).

    The Expected Number of Maximal Points of the Convolution of Two 2-D Distributions

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    The Maximal points in a set S are those that are not dominated by any other point in S. Such points arise in multiple application settings and are called by a variety of different names, e.g., maxima, Pareto optimums, skylines. Their ubiquity has inspired a large literature on the expected number of maxima in a set S of n points chosen IID from some distribution. Most such results assume that the underlying distribution is uniform over some spatial region and strongly use this uniformity in their analysis. This research was initially motivated by the question of how this expected number changes if the input distribution is perturbed by random noise. More specifically, let B_p denote the uniform distribution from the 2-dimensional unit ball in the metric L_p. Let delta B_q denote the 2-dimensional L_q-ball, of radius delta and B_p + delta B_q be the convolution of the two distributions, i.e., a point v in B_p is reported with an error chosen from delta B_q. The question is how the expected number of maxima change as a function of delta. Although the original motivation is for small delta, the problem is well defined for any delta and our analysis treats the general case. More specifically, we study, as a function of n,delta, the expected number of maximal points when the n points in S are chosen IID from distributions of the type B_p + delta B_q where p,q in {1,2,infty} for delta > 0 and also of the type B_infty + delta B_q where q in [1,infty) for delta > 0. For fixed p,q we show that this function changes "smoothly" as a function of delta but that this smooth behavior sometimes transitions unexpectedly between different growth behaviors

    Smoothed analysis of the simplex method

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    In this chapter, we give a technical overview of smoothed analyses of the shadow vertex simplex method for linear programming (LP). We first review the properties of the shadow vertex simplex method and its associated geometry. We begin the smoothed analysis discussion with an analysis of the successive shortest path algorithm for the minimum-cost maximum-flow problem under objective perturbations, a classical instantiation of the shadow vertex simplex method. Then we move to general linear programming and give an analysis of a shadow vertex based algorithm for linear programming under Gaussian constraint perturbations

    La triangulation de Delaunay d'un échantillon aléatoire d'un bon échantillon a une taille linéaire

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    A good sample is a point set such that any ball of radius Ï”\epsilon contains a constant number of points. The Delaunay triangulation of a good sample is proved to have linear size, unfortunately this is not enough to ensure a good time complexity of the randomized incremental construction of the Delaunay triangulation. In this paper we prove that a random Bernoulli sample of a good sample has a triangulation of linear size. This result allows to prove that the randomized incremental construction needs an expected linear size and an expected O(nlog⁥n)O(n\log n) time.Un bon Ă©chantillon est un ensemble de points tel que toute boule de rayon Ï”\epsilon contienne un nombre constant de points.Il est dĂ©montrĂ© que la triangulation de Delaunay d'un bon Ă©chantillon a une taille linĂ©aire, malheureusement cela ne suffit pas Ă  assurerune bonne complexitĂ© Ă  la construction incrĂ©mentale randomisĂ©e de latriangulation de Delaunay.Dans ce rapport, nous dĂ©montrons que la triangulation d'un Ă©chantillon alĂ©atoire de Bernoullid'un bon Ă©chantillon a une taille linĂ©aire. Nous en dĂ©duisonsque la construction incrĂ©mentale randomisĂ©e peut ĂȘtre faite en tempsO(nlog⁥n)O(n \log n) et espace O(n)O(n)

    Upper and Lower Bounds on the Smoothed Complexity of the Simplex Method

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    The simplex method for linear programming is known to be highly efficient in practice, and understanding its performance from a theoretical perspective is an active research topic. The framework of smoothed analysis, first introduced by Spielman and Teng (JACM '04) for this purpose, defines the smoothed complexity of solving a linear program with dd variables and nn constraints as the expected running time when Gaussian noise of variance σ2\sigma^2 is added to the LP data. We prove that the smoothed complexity of the simplex method is O(σ−3/2d13/4log⁥7/4n)O(\sigma^{-3/2} d^{13/4}\log^{7/4} n), improving the dependence on 1/σ1/\sigma compared to the previous bound of O(σ−2d2log⁥n)O(\sigma^{-2} d^2\sqrt{\log n}). We accomplish this through a new analysis of the \emph{shadow bound}, key to earlier analyses as well. Illustrating the power of our new method, we use our method to prove a nearly tight upper bound on the smoothed complexity of two-dimensional polygons. We also establish the first non-trivial lower bound on the smoothed complexity of the simplex method, proving that the \emph{shadow vertex simplex method} requires at least Ω(min⁥(σ−1/2d−1/2log⁡−1/4d,2d))\Omega \Big(\min \big(\sigma^{-1/2} d^{-1/2}\log^{-1/4} d,2^d \big) \Big) pivot steps with high probability. A key part of our analysis is a new variation on the extended formulation for the regular 2k2^k-gon. We end with a numerical experiment that suggests this analysis could be further improved.Comment: 41 pages, 5 figure
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