Incremental and Decremental Maintenance of Planar Width

We present an algorithm for maintaining the width of a planar point set dynamically, as points are inserted or deleted. Our algorithm takes time O(kn^epsilon) per update, where k is the amount of change the update causes in the convex hull, n is the number of points in the set, and epsilon is any arbitrarily small constant. For incremental or decremental update sequences, the amortized time per update is O(n^epsilon).Comment: 7 pages; 2 figures. A preliminary version of this paper was presented at the 10th ACM/SIAM Symp. Discrete Algorithms (SODA '99); this is the journal version, and will appear in J. Algorithm

On computing the diameter of a point set in high dimensional Euclidean space

We consider the problem of computing the diameter of a set of $n$ points in $d$-dimensional Euclidean space under Euclidean distance function. We describe an algorithm that in time $O(dnlog n +n^{2})$ finds with high probability an arbitrarily close approximation of the diameter. For large values of $d$ the complexity bound of our algorithm is a substantial improvement over the complexity bounds of previously known exact algorithms. Computing and approximating the diameter are fundamental primitives in high dimensional computational geometry and find practical application, for example, in clustering operations for image databases

Fast algorithms for collision and proximity problems involving moving geometric objects

Consider a set of geometric objects, such as points, line segments, or axes-parallel hyperrectangles in \IR^d, that move with constant but possibly different velocities along linear trajectories. Efficient algorithms are presented for several problems defined on such objects, such as determining whether any two objects ever collide and computing the minimum inter-point separation or minimum diameter that ever occurs. The strategy used involves reducing the given problem on moving objects to a different problem on a set of static objects, and then solving the latter problem using techniques based on sweeping, orthogonal range searching, simplex composition, and parametric search

Reconnaissance de morceaux de plans discrets bruités

mémoire de DEA, 37 pagesLa reconnaissance d'objets discrets est un sujet important en géométrie discrète et de nombreux travaux concernant les droites et les plans ont été réalisés. Nous nous intéressons à la notion d'objets discrets flous, correspondant à des objets discrets bruités, et à leur détection. Ces objets sont définis analytiquement et les algorithmes développés à partir de ceux-ci s'adaptent aux données bruitées par les méthodes d'acquisition, par exemple les données médicales issues de scanners, d'échographie, d'IRM... Dans ce mémoire nous nous intéresserons plus particulièrement aux plans discrets bruités. En introduisant la définition d'un morceau de plan discret flou, nous montrerons que le problème de le reconnaissance d'un tel objet est équivalent à celui du calcul de l'épaisseur d'un ensemble de points en dimension 3. Après une étude des méthodes existantes résolvant le calcul de l'épaisseur, un algorithme connu sera alors adapté pour résoudre le problème de la reconnaissance incrémentale des morceaux de plans discrets flous