Kinetic collision detection for low-density scenes in the black-box model

We present an efficient method for collision detection in the black-box KDS model for a set S of n objects in the plane. In this model we receive the object locations at regular time steps and we know a bound dmax on the maximum displacement of any object within one time step. Our method maintains, in O((¿+k)n) time per time step, a compressed quadtree on the bounding-box vertices of the objects; here ¿ denotes the density of S and k denotes the maximum number of objects that can intersect any disk of radius dmax. Collisions can then be detected by testing O((¿+k)2n) pairs of objects for intersection

Kinetic Euclidean 2-centers in the black-box model

We study the 2-center problem for moving points in the plane. Given a set P of n points, the Euclidean 2-center problem asks for two congruent disks of minimum size that together cover P. Our methods work in the black-box KDS model, where we receive the locations of the points at regular time steps and we know an upper bound d_max on the maximum displacement of any point within one time step. We show how to maintain a (1 + e)-approximation of the Euclidean 2-center in amortized sub-linear time per time step, under certain assumptions on the distribution of the point set P. In many cases --namely when the distance between the centers of the disks is relatively large or relatively small-- the solution we maintain is actually optimal

Kinetic Euclidean 2-centers in the black-box model

We study the 2-center problem for moving points in the plane. Given a set P of n points, the Euclidean 2-center problem asks for two congruent disks of minimum size that together cover P. Our methods work in the black-box KDS model, where we receive the locations of the points at regular time steps and we know an upper bound d_max on the maximum displacement of any point within one time step. We show how to maintain a (1 + e)-approximation of the Euclidean 2-center in amortized sub-linear time per time step, under certain assumptions on the distribution of the point set P. In many cases --namely when the distance between the centers of the disks is relatively large or relatively small-- the solution we maintain is actually optimal

Kinetic 2-centers in the black-box model

We study two versions of the 2-center problem for moving points in the plane. Given a set P of n points, the Euclidean 2-center problem asks for two congruent disks of minimum size that together cover P; the rectilinear 2-center problem correspondingly asks for two congruent axis-aligned squares of minimum size that together cover P. Our methods work in the black-box KDS model, where we receive the locations of the points at regular time steps and we know an upper bound d_{max} on the maximum displacement of any point within one time step. We show how to maintain the rectilinear 2-center in amortized sub-linear time per time step, under certain assumptions on the distribution of the point set P. For the Euclidean 2-center we give a similar result: we can maintain in amortized sub-linear time (again under certain assumptions on the distribution) a (1+e)-approximation of the optimal 2-center. In many cases---namely when the distance between the centers of the disks is relatively large or relatively small---the solution we maintain is actually optimal. We also present results for the case where the maximum speed of the centers is bounded. We describe a simple scheme to maintain a 2-approximation of the rectilinear 2-center, and we provide a scheme which gives a better approximation factor depending on several parameters of the point set and the maximum allowed displacement of the centers. The latter result can be used to obtain a 2.29-approximation for the Euclidean 2-center; this is an improvement over the previously best known bound of 8/p approx 2.55. These algorithms run in amortized sub-linear time per time step, as before under certain assumptions on the distribution

Kinetic convex hulls and Delaunay triangulations in the black-box model

Over the past decade, the kinetic-data-structures framework has become the standard in computational geometry for dealing with moving objects. A fundamental assumption underlying the framework is that the motions of the objects are known in advance. This assumption severely limits the applicability of KDSs. We study KDSs in the black-box model, which is a hybrid of the KDS model and the traditional time-slicing approach. In this more practical model we receive the position of each object at regular time steps and we have an upper bound on dmax, the maximum displacement of any point in one time step. We study the maintenance of the convex hull and the Delaunay triangulation of a planar point set P in the black-box model, under the following assumption on dmax: there is some constant k such that for any point p ¿ P the disk of radius dmax contains at most k points. We analyze our algorithms in terms of ¿k , the so-called k-spread of P. We show how to update the convex hull at each time step in O(k¿k log2 n) amortized time. For the Delaunay triangulation our main contribution is an analysis of the standard edge-flipping approach; we show that the number of flips is O(k2 ¿k2) at each time step

Finding structures on imprecise points

An imprecise point is a point in R^2 of which we do not know the location exactly; we only know for each point a region in R^2 containing it. On such a set of imprecise points, structures like the closest pair or convex hull are not uniquely defined. This leads us to study the following problem: Given a structure of interest, a set R of regions and a subset C ¿ R, we want to determine if it is possible to place a point in each region of R such that the points placed in regions of C form the structure of interest. We study this problem for the convex hull, with various types of regions. For each variant we either give a NP-hardness proof or a polynomial-time algorithm

Kinetic convex hulls in the black-box model

Over the past decade, the kinetic-data-structures framework has become the standard in computational geometry for dealing with moving objects. A fundamental assumption underlying the framework is that the motions of the objects are known in advance. This assumption severely limits the applicability of KDSs. We study KDSs in the black-box model, which is a hybrid of the KDS model and the traditional time-slicing approach. In this more practical model we receive the position of each object at regular time steps and we have an upper bound on d_max, the maximum displacement of any point in one time step. We study the maintenance of the convex hull of a planar point set P in the black-box model, under the following assumption on d_max: there is some constant k such that for any point p in P the disk of radius d_max contains at most k points. We analyze our algorithms in terms of \Delta_k, the so-called k-spread of P. We show how to update the convex hull at each time step in O(k \Delta_k log^2 n) amortized time