Mjm Marcel Roeloffzen

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+ε)-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/π 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.

Kinetic compressed quadtrees in the black-box model with applications to collision detection for low-density scenes

We present an efficient method for maintaining a compressed quadtree for a set of moving points in R d . Our method works in the black-box KDS model, where we receive the locations of the points at regular time steps and we know a bound d max on the maximum displacement of any point within one time step. When the number of points within any ball of radius d max is at most k at any time, then our update algorithm runs in O(nlogk) time. We generalize this result to constant-complexity moving objects in R d . The compressed quadtree we maintain has size O(n); under similar conditions as for the case of moving points it can be maintained in O(n log?) time per time step, where ? is the density of the set of objects. The compressed quadtree can be used to perform broad-phase collision detection for moving objects; it will report in O((??+?k)n) time a superset of all intersecting pairs of objects.

Distance-Sensitive planar point location

Let

Distance-sensitive planar point location

\mathcal{S}

Kinetic data structures in the black-box model

be a connected planar polygonal subdivision with n edges and of total area 1. We present a data structure for point location in

Finding structures on imprecise points

\mathcal{S}

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

where queries with points far away from any region boundary are answered faster. More precisely, we show that point location queries can be answered in time

Faster DBScan and HDBScan in low-dimensional euclidean spaces

O(1+\min(\log \frac{1}{\Delta_{p}}, \log n))

Fully-Dynamic and Kinetic Conflict-Free Coloring of Intervals with Respect to Points

, where Δp is the distance of the query point p to the boundary of the region containing p. Our structure is based on the following result: any simple polygon P can be decomposed into a linear number of convex quadrilaterals with the following property: for any point p∈P, the quadrilateral containing p has area