Julien Basch

Data structures for mobile data

Akinetic data structure(KDS) maintains an attribute of interest in a system of geometric objects undergoing continuous motion. In this paper we develop a concentual framework for kinetic data structures, we propose a number of criteria for the quality of such structures, and we describe a number of fundamental techniques for their design. We illustrate these general concepts by presenting kinetic data structures for maintaining the convex hull and the closest pair of moving points in the plane; these structures behave well according to the proposed quality criteria for KDSs.

Deformable Free-Space Tilings for Kinetic Collision Detection†:

We present kinetic data structures for detecting collisions between a set of polygons that are moving continuously. Unlike classical collision detection methods that rely on bounding volume hierarchies, our method is based on deformable tilings of the free space surrounding the polygons. The basic shape of our tiles is that of a pseudo-triangle, a shape sufficiently flexible to allow extensive deformation, yet structured enough to make detection of self-collisions easy. We show different schemes for maintaining pseudo-triangulations as a kinetic data structure, and we analyze their performance. Specifically, we first describe an algorithm for maintaining a pseudo-triangulation of a point set, and show that the pseudo-triangulation changes only quadratically many times if points move along algebraic arcs of constant degree. In addition, by refining the pseudo-triangulation, we show triangulations of points that only change about O(n 7 / 3 ) times for linear motion. We then describe an algorithm for maintaining a pseudo-triangulation of a set of convex polygons. Finally, we extend our algorithm to the general case of maintaining a pseudo-triangulation of a set of moving or deforming simple polygons.

Proximity problems on moving points

A kinetic data structure for the maintenance of a multidimensional range search tree is introduced. This structure is used as a building block to obtain kinetic data structures for two classical geometric proximity problems in arbitrary dlmensions: the first structure maintains the closest pair of a set of continuously moving points, and is provably efficient. The second structure maintains a spanning tree of the moving points whose cost remains within some prescribed factor of the minimum spanning tree.

A practical evaluation of kinetic data structures

In many applications of computational geometry to modeling objects and processes in the physical world, the participating objects are in a state of continuous change. Motion is the most ubiquitous kind of continuous transformation but others, such as shape deformation, are also possible. In a recent paper, Baech, Guibas, and Hershberger [BGH97] proposed the framework of kinetic data structures (KDSS) as a way to maintain, in a completely on-line fashion, desirable information about the state of a geometric system in continuous motion or change. They gave examples of kinetic data structures for the maximum of a set of (changing) numbers, and for the convex hull and closest pair of a set of (moving) points in the plane. The KDS frameworkallowseach object to change its motion at will according to interactions with other moving objects, the environment, etc. We implemented the KDSSdescribed in [BGH97],es well as came alternative methods serving the same purpose, as a way to validate the kinetic data structures framework in practice. In this note, we report some preliminary results on the maintenance of the convex hull, describe the experimental setup, compare three alternative methods, discuss the value of the measures of quality for KDSS proposed by [BGH97],and highlight some important numerical issues.

Kinetic collision detection between two simple polygons

We design a kinetic data structure for detecting collisions between two simple polygons in motion. In order to do so, we create a planar subdivision of the free space between the two polygons, called the external relative geodesic triangulation, which certifies their disjointness. We show how this subdivision can be maintained as a kinetic data structure when the polygons are moving, and analyze its performance in the kinetic setting.

Sweeping lines and line segments with a heap

Given n line segments in the plane, the Bentley-Ottmann sweep maintains the exact ordering of the intersections of the segments with a vertical liie, as this line sweeps the plane from left to right. To accomplish thk, every intersection between two segments must be processed, and the running time of the sweep can be fl(nz ). In this paper, it is shown how a heap on the intersections can be maintained during the sweep. ThM new type of

Lower bounds for kinetic planar subdivisions

We revisit the notion of kinetic efficiency for noncanonically defined discrete attributes of moving data, like binary space partitions and triangulations. Under reasonable computational models, we obtain lower bounds on the minimum amount of work required to maintain any binary space partition of moving segments in the plane or any Steiner triangulation of moving points in the plane. Such lower bounds—the first to be obtained in the kinetic context—are necessary to evaluate the efficiency of kinetic data structures when the attribute to be maintained is not canonically defined.

Probabilistic analysis for combinatorial functions of moving points

We initiate a probabilistic study of configuration functions of moving points. In our probabilistic model, a particle is given an initiaf position and a velocity drawn independently at random from the same distribution D. We show that if n particles are drawn independently at random from the uniform distribution on the square, their convex hull undergoes El(logz n) combinatorial changes in expectation, their Voronoi diagram undergoes e(n312 ) combinatorial changes, and their closest pair undergoes El(n) combinatorial changes.

Kinetic data structures: animating proofs through time

Kinetic Data Structures (KDS for short) are a new class of data structures aimed at keeping track of attributes of interest in systems of moving objects [1, 2]. ln this video we illustrate the principies of KDS design in the context of some classical geometric problems, such as the calculation of the convex hull or closest pair of moving points in the plane. ln an ordinary simulation the positions of the points are advanced by a fixed time step, and then the attribute of interest in incrementally recomputed. Since the combii:J.atorial structure of the attribute changes irregularly, it is quite difficult to select a time-step size that avoids both oversampling and undersampling the system. Unlike such fixed step methods, a kinetic data structure performs an event-driven simulation where only events relevant to the attribute of interest are generated and processed. The result is a simulation that is always accurate and with a computation cost dose to the minimum possible. This video presents animations for and contains the description of a few two-dimensional kinetic geometric algorithms:

Reporting Red-Blue Intersections between Two Sets of Connected Line Segments

We present a new line sweep algorithm, HeapSweep, for reporting bichromatic (‘purple’) intersections between a red and a blue family of line segments. If the union of the segments in each family is connected as a point set, HeapSweep reports all k purple intersections in time O((n+k)α(n) log 3n), where n is the total number of input segments and α(n) is the familiar inverse Ackermann function. To achieve these bounds, the algorithm keeps only partial information about the vertical ordering between segments of the same color, using a new data structure called a kinetic queue. In order to analyze the running time of HeapSweep, we also show that a simple polygon containing a set of n line segments can be partitioned into monotone regions by lines cutting these segments Θ(n log n) times.