P. K. Agarwal, L. J. Guibas, J. Hershberger, and E. Veach. Maintaining the extent of a moving set of points. In 5th Workshop on Algorithms & Data Structures, 1997, 31--44. 10  Home/Search   Document Not in Database   Summary   Related Articles   Check

....then whenever a new vertex appears on the convex hull boundary of S(t) or an existing vertex disappears from it, any triangulation of S has to change. Since it is possible to construct a set of n points, each moving at constant velocity, so that their convex hull changes Omega Gamma ) times [4], any KDS maintaining a triangulation has to update the triangulation Omega Gamma ) times. This argument, however, does not apply if we allow Steiner points, that is, additional moving points that are not part of the original point set, as we can enclose the moving points in a big bounding ....

....arise, and each Steiner point is free to move along an arbitrary (continuous) path. In this model, we construct a scenario of O(n) points in linear motion such that any linear space kinetic Steiner triangulation requires a processing cost of Omega Gamma n ) A technique of Agarwal et al. [4] (described later in this section) allows us to simulate a circular motion with linearly moving points, in the following sense: We can create a set S of points with positions S(t) at time t, such that each point moves at constant velocity, and the set is always cocircular along a circle of fixed ....

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P. K. Agarwal, L. J. Guibas, J. Hershberger, and E. Veach. Maintaining the extent of a moving set of points. In 5th Workshop on Algorithms & Data Structures, 1997, 31--44. 10

....when an object changes it motion law. KDSs have been primarily studied in the context of geometric problems that arise in virtual reality simulations. Good KDSs have been developed for a variety of spatial proximity [9, 1, 21, 17] e.g. collision detection, closest pair, clustering) extent [4, 8] (e.g. diameter, convex hull) visibility [6, 5] binary space partitions, occlusion) and connectivity [2, 23] e.g. minimum spanning trees, sparse spanners) problems. For example, the three frames below from a kinetic convex hull simulation (Figure 12) illustrate a combinatorial change to the ....

P. Agarwal, L. Guibas, J. Hershberger, and E. Veach. Maintaining the extent of a moving set of points. In 5-th Int. Workshop on Algorithms & Data Structures (WADS), pages 31--44, 1997.

....From these considerations it easily follows that the structure is local, compact, and responsive. Figure 1 shows a simple example of this process. How about efficiency Even with straight lines motions, we can show that the convex hull can change combinatorially Omega Gamma n 2 ) times [5]. Because of our algebraic trajectory assumption, it is clear that the number of events corresponding to s certificate failures is O(n 2 ) but the corresponding counts for y and x certificates are O(n 3 ) and O(n 4 ) respectively. However, by considering the surfaces swept by the lines ....

....We expect progress on these fundamental structures to enable progress on other higher dimensional kinetic problems as well. Diameter, width, minimum spanning circle, etc. Once we can maintain the convex hull of moving points in 2 d, we can also solve a number of other related extent problems. In [5] we show how to maintain the diameter, width, and various flavors of bounding boxes (minimum area, minimum perimeter) with the same overall kinetic performance as the convex hull. These applications show the power of composing kinetic data structures: once we have the convex hull we can maintain ....

P. K. Agarwal, L. J. Guibas, J. Hershberger, and E. Veach. Maintaining the extent of a moving set of points. In 5-th workshop on algorithms & data structures (WADS), 1997.

....a new vertex appears on the convex hull boundary of S(t) or an existing vertex disappears from the hull, any triangulation of S has to change. Since it is possible to construct a set of n points, each moving at constant velocity, so that their convex hull changes Omega Gamma n 2 ) times [4], any KDS maintaining a triangulation has to update the triangulation Omega Gamma n 2 ) times. The question of effectively finding a kinetic triangulation that achieves these bounds is addressed in [6] This argument, however, does not apply if we allow Steiner points, that is, additional ....

....arise, and each Steiner point is free to move along an arbitrary (continuous) path. In this model, we construct a scenario of O(n) points in linear motion such that any linear space kinetic Steiner triangulation requires a processing cost of Omega Gamma n 2 ) A technique of Agarwal et al. [4] allows us to simulate a circular motion with linearly moving points, in the following sense: We can create a set S of points with position S(t) at time t, such that each Page 5 b b i 1 r i t Figure 5: The static blue points (hollow) and the moving red points (filled) In reality, the inner and ....

[Article contains additional citation context not shown here]

P. K. Agarwal, L. J. Guibas, J. Hershberger, and E. Veach. Maintaining the extent of a moving set of points. In 5th Workshop on Algorithms & Data Structures, page 31--44, 1997.

....then whenever a new vertex appears on the convex hull boundary of S(t) or an existing vertex disappears from the hull, any triangulation of S has to change. Since it is possible to construct a set of n points, each moving at constant velocity, so that their convex hull changes n 2 ) times [4], any KDS maintaining a triangulation has to update the triangulation n 2 ) times. The question of e ectively nding a kinetic triangulation that achieves these bounds is addressed in [6] This argument, however, does not apply if we allow Steiner points, that is, additional moving points that ....

....that ever arise, and each Steiner point is free to move along an arbitrary (continuous) path. In this model, we construct a scenario of O(n) points in linear motion such that any linear space kinetic Steiner triangulation requires a processing cost of n 2 ) A technique of Agarwal et al. [4] allows us to simulate a circular motion with linearly moving points, in the following sense: We can create a set S of points with position S(t) at time t, such that each point moves at constant velocity, and the set is always cocircular along a circle of xed center (but with varying radius) For ....

[Article contains additional citation context not shown here]

P. K. Agarwal, L. J. Guibas, J. Hershberger, and E. Veach. Maintaining the extent of a moving set of points. In 5th Workshop on Algorithms & Data Structures, page 31-44, 1997.

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