J. Basch, L. Guibas, and L. Zhang. Proximity problems on moving points. In Proceedings of the 13th Symposium of Computational Geometry, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....that hosts maintain their neighbors throughout the experiment (that is, the topology of the minimum spanning tree does not change. Extensions to dynamically changing minimum spanning trees can be done using the previous algorithms for dynamically constructing a minimum spanning trees derived in [8, 1]. Suppose there are two hosts which have to communicate with each other, but they are out of transmission range. There are two types of message exchanges: 1) an actual message and (2) a location update message. Each host has its own task to carry out which may require movement. We would like to ....

Julien Basch, Leonidas J. Guibas, and Li Zhang. Proximity problems on moving points. In the 13th Symposium of Computational Geometry, pages 344--351, 1997.

....result on multiple chains for pseudo parabolas was too weak. By Agarwal et al. s recently improved cut theorem for parabolas [5] the bound can be further reduced to O(n 8=3 For algorithms on the parametric kinetic minimum spanning tree problem in both its graph and geometric settings, see [3, 13]. Very recently, our Theorem 3.3 was used by Agarwal, Aronov, and Sharir [2] to bound the combinatorial complexity of multiple faces in arrangements of pseudo segments and of circles. Our Theorem 6.2 7.5 was also used by Aronov and Sharir [10] to derive improved bounds on the number of incidences ....

J. Basch, L. Guibas, and L. Zhang. Proximity problems on moving points. In Proc. 13th ACM Sympos. Comput. Geom., pages 344-351, 1997.

....any explicit modification in the input data. It is allowed however, that the motion function can be modified, in which case an explicit modification in the database is reported. A typical example for a kinetic data structure is to maintain the closest pair of balls in a billiard simulation [5]. In such an application the closest pair of balls may change at certain discrete points of time which are called (external) events. Possible future events are stored in an event queue and a kinetic data structure always processes the next event in the event queue. It may be necessary to have ....

....and a kinetic data structure always processes the next event in the event queue. It may be necessary to have additional events that are needed to keep control of the system. These events are called internal. In recent years kinetic data structures have been applied to many problems (see, e.g. [2, 5, 6, 21]) The previous research has focused on the case when the objects motion is described by some simple functions which are known to the system. In many applications, however, the motion of the objects is either completely unpredictable in time or is unknown to the system (think, for example, on ....

J. Basch, L. J. Guibas, and Li Zhang. Proximity problems on moving points. In Proceedings of the 13th Annual ACM Symposium on Computational Geometry, pp. 344--351, 1997.

....a Q1 query using O(log B n k) I Os, provided that the queries arrive in chronological order. This is achieved using the kinetic framework on an external range tree [6] Kinetic range trees (with slightly worse performance) were first developed in the internal memory setting by Basch et al. [9]. Our structure uses O(n log B n= log B log B n) disk blocks. If the points move with fixed velocity, then it processes O(N 2 ) events, each of which can be processed in O(log 2 B n= log B log B n) I Os. Our structures works even if the trajectories of the points are polynomials of fixed ....

....structure to change over time, and if we allow queries only at the current time. The approach Indexing Moving Points 14 can be extended to handle queries in future time as long as they arrive in chronological order. We develop our results in the kinetic data structure framework of Basch et al. [8, 9]. The main idea is to store only a combinatorial snapshot of the moving points at any time. Although the points are moving continuously, the data structure itself only depends on certain combinatorial properties (such as sorted order of points along x and y axes) and changes only at discrete ....

[Article contains additional citation context not shown here]

J. Basch, L. Guibas, and L. Zhang, Proximity problems on moving points, Proc. 15th Annu. ACM Sympos. Comput. Geom., 344--351, 1997.

....3 2 (n) bound [31] Tamaki and Tokuyama [43] were unable to obtain such an improvement because their result on multiple chains for pseudo parabolas was too weak. For recent algorithms on the parametric kinetic minimum spanning tree problem in both its graph and geometric settings, see [2, 10]. 8. Open problems We close with some interesting questions. 1. Can we improve the O(n log n) bound for cutting pseudo segments into extendible pseudo segments, or is there a superlinear lower bound 2. Can we improve Tamaki and Tokuyama s O(n 5=3 ) bound for cutting pseudo parabolas (or ....

J. Basch, L. Guibas, and L. Zhang. Proximity problems on moving points. In Proc. 13th ACM Sympos. Comput. Geom., pages 344--351, 1997.

....queries arrive in chronological order. This is achieved using the kinetic framework on an external range tree [5] Our structure uses O(n log B n= log B log B n) disk blocks. Kinetic range trees (with slightly worse performance) were first developed in the internal memory setting by Basch et al. [8]. We also show how one can combine kinetic range trees with partition trees to obtain a tradeoff between the query time and the number of events at which the kinetic index needs to be updated. Given a parameter NB Delta N 2 , we can answer a query in O(N 1 = p Delta k) I Os, and ....

....time, and if we allow queries only at the current time. The approach can be extended to handle queries in future time as long as they arrive in chronological order. Our so called kinetic range tree is based on an internal memory data structure with suboptimal query time developed by Basch et al. [8] and a query optimal external range tree developed in [5] We will present our range tree in the primal setting. We give a brief overview of the external range tree in Section 4.1, and then discuss how to kinetize it in Section 4.2. The main idea will be to store only a snapshot of the moving ....

[Article contains additional citation context not shown here]

J. Basch, L. Guibas, and L. Zhang, Proximity problems on moving points, Proc. 15th Annu. ACM Sympos. Comput. Geom., 344--351, 1997.

....are applicable in any domain with moving objects, such as robotics and computer vision. In fact, in addition to the kinetic data structures we have mentioned earlier in the thesis, there has already been a lot of research on using kinetic data structures to solve various geometric problems [3, 13, 14, 62]. Kinetic data structures promise to significantly impact any problem areas that deal with moving objects or continuous change. In this thesis, we have applied the themes of object complexity, geometric complexity, and kinetic data structures to develop a set of efficient algorithms for ....

J. Basch, L. J. Guibas, and L. Zhang, Proximity problems on moving points, Proc. 13th Annu. ACM Sympos. Comput. Geom., 1997, pp. 344--351.

....variations of this problem have been surveyed by Shamos [57] The trivial solution is to repeatedly check every pair of balls to nd each successive collision; this requires O(n 2 ) time per collision. Very few other algorithms are known with theoretical guarantees of any kind. Basch et al. [12] observe that only the closest pair of balls can collide, and suggest simulating collisions using a kinetic data structure to eciently maintain the closest pair. See also [11, 39] In the worst case, however, the closest pair changes (n 2 ) times without a single collision, which makes the ....

J. Basch, L. J. Guibas, and L. Zhang. Proximity problems on moving points. Proc. 13th Annu. ACM Sympos. Comput. Geom., pp. 344-351. 1997.

....are not directly applicable to the set of objects moving continuously. The dynamic algorithms allow insertions and deletions from a set of objects. It is not immediately obvious how to use these classical algorithms to maintain the closest point pair as the points move continuously. Guibas et al.[5] proposed a kinetic data structure to solve this kind of problems by capturing the continuous moving property. They assume that each object follows a posted flight plan, but the plan can be changed at any moment through a flight plan update. A kinetic data structure maintains a configuration ....

....switch from being true to being false. When a certificate fails, the proof structure needs to be modified and the combinatorial description of the configuration function may need to be updated also. A good data structure for moving objects should have the following desiderata conditions[5]. ffl Responsible: the kinetic data structure takes advantage of continuity of the object motion to select certificate structures which are easy to update at these critical events. ffl efficient: meaning that the number of events it preprocesses is not much greater than the number of ....

J.Basch, L.J.Guibas, and L.Zhang. Proximity Problems on Moving Points. Proc. of 13th Computational Geometry, 1997,344-351.

....balls, each with an initial position and velocity, and are asked to compute the sequence of collisions that occur as the balls move and bounce off each other, following the standard laws of classical physics. Several variations of this problem have been surveyed by Shamos [51] Basch et al. [12] observe that only the closest pair of balls can collide, and suggest simulating collisions using a kinetic data structure to efficiently maintain the closest pair. See also [11, 33] In the worst case, however, the closest pair changes Theta(n 2 ) times without a single collision, which ....

J. Basch, L. J. Guibas, and L. Zhang. Proximity problems on moving points. Proc. 13th Annu. ACM Sympos. Comput. Geom., pp. 344--351. 1997.

....and higher dimensions. In recent work, related to the Delaunay approach, Basch et al. 2, 3] have proposed a new set of kinetic data structures designed to maintain efficiently the closest pair (among the Gamma n 2 Delta possibilities) for n points in continuous motion in the plane. See [4] for results in d . 3.2 A Simple Geometric Hashing Approach While the Delaunay approach has some advantages and a theoretical basis, it also has limitations: the need to solve high degree equations (degree 4k for trajectories of degree k) which is prone to numerical errors, and the ....

J. Basch, L. Guibas, and L. Zhang. Proximity problems on moving points. In Proc. 13th Annu. ACM Sympos. Comput. Geom., pages 344--351, 1997.

....To maintain the range search trees kinetically, we keep sorted lists of the x and y coordinates of the points themselves, in addition to the sorted lists containing the extrema of the squares on each level. A range tree can be updated by deleting a point and re inserting it in the right place [5]. For the hierarchical algorithm, we need to maintain these structures for each level. In addition, we also need to insert or delete a point to or from a level, as a consequence of an event happening at a lower level. This requires the sorted lists and range search trees used in the basic ....

J. Basch, L. J. Guibas, and L. Zhang. Proximity problems on moving points. In Proc. 13th Annu. ACM Sympos. Comput. Geom., pages 344--351, 1997.

....this condition makes it easier to re estimate certificate failure times 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 ....

J. Basch, L. J. Guibas, and L. Zhang. Proximity problems on moving points. In Proc. 13th Annu. ACM Sympos. Comput. Geom., pages 344--351, 1997.

....may be significantly more efficient. We have been able to develop two different kinetic data structures for maintaining the closest pair among n moving points. One structure is for d = 2 [12] though extensions to higher dimensions should be possible) while the other works for all dimensions [15]. Both structures are based on a common insight: to find the closest pair is is sufficient to examine a linear number of point pairs defined by partitioning the space around each point into a fixed number of congruent cones having the point as their apex. These cones have a central axis and each ....

....of the moving points along a fixed (but of size roughly 2 d ) set of directions in the space. Thus these methods clearly process a quadratic set of events in the worst case. Keeping track of which points lie in which cones is accomplished by using a kinetic multidimensional range search tree [15]. With some further insights that we do have not the space to discuss here, these ideas lead to closest pair KDSs which are responsive, efficient, local, and compact. Several variations on the closest pair problem are interesting and open, even for d = 2. First, we do not have a good KDS for the ....

[Article contains additional citation context not shown here]

J. Basch, L. J. Guibas, and L. Zhang. Proximity problems on moving points. In Proc. 13th Annu. ACM Sympos. Comput. Geom., pages 344--351, 1997.

....To maintain the range search trees kinetically, we keep sorted lists of the x and y coordinates of the points themselves, in addition to the sorted lists containing the extrema of the squares on each level. A range tree can be updated by deleting a point and re inserting it in the right place [6]. 12 For the hierarchical algorithm, we need to maintain these structures for each level. In addition, we also need to insert or delete a point to or from a level, as a consequence of an event happening at a lower level. This requires the sorted lists and range search trees used in the basic ....

J. Basch, L. J. Guibas, and L. Zhang. Proximity problems on moving points. In Proc. 13th Annu. ACM Sympos. Comput. Geom., pages 344--351, 1997.

....To maintain the range search trees kinetically, we keep sorted lists of the x and y coordinates of the points themselves, in addition to the sorted lists containing the extrema of the squares on each level. A range tree can be updated by deleting a point and re inserting it in the right place [6]. For the hierarchical algorithm, we need to maintain these structures for each level. In addition, we also need to insert or delete a point to or from a level, as a consequence of an event happening at a lower level. This requires the sorted lists and range search trees used in the basic ....

J. Basch, L. J. Guibas, and L. Zhang. Proximity problems on moving points. In Proc. 13th Annu. ACM Sympos. Comput. Geom., pages 344--351, 1997.

....optimization problems have been studied for several other graph problems as well; see [13, 28] for a sample of such results. There are no known previous algorithms for the kinetic minimum spanning tree problem. A related problem, which has been studied, is the kinetic Euclidean MST problem [4], in which we want to list all different Euclidean minimum spanning trees of a set of points, each of which is moving along a line or curve. Let p denote the number of edge insertions, edge deletions, or minimum spanning tree topology changes. Here we show the following results, substantially ....

J. Basch, L. J. Guibas, and L. Zhang. Proximity problems on moving points. In Proc. 13th ACM Symp. Computational Geometry, 1997, 344--351.

....moving objects. An approach to this problem is to surround each object by a bounding ball and then detect collisions between these balls. If we can continuously keep track of the closest pair of balls as the objects move, we will have a way to detect such collisions. This setting is examined in [BGZ97], where a kinetic data structure ( BGH97] is given for the case when all the balls have similar sizes. However, that method doesn t apply to the case when there is no bound on the ratio on the largest radius to the smallest radius. The result of this note implies that another way to maintain the ....

J. Basch, L. Guibas, and L. Zhang. Proximity problems on moving points. In Proc. 13th Annu. ACM Sympos. Comput. Geom., pages 344--351, 1997.

....plans for further work. Following the publication of the conference version of this paper [9] several kinetic data structures have been developed for the maintenance of a variety of structures: binary space partitions [1, 3] closest pair and minimum spanning trees in arbitrary dimensions [12], and diameter and width [2] The framework has also been applied to the problem of collision detection between polygons in two dimensions [8, 20] 2 2 D convex hull In this section, we present an efficient kinetic data structure to maintain the convex hull of a set of moving points in the plane. ....

J. Basch, L. J. Guibas, and L. Zhang. Proximity problems on moving points. In Proc. 13th Annu. ACM Sympos. Comput. Geom., pages 344--351, 1997.

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J. Basch, L. Guibas, and L. Zhang. Proximity problems on moving points. In Proceedings of the 13th Symposium of Computational Geometry, 1997.

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Julien Basch, Leonidas J. Guibas, and Li Zhang. Proximity problems on moving points. In the 13th Symposium of Computational Geometry, pages 344--351, 1997.

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J. Basch, L. J. Guibas, and L. Zhang, "Proximity problems on moving points," In 13th Symposium of Computational Geometry, 1997, 344--351.

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BASCH, J., GUIBAS, L. J., AND ZHANG, L. Proximity problems on moving points. In Proceedings of the 13th Annual ACM Symposium on Computational Geometry (1997), pp. 344-- 351.

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L. Z. Julien Basch, Leonidas J. Guibas. Proximity problems on moving points. pages 344351, 1997.

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BASCH, J., GUIBAS, L. J., AND ZHANG, L. Proximity problems on moving points. In Proceedings of the 13th Annual ACM Symposium on Computational Geometry (1997), pp. 344-- 351.

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