P.K. Agarwal, L.J. Guibas, T.M. Murali, and J.S. Vitter. Cylindrical static and kinetic binary space partitions. In Proc. 13th Annu. ACM Sympos. Comput. Geom., pages 39-48, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... by Roos [87] on Voronoi diagrams for an approach that is closely related to KDS) Kinetic connectivity structures that maintain the connected components of sets of mov ing disks, rectangles or hypercubes [38, 44, 45, 54] as well as binary space partitions (BSP) for sets of moving line segments [7, 5, 21] have also been the subject of extensive study. Some attributes like Voronoi diagrams or convex hulls are uniquely defined for a collection of objects while others like triangulations or binary space partitions are not. To evaluate the efficiency of kinetic data structures that monitor ....

Pankaj K. Agarwal, Leonidas J. Guibas, T. M. Murali, and Jeffrey Scott Vitter. Cylin- drical static and kinetic binary space partitions. In Proc. 13th Annu. ACM Sympos. Cornput. Geom., pages 39-48, 1997.

....later) however, occur only at certain discrete times. We explicitly takeadvantage of the continuity of the motion of the objects involved so as to update the BSP only when actual events cause the BSP to change combinatorially. Such an approachwas first used in a recent paper by Agarwal et al. [1]tomaintain the BSP of asetofdisjoint segments in the plane. In this paper we extend their approach to efficiently maintain a BSP for the considerably more complex case of disjoint triangles in IR 3 . Toachieve this result, wefirst develop a method to efficiently maintain a BSP for intersecting ....

....log 2 n k 0 ) in expected time O(n log 3 n k 0 log n) for a set Delta of n disjoint triangles in space# here k 0 is the number of intersection points among the edges of the xy projections of triangles in Delta. The previous bestknown algorithms for triangles are by Agarwal et al. [1]. They present a randomized algorithm that constructs a BSP of size O(n 2 ) in expected time O(n 2 log 2 n) and a deterministic algorithm that constructs a BSP of size O( n k 0 )logn) in time O( n k 0 )log 2 n) A shortcoming of all known algorithms (including ours) for ....

[Article contains additional citation context not shown here]

P.K.Agarwal, L. J. Guibas, T. M. Murali, and J. S. Vitter. Cylindrical static and kinetic binary space partitions. Proc. 13th Annu. ACM Sympos. Comput. Geom., pp. 39--48, 1997.

....happened in the past. Self customizing is premised on this temporal coherence principle alone. It is this minimalist approach that makes it particularly attractive. Just as spatial coherence is essential to fast rendering, temporal coherence has proven useful for collision and visibility, eg, [1, 8]. The main difference in our approach is that of scale. Our assumption is not that of micro coherence (a fancy way of saying that functions should be piecewise smooth) but of coherence over longer periods of time. In our model, request distributions can be arbitrary and they can change over ....

.... t Gamma1 i ) T ( Sigma t Gamma1 i ) Gamma1 (x j Gamma t Gamma1 i ) P w t Gamma1 e Gamma 1 2 (x j Gamma t Gamma1 ) T ( Sigma t Gamma1 ) Gamma1 (x j Gamma t Gamma1 ) 3 Tree Configuration Dynamic updating of bsp trees has been extensively studied [1, 7, 27, 30], and there is no need here for detailed discussion of the primitive geometric operations involved. Given a node v of a bsp tree for S, let P v be the convex polygon formed by the intersection of the cutting plane v with the convex polyhedron C v . The traversal cost of a directed line in a bsp ....

Agarwal, P.K., Guibas, L.J., Murali, T.M., Vitter, J.S. Cylindrical static and kinetic binary space partitions, Proc. 13th Annu. Symp. Comput. Geom. (1997), 39-48.

....v and each portion will be stored in the corresponding subtree. There is no special rule for selecting the cutting hyperplanes for a BSP tree. However, the choice of cutting hyperplanes affects the size and maximum depth of the BSP tree and the number of object fragments that arise. Several works [1, 2, 4, 14, 15] have been devoted to the problem of selecting the cutting hyperplanes so as to minimize the complexity of the resulting BSP tree. For example, if there is a facet of an object lying on a hyperplane which does not intersect the interior of any other objects of this node, then this is a good ....

Pankaj K. Agarwal, Leonidas J. Guibas, T. M. Murali, and Jeffrey Scott Vitter. Cylindrical static and kinetic binary space partitions. In Proc. 13th Annu. ACM Sympos. Comput. Geom., pages 39--48, 1997.

....later) however, occur only at certain discrete times. We explicitly take advantage of the continuity of the motion of the objects involved so as to update the BSP only when actual events cause the BSP to change combinatorially. Such an approach was first used in a recent paper by Agarwal et al. [1] to maintain the BSP of a set of disjoint segments in the plane. In this paper we extend their approach to efficiently maintain a BSP for the considerably more complex case of disjoint triangles in IR 3 . To achieve this result, we first develop a method to efficiently maintain a BSP for ....

....log 2 n k 0 ) in expected time O(n log 3 n k 0 log n) for a set Delta of n disjoint triangles in space; here k 0 is the number of intersection points among the edges of the xy projections of triangles in Delta. The previous bestknown algorithms for triangles are by Agarwal et al. [1]. They present a randomized algorithm that constructs a BSP of size O(n 2 ) in expected time O(n 2 log 2 n) and a deterministic algorithm that constructs a BSP of size O( n k 0 ) log n) in time O( n k 0 ) log 2 n) A shortcoming of all known algorithms (including ours) for ....

[Article contains additional citation context not shown here]

P. K. Agarwal, L. J. Guibas, T. M. Murali, and J. S. Vitter. Cylindrical static and kinetic binary space partitions. Proc. 13th Annu. ACM Sympos. Comput. Geom., pp. 39--48, 1997.

....maintains the pseudotriangulation that the static algorithm would have constructed on P(t) This way, the structure is canonical, and it is more convenient for the analysis. Such approach has been taken in maintaining other structures, for example, the binary space partitioning structure in [1]. To maintain this invariant, we need additional certificates. When these certificates fail, we usually update the structure by the flipping operation: For any diagonal edge e, consider the two pseudo triangles # 1 , # 2 that contain e. The union of # 1 , # 2 forms a pseudo quadrangle 3 a cell ....

P. K. Agarwal, L. J. Guibas, T. M. Murali, and J. S. Vitter. Cylindrical static and kinetic binary space partitions. In Proc. 13th Annu. ACM Sympos. Comput. Geom., pages 39--48, 1997.

....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 hull in a larger example, based on the algorithm ....

P. K. Agarwal, L. J. Guibas, T. Murali, and J. Vitter. Cylindrical static and kinetic binary space partitions. In Proc. 13th Annu. ACM Sympos. Comput. Geom., pages 39--48, 1997.

....objects from their earlier positions and reinserting them in their current positions after some time interval has elapsed. Such approaches suffer from all the problems discussed in Section 1. We have recently been able to obtain kinetic data structures for disjoint moving segments in the plane [6], and disjoint moving triangles in space [3] These methods are based on defining a BSP by cuts along the given objects or parallel to a particular axis; the cuts are generated according to a random ordering of the objects. The resulting BSP has expected size O(n log n) and depth O(log n) in 2 d, ....

....the current BSP structure becomes invalid due to object motion and update the tree at a cost of O(log n) per event for the 2 d BSP, and O(log 2 n) for the 3 d BSP. In the easier to explain 2 d algorithm the events correspond to times when certain critical trapezoids, called transient trapezoids [6], collapse by having their two parallel sides coincide. The event counts are O(n 2 ) and O(n 2 s (n) in the 2 d and 3 d cases respectively. These structures are responsive and strongly efficient. Their size is bounded by the BSP size, but they are not local. Kinetic Data Structures Many ....

P. K. Agarwal, L. J. Guibas, T. Murali, and J. Vitter. Cylindrical static and kinetic binary space partitions. In Proc. 13th Annu. ACM Sympos. Comput. Geom., pages 39--48, 1997.

....However, Agarwal et al. 1] have shown that there are con gurations of n moving segments, such that any BSPs must undergo n p n) changes during some smooth motion. BSPs have already been studied in a kinetic setting. For a set of n disjoint moving line segments in the plane, Agarwal et al. [3] present a kinetic BSP of expected size O(n log n) whose expected response time is O(log n) The expectation is with respect to a random order on the segments, which is xed at the beginning. It is assumed that there is no correlation between the motion of the segments and the random order, so ....

....another 3 dimensional BSP, which maintains the vertical decomposition of a set of non intersecting triangles in 3 space, and gives a detailed description of its implementation. In this paper we describe a new kinetic BSP, which improves in several ways over the approach of Agarwal et al. [3]. The main advantage is that the response time is O(log 2 n) in the worst case, compared to the (n) worst case response time for the previous approach. The expected response time remains O(log n) Another advantage is that the O(n log n) bound on its size is deterministic. Like the structure of ....

[Article contains additional citation context not shown here]

P.K. Agarwal, L.J. Guibas, T.M. Murali, and J.S. Vitter. Cylindrical static and kinetic binary space partitions. In Proc. 13th Annu. ACM Sympos. Comput. Geom., pages 39-48, 1997.

....make free cuts through the non orthogonal polygons too. In Step 3, if the number of triangles at a node is greater than the number of fat rectangles, we use the algorithm of Agarwal et al. for triangles in R 3 to construct a BSP of size quadratic in the number of triangles in near quadratic time [4]. Proceeding as in the previous section, we can prove the following theorem: Theorem 4.6.2 A BSP of size np2 O( p log n ) can be constructed in np 2 2 O( p log n ) time for n polygons in R 3 , of which p 1 are non orthogonal and the rest are fat rectangles. The constants of ....

P. K. Agarwal, L. J. Guibas, T. M. Murali, and J. S. Vitter, Cylindrical static and kinetic binary space partitions, Proc. 13th ACM Sympos. Comput. Geom., 1997, pp. 39--48.

....maintains the pseudotriangulation that the static algorithm would have constructed on P(t) This way, the structure is canonical, and it is more convenient for the analysis. Such approach has been taken in maintaining other structures, for example, the binary space partitioning structure in [1]. To maintain this invariant, we need additional certificates. When these certificates fail, we usually update the structure by the flipping operation: For any diagonal edge e, consider the two pseudo triangles # 1 , # 2 that contain e. The union of # 1 , # 2 forms a pseudo quadrangle 3 a cell ....

P. K. Agarwal, L. J. Guibas, T. M. Murali, and J. S. Vitter. Cylindrical static and kinetic binary space partitions. In Proc. 13th Annu. ACM Sympos. Comput. Geom., pages 39--48, 1997.

.... generation [10, 11] visibility problems [4, 30] solid modeling [23, 25, 31] geometric data repair [20] ray tracing [22] robotics [5] and approximation algorithms for network design [19] and surface simplification [3] Algorithms have also been developed to construct BSPs for moving objects [1, 12, 24, 32]. Informally, a BSP B for a set of objects is a binary tree. Each node v of B is associated with a convex region R v . The regions associated with the children of v are obtained by splitting R v with a hyperplane. If v is a leaf of B, then the interior of R v does not intersect any object. The ....

....make free cuts through the non orthogonal polygons too. In Step 3, if the number of triangles at a node is greater than the number of fat rectangles, we use the algorithm of Agarwal et. al for triangles in R 3 to construct a BSP of size quadratic in the number of triangles in near quadratic time [1]. Proceeding as in the previous section, we can prove the following theorem: Theorem 7.2 A BSP of size np2 O( p log n ) can be constructed in np2 O( p log n ) time for n objects in R 3 , of which p are non orthogonal and the rest are fat rectangles. The constants of proportionality in ....

P. K. Agarwal, L. J. Guibas, T. M. Murali, and J. S. Vitter, Cylindrical static and kinetic binary space partitions, To appear in the 13th Annual Symp. on Comp. Geom., 1997.

....some further issues generated by this framework for mobile data and present 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 ....

P. K. Agarwal, L. J. Guibas, T. Murali, and J. Vitter. Cylindrical static and kinetic binary space partitions. In Proc. 13th Annu. ACM Sympos. Comput. Geom., pages 39--48, 1997.

.... generation [11, 12] visibility problems [5, 29] solid modeling [22, 24, 30] geometric data repair [19] ray tracing [21] robotics [6] and approximation algorithms for network design [18] and surface simplification [4] Algorithms have also been developed to construct BSPs for moving objects [2, 13, 23, 31]. Informally, a BSP B for a set of (d Gamma 1) dimensional objects in R d is a binary tree. Each node v of B is associated with a convex region R v . The regions associated with the children of v are obtained by splitting R v with a hyperplane. If v is a leaf of B, then the interior of R v does ....

....make free cuts through the non orthogonal polygons too. In Step 3, if the number of triangles at a node is greater than the number of fat rectangles, we use the algorithm of Agarwal et al. for triangles in R 3 to construct a BSP of size quadratic in the number of triangles in near quadratic time [2]. Proceeding as in the previous section, we can prove the following theorem: Theorem 6.2 A BSP of size np2 O( p log n ) can be constructed in np2 O( p log n ) time for n polygons in R 3 , of which p 1 are non orthogonal and the rest are fat rectangles. The constants of ....

P. K. Agarwal, L. J. Guibas, T. M. Murali, and J. S. Vitter, Cylindrical static and kinetic binary space partitions, Proc. 13th ACM Sympos. Comput. Geom., 1997, pp. 39-- 48.

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P.K. Agarwal, L.J. Guibas, T.M. Murali, and J.S. Vitter. Cylindrical static and kinetic binary space partitions. In Proc. 13th Annu. ACM Sympos. Comput. Geom., pages 39-48, 1997.

No context found.

Agarwal, P.K., Guibas, L.J., Murali, T.M., Vitter, J.S. Cylindrical static and kinetic binary space partitions, Proc. 13th Annu. Symp. Comput. Geom. (1997), 3948.

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Agarwal, P., Guibas, L., Murali, T. and Vitter, J. 1997. "Cylindrical static and kinetic binary space partitions," in Proceedings of the 13th Annual Symposium on Computational Geometry, pp. 39-48.

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