P. Agarwal, E. Grove, T. Murali, and J. Vitter, Binary space partitions for fat rectangles, in Proceedings of the 37th Annual Symposium on Foundations of Computer Science, IEEE Computer Society Press, Los Alamitos, CA, 1996, pp. 482--491.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....in [29] in their seminal paper; subsequent improvements have led to the previous best upper bound of 4n [13, 8] Our result above improves this by a factor of 3 in general, and a factor of 2 in the tiling case. Lately, there has been a great interest in BSPs and their applications in general [1, 10, 11] and in BSPs for rectangles in particular [11] Since we proved this upper bound for the tiling case in a preliminary writeup [4] Dumitrescu et al. 11] proved a lower bound of 2n Gamma o(n) matching our upper bound within a lower order term . We use our BSP results in two ways for tiling ....

....algorithm, while being polynomial Tradeoff between the constants 2 and 4 is also possible; see Theorem 4.4 for details. There has been a considerable amount of work of late in proving efficient solutions for geometric problems where the objects are uniform in some local sense only (e.g. see [1, 9]) However, it is easy to see that our technique will not provide any better solutions if we instead assume that the aspect ratios of individual tiles are bounded. in N , is too prohibitive to be practical for small values of . This result however is of theoretical significance: the dual of ....

[Article contains additional citation context not shown here]

P. Agarwal, E. Grove, T. Murali and J. Vitter. Binary space partitions for fat rectangles. Proc. Symp. Found. of Comp. Sci. (FOCS), 1996.

....case Theta(n) However, most of randomly created BSP trees have reasonable behavior for practical scenes. Their sizes are considerably smaller than the worst case determined boundary. Modern algorithms try to use these properties to construct nearly linear BSP trees. Pankaj K. Agarwal et al. [1] solved the problem of a construction of BSP tree for a set of fat orthogonal rectangles (the fat objects are intuitive objects without extremely skinny and long 2 parts) Their algorithm creates BSP trees of n2 O( p log n) size for scene of n fat rectangles and of n p m2 O( p log n) ....

....number ( of other segments in our algorithm. For algorithm s intentions, we suppose that any object (hyperrectangle) has extended low directional density dened in the next part of this paper. We believe, that many realistic scenes t to this condition. 2 We take up the denition of Agarwal [1] 4 free cut l s 1 s l 2 = 1 ( ffi) free cut l 1 s l ffi l ffi Figure 1: Free cut Denition 2.2: Let r be a rectangle in the space with vertices r[X j ] j 2 f1; 4g and side vectors r u = X 2 Gamma X 1 and r v = X 3 Gamma X 2 , as you can see on the gure 2. Point C ....

P. K. Agarwal, E. F. Grove, T. M. Murali, and J. S. Vitter. Binary space partitions for fat rectangles. In Proc. 37th Annu. IEEE Sympos. Found. Comput. Sci., pages 482491, October 1996.

....worst case #(n) However, most of randomly created BSP trees have reasonable behavior for practical scenes. Their sizes are considerably smaller than the worst case determined boundary. Modern algorithms try to use these properties to construct nearly linear BSP trees. Pankaj K. Agarwal et al. [1] solved the problem of a construction of BSP tree for a set of fat orthogonal rectangles (the fat objects are intuitive objects without extremely skinny and long 2 parts) Their algorithm creates BSP trees of n2 O( # log n) size for scene of n fat rectangles and of n # m2 O( # log n) ....

....number (#) of other segments in our algorithm. For algorithm s intentions, we suppose that any object (hyperrectangle) has extended low directional density defined in the next part of this paper. We believe, that many realistic scenes fit to this condition. 2 We take up the definition of Agarwal [1] 4 free cut l s 1 s l 2 # =1 (#, #) free cut l 1 s l # # l # # Figure 1: Free cut Definition 2.2: Let r be a rectangle in the space with vertices r[X j ] j # 1, 4 and side vectors #r u = X 2 X 1 and #r v = X 3 X 2 , as you can see on the figure 2. Point C be center of ....

P. K. Agarwal, E. F. Grove, T. M. Murali, and J. S. Vitter. Binary space partitions for fat rectangles. In Proc. 37th Annu. IEEE Sympos. Found. Comput. Sci., pages 482--491, October 1996.

....that are placed so as to divide the set of points associated with a node more or less in half. Such trees have excellent depth properties, in that their depth is always O(log n) Unfortunately, since the objects in S are points, which are not themselves fat (as with the sets of objects studied in [2, 12, 21]) the regions associated with vertices in a k d tree can have arbitrarily large aspect ratios. This unbounded aspect ratio property of k d trees results in worst case running times for most searching problems, such as nearest neighbor searching, that are Omega Gamma n) even for approximation ....

P. K. Agarwal, E. F. Grove, T. M. Murali, and J. S. Vitter. Binary space partitions for fat rectangles. In Proc. 37th Annu. IEEE Sympos. Found. Comput. Sci., pages 482--491, Oct. 1996.

....case Theta(n) However, most of randomly created BSP trees have reasonable behaviour for practical scenes. Their size are considerably smaller than the worst case determined boundary. Modern algorithms try to use these properties to construct nearly linear BSP trees. Pankaj K. Agarwal et al. [Agarw96] solve the problem of a construction of BSP tree for a set of fat orthogonal rectangles (the fat objects are intuitive objects without extremely skinny and long parts) Their algorithm creates BSP trees of n2 O( p log n) size for scene of n fat rectangles and of n p m2 O( p log n) size ....

....Denition 3: Let S be a set of segments in the plane, s i 2 S be a segment with endpoints s i [X] a s i [Y ] In additional let Omega 1 = Omega Gamma 1; s i [X] Omega 2 = Omega Gamma 1; s i [Y ] and be a integer constant. We say that segment s i is: 2 We take up the denition of Agarwal [Agarw96] 4 ffl free, if 8j 2 f1; 2g : Omega j S = ffl free, if 8j 2 f1; 2g : j Omega j Sj Denition 4: We say that a segment s has ( ffi) low directional density, if the following holds: j Omega Gamma ffi; s[X] Sj ) j Omega Gamma ffi; s[Y ] Sj ) whereas is a integer ....

P. K. Agarwal, E. F. Grove, T. M. Murali, and J. S. Vitter. Binary space partitions for fat rectangles. In Proc. 37th Annu. IEEE Sympos. Found. Comput. Sci., pages 482491, October 1996.

....worst case #(n) However, most of randomly created BSP trees have reasonable behaviour for practical scenes. Their size are considerably smaller than the worst case determined boundary. Modern algorithms try to use these properties to construct nearly linear BSP trees. Pankaj K. Agarwal et al. [Agarw96] solve the problem of a construction of BSP tree for a set of fat orthogonal rectangles (the fat objects are intuitive objects without extremely skinny and long parts) Their algorithm creates BSP trees of n2 O( # log n) size for scene of n fat rectangles and of n # m2 O( # log n) size ....

....s i [X] # ## #, s i [Y ] Definition 3: Let S be a set of segments in the plane, s i # S be a segment with endpoints s i [X] a s i [Y ] In additional let# 1=# (#,s i [X] # 2 = ## #,s i [Y] and # be a integer constant. We say that segment s i is: 2 We take up the definition of Agarwal [Agarw96] 4 . free, if #j # 1,2 :# j #S=#. # free, if #j # 1,2 : # j #S # Definition 4: We say that a segment s has (#, #) low directional density, if the following holds: ## #, s[X] #S ##)#( ## #, s[Y ] #S ##) whereas # is a integer constant and # is a real constant. We say, that a set of ....

P. K. Agarwal, E. F. Grove, T. M. Murali, and J. S. Vitter. Binary space partitions for fat rectangles. In Proc. 37th Annu. IEEE Sympos. Found. Comput. Sci., pages 482--491, October 1996.

....that are placed so as to divide the set of points associated with a node more or less in half. Such trees have excellent depth properties, in that their depth is always O(log n) Unfortunately, since the objects in S are points, which are not themselves fat (as with the sets of objects studied in [1, 12, 21]) the regions associated with vertices in a k d tree can have arbitrarily large aspect 2 ratios. This unbounded aspect ratio property of k d trees partly accounts for why there are few simple theoretical results better than the O(n 1 Gamma1=d ) average running time, even for approximation ....

P. K. Agarwal, E. F. Grove, T. M. Murali, and J. S. Vitter, Binary space partitions for fat rectangles, in Proc. 37th Annu. IEEE Sympos. Found. Comput. Sci., Oct. 1996, pp. 482--491.

....structure. If these quantities are comparable, the kinetic structure is said to be efficient. See [5, 13, 15] for some other models for addressing problems involving motion. In the context of constrained subdivisions, the kinetic approach has been successfully applied by Agarwal et al. [3] for maintaining a valid BSP of segments moving in the plane. This work is extended to triangles moving in space in [2] Basch et al. 7] obtain kinetic solutions to the problem of collision detection between two polygons by maintaining a pseudo triangulation of their convex hull. In both ....

....polygons by maintaining a pseudo triangulation of their convex hull. In both cases, however, the notion of efficiency is not clear, as there is no canonical discrete attribute against which to compare the performance of the kinetic data structures. In the context of kinetic BSPs, Agarwal et al. [3] specify a static algorithm and ensure that at every moment the BSP they maintain is precisely the same as the one that the static algorithm would have computed for the current position of the input objects. They show that for any pseudo algebraic motion, the number of events is quadratic in the ....

P. K. Agarwal, E. F. Grove, T. M. Murali, and J. S. Vitter. Binary space partitions for fat rectangles. In Proc. 37th Annu. IEEE Sympos. Found. Comput. Sci., 1996, 482--491.

....size O(n ) in time O(n They also show that their algorithm can be made deterministic without affecting its asymptotic running time. It has been an open problem whether a BSP for n triangles in R can be constructed in near quadratic time. Sub quadratic bounds are known for special cases [2, 14, 25]. However, none of these approaches lead to a near quadratic algorithm for triangles in R . We present a randomized algorithm (in Section 4) that constructs a BSP for S of expected size O(n in O(n The bottleneck in analyzing the expected running time of the Paterson Yao algorithm is ....

P. K. Agarwal, E. F. Grove, T. M. Murali, and J. S. Vitter, Binary space partitions for fat rectangles, Proceedings of the 37th IEEE Annual Symposium on foundations of Computer Science (FOCS '96), October 1996.

....It has been an open problem whether a BSP for n triangles in R 3 can be constructed in near quadratic time. Sub quadratic bounds are known for special cases: Paterson and Yao s algorithm for orthogonal rectangles [29] de Berg s result for fat polyhedra [16] and the technique of Agarwal et al. [2] for fat orthogonal rectangles. However, none of these approaches lead to a 2 near quadratic time algorithm for triangles in R 3 . The bottleneck in analyzing the expected running time of the Paterson Yao algorithm is that no nontrivial bound is known on the number of vertices in the convex ....

P. K. Agarwal, E. F. Grove, T. M. Murali, and J. S. Vitter, Binary space partitions for fat rectangles, Proc. 37th Annu. IEEE Sympos. Found. Comput. Sci., October 1996, pp. 482-491.

....structure. If these quantities are comparable, the kinetic structure is said to be efficient. See [5, 13, 15] for some other models for addressing problems involving motion. 1 ) In the context of constrained subdivisions, the kinetic approach has been successfully applied by Agarwal et al. [3] for maintaining a valid BSP of segments moving in the plane. This work is extended to triangles moving in space in [2] Basch et al. 7] obtain kinetic solutions to the problem of collision detection between two polygons by maintaining a pseudo triangulation of their convex hull. In both ....

....polygons by maintaining a pseudo triangulation of their convex hull. In both cases, however, the notion of efficiency is not clear, as there is no canonical discrete attribute against which to compare the performance of the kinetic data structures. In the context of kinetic BSPs, Agarwal et al. [3] specify a static algorithm and ensure that at every moment the BSP they maintain is precisely the same as the one that the static algorithm would have computed for the current position of the input objects. They show that for any pseudo algebraic motion, the number of events is quadratic in the ....

P. K. Agarwal, E. F. Grove, T. M. Murali, and J. S. Vitter. Binary space partitions for fat rectangles. In Proc. 37th Annu. IEEE Sympos. Found. Comput. Sci., pages 482--491, Oct. 1996.

....structure. If these quantities are comparable, the kinetic structure is said to be ecient. See [5, 13, 15] for some other models for addressing problems involving motion. 1 ) In the context of constrained subdivisions, the kinetic approach has been successfully applied by Agarwal et al. [3] for maintaining a valid BSP of segments moving in the plane. This work is extended to triangles moving in space in [2] Basch et al. 7] obtain kinetic solutions to the problem of collision detection between two polygons by maintaining a pseudo triangulation of their convex hull. In both ....

....two polygons by maintaining a pseudo triangulation of their convex hull. In both cases, however, the notion of eciency is not clear, as there is no canonical discrete attribute against which to compare the performance of the kinetic data structures. In the context of kinetic BSPs, Agarwal et al. [3] specify a static algorithm and ensure that at every moment the BSP they maintain is precisely the same as the one that the static algorithm would have computed for the current position of the input objects. They show that for any pseudo algebraic motion, the number of events is quadratic in the ....

P. K. Agarwal, E. F. Grove, T. M. Murali, and J. S. Vitter. Binary space partitions for fat rectangles. In Proc. 37th Annu. IEEE Sympos. Found. Comput. Sci., pages 482-491, Oct. 1996.

....Email: pankaj,tmax,jsv cs.duke.edu Abstract We present the rst systematic comparison of the performance of algorithms that construct Binary Space Partitions for orthogonal rectangles in R 3 . We compare known algorithms with our implementation of a recent algorithm of Agarwal et al. [1]. We show via an empirical study that their algorithm constructs BSPs of near linear size in practice and performs better than most of the other algorithms in the literature. Support was provided by National Science Foundation research grant CCR9301259, by Army Research OOEce MURI grant ....

....orthogonal bounding boxes [12] Paterson and Yao [22] show that a BSP of size O(n p n) can be constructed for n non intersecting, orthogonal rectangles in R 3 . This bound is also optimal in the worst case. If all but m of the rectangles have aspect ratio bounded by a constant, Agarwal et al. [1] describe an algorithm that constructs a BSP of size n p m2 O( p log n ) A related result of de Berg shows that a BSP of linear size can be constructed for fat polyhedra in R d [10] We have implemented the recent algorithm of Agarwal et al. 1] to study its performance on irealj data ....

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P. K. Agarwal, E. F. Grove, T. M. Murali, and J. S. Vitter, Binary space partitions for fat rectangles, Proc. of the 37th Annual IEEE Symp. on the Found. of Comp. Sci., 1996, pp. 482491.

....O(n 2 ) in time O(n 3 ) They also show that their algorithm can be made deterministic without affecting its asymptotic running time. It has been an open problem whether a BSP for n triangles in R 3 can be constructed in near quadratic time. Sub quadratic bounds are known for special cases [2, 14, 25]. However, none of these approaches lead to a near quadratic algorithm for triangles in R 3 . We present a randomized algorithm (in Section 4) that constructs a BSP for S of expected size O(n 2 ) in O(n 2 log 2 n) time. The bottleneck in analyzing the expected running time of the ....

P. K. Agarwal, E. F. Grove, T. M. Murali, and J. S. Vitter, Binary space partitions for fat rectangles, Proceedings of the 37th IEEE Annual Symposium on foundations of Computer Science (FOCS '96), October 1996.

....fpankaj,tmax,jsvg cs.duke.edu Abstract We present the first systematic comparison of the performance of algorithms that construct Binary Space Partitions for orthogonal rectangles in R 3 . We compare known algorithms with our implementation of a variant of a recent algorithm of Agarwal et al. [1]. We show via an empirical study that their algorithm constructs BSPs of near linear size in practice and performs better than most of the other algorithms in the literature. 1 Introduction The Binary Space Partition (BSP) 3, 5] is a versatile and popular data structure, with applications in ....

.... Introduction The Binary Space Partition (BSP) 3, 5] is a versatile and popular data structure, with applications in many problems hidden surface removal, global illumination, shadow generation, solid geometry, geometric data repair, ray tracing, network design, and surface simplification; see [1] for a detailed list of references. The efficiency of most BSP based algorithms depends on the size and or the depth of the BSP (we formally define the size of a BSP later) Therefore, several algorithms have been developed to construct BSPs of small size and depth; see [1] for a list of ....

[Article contains additional citation context not shown here]

P. K. Agarwal, E. F. Grove, T. M. Murali, and J. S. Vitter, Binary space partitions for fat rectangles, Proceedings of the 37th IEEE Annual Symposium on foundations of Computer Science (FOCS '96), October 1996.

....on the size of a BSP were developed by Paterson and Yao. They show that a BSP of size Theta(n 2 ) can be constructed for n disjoint triangles in R 3 [17] and that a BSP of size Theta(n p n) can be constructed for n non intersecting, orthogonal rectangles in R 3 [18] Agarwal et al. [1] consider the problem of constructing BSPs for fat rectangles. A rectangle is said to be fat if its aspect ratio is at most ff, for some constant ff 1; otherwise, it is said to be thin. If m rectangles are thin and the rest are fat, they present an algorithm that constructs a BSP of size n p ....

....First, we develop and implement a simple technique for constructing a BSP for orthogonal rectangles in R 3 . Our algorithm has the useful property that it tunes its performance to the geometric structure present in the input, e.g. the aspect ratios of the input rectangles. While Agarwal et al. [1] use similar ideas, our algorithm is considerably simpler than theirs and is much more easy to implement. Moreover, our 1 For each internal node v, we store the description of the polytope Rv . However, if v is a leaf, we do not store Rv , since it is completely defined by Rw and the cutting ....

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P. K. Agarwal, E. F. Grove, T. M. Murali, and J. S. Vitter, Binary space partitions for fat rectangles, Proc. 37th Annu. IEEE Sympos. Found. Comput. Sci., October 1996, pp. 482--491.

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P. Agarwal, E. Grove, T. Murali, and J. Vitter, Binary space partitions for fat rectangles, in Proceedings of the 37th Annual Symposium on Foundations of Computer Science, IEEE Computer Society Press, Los Alamitos, CA, 1996, pp. 482--491.

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