P. K. Agarwal and S. Suri, Surface approximation and geometric partitions, Proc. 5th ACMSIAM Sympos. Discrete Algorithms, 1994, pp. 24--33.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....One way of solving the problem is to run a binary search on the number of vertices of the approximating surface. We then need to solve the decision problem of determining whether there exists an approximation with at most k vertices, for some given k. Unfortunately, this problem is NP Hard [21], so one seeks efficient techniques for computing an approximation of size (number of vertices) close to kOPT , where kOPT is the minimum size of an approximation. Although several ad hoc algorithms have been developed for computing an approximation [87, 88, 158, 159, 196] none of them ....

....function) In most of the applications, P is represented as a finite set of n points, sampled from the input surface, and the goal is to compute a polyhedral terrain Q with the minimum number of vertices, such that the vertical distance between any point of P and Q is at most . Agarwal and Suri [21] showed that this problem is NP Hard. They also gave a polynomial time algorithm for computing an approximation of size O(kOPT log kOPT ) by reducing the problem to a geometric set cover problem, but the running time of their algorithm is O(n 8 ) which is rather high. Agarwal and Desiken [6] ....

P. K. Agarwal and S. Suri, Surface approximation and geometric partitions, Proc. 5th ACMSIAM Sympos. Discrete Algorithms, 1994, pp. 24--33.

....choice of balls covers D. For the planar case, Drezner [92] gave an improved O(n ) time algorithm, which was subsequently improved by Hwang et al. 166] to n . Hwang et al. 165] have given another n time algorithm for computing a discrete p center. Recently, Agarwal and Procopiuc [12] presented an n time algorithm for computing a p center Of n points in R . Therefore, for a fixed value of p, the Euclidean p center (and also the Euclidean discrete p center) problem can be solved in polynomial time in any fixed dimension. However, either of these problems is NP Complete ....

....minfNN Dist(j) ffi(s i ; d j )g; return fs 1 ; s p g; Figure 3: Greedy algorithm for approximate p center. This algorithm works equally well for any metric and for the discrete p center problem. The running time was improved to O(n log p) by Feder and Green [118] Agarwal and Procopiuc [12] showed that, for any 0, a set S of p supply points with c(D; S) 1 )r can be computed in time O(n log p) p= Dyer and Freeze [103] modified the greedy algorithm to obtain an approximation algorithm for the weighted k center problem, in which a weight w(p) is associated with ....

P. K. Agarwal and C. M. Procopiuc, Exact and approximation algortihms for clustering, Proc. 9th ACM-SIAM Sympos. Discrete Algorithms, 1998, pp. 658--667.

.... of space widely used in several areas, including computer graphics (global illumination [7] shadow generation [10, 11] visibility determination [4, 28] and ray tracing [21] solid modeling [22, 20, 29] geometric data repair [19] robotics [5] network design [18] and surface simplification [3]. Key to Support was provided by National Science Foundation research grant CCR 93 01259, by Army Research Office MURI grant DAAH04 96 1 0013, by a Sloan fellowship, by a National Science Foundation NYI award and matching funds from Xerox Corp, and by a grant from the U.S. Israeli ....

P. K. Agarwal and S. Suri, Surface approximation and geometric partitions, Proc. 5th ACM-SIAM Sympos. Discrete Algorithms, 1994, pp. 24--33.

....function) In most of the applications, P is represented as a nite set of n points, sampled from the input surface, and the goal is to compute a polyhedral terrain Q with the minimum number of vertices, such that the vertical distance between any point of P and Q is at most . Agarwal and Suri [14] showed that this problem is NP Hard. Agarwal and Desiken [6] have shown that Clarkson s randomized algorithm can be extended to compute a polyhedral terrain of size O(k 2 OPT log 2 kOPT ) in expected time O(n 2 k 3 OPT log 3 kOPT ) The survey paper by Heckbert and Garland [96] ....

P. K. Agarwal and S. Suri, Surface approximation and geometric partitions, Proc. 5th ACMSIAM Sympos. Discrete Algorithms, 1994, pp. 24-33.

.... of space widely used in several areas, including computer graphics (global illumination [9] shadow generation [12, 13] visibility determination [4, 33] and ray tracing [24] solid modeling [27, 25, 34] geometric data repair [23] robotics [5] network design [21] and surface simpli cation [3]. Key to the BSP s success is that it serves both as a model for an object (or a set of objects) and as a data structure for querying the object. Informally, a BSP B for a set of objects is a binary tree, where each node v is associated with a convex region v . The regions associated with the ....

P. K. Agarwal and S. Suri, Surface approximation and geometric partitions, Proc. 5th ACM-SIAM Sympos. Discrete Algorithms, 1994, pp. 24-33.

....One way of solving the problem is to run a binary search on the number of vertices of the approximating surface. We then need to solve the decision problem of determining whether there exists an approximation with at most k vertices, for some given k. Unfortunately, this problem is NP Hard [20], so one seeks efficient techniques for computing an approximation of size (number of vertices) close to kOPT , where kOPT is the minimum size of an approximation. Although several ad hoc algorithms have been developed for computing an approximation [74, 75, 135, 136, 168] none of them ....

....function) In most of the applications, P is represented as a finite set of n points, sampled from the input surface, and the goal is to compute a polyhedral terrain Q with the minimum number of vertices, such that the vertical distance between any point of P and Q is at most . Agarwal and Suri [20] showed that this problem is NP Hard. They also gave a polynomial time algorithm for computing an approximation of size O(kOPT log kOPT ) by reducing the problem to a geometric set cover problem, but the running time of their algorithm is O(n 8 ) which is rather high. Agarwal and Desiken [6] ....

P. K. Agarwal and S. Suri, Surface approximation and geometric partitions, Proc. 5th ACMSIAM Sympos. Discrete Algorithms, 1994, pp. 24--33.

....function) In most of the applications, P is represented as a finite set of n points, sampled from the input surface, and the goal is to compute a polyhedral terrain Q with the minimum number of vertices, such that the vertical distance between any point of P and Q is at most . Agarwal and Suri [14] showed that this problem is NP Hard. Agarwal and Desikan [6] have shown that Clarkson s randomized algorithm can be extended to compute a polyhedral terrain of size O(k 2 OPT log 2 kOPT ) in expected time O(n 2 ffi k 3 OPT log 3 kOPT ) The survey paper by Heckbert and Garland [101] ....

P. K. Agarwal and S. Suri, Surface approximation and geometric partitions, Proc. 5th ACMSIAM Sympos. Discrete Algorithms, 1994, pp. 24--33.

....clustering problems. Previous results. Clustering has been widely studied in several areas of computer science, including database systems [24, 59, 28, 13, 7, 55, 38, 52] information retrieval, image processing, data compression, combinatorial optimization, and computational geometry; see [1, 4, 8, 56, 15, 16, 17, 23, 26, 13, 20, 28, 35, 54] and references therein for a small sample of results. Although many theoretical results are known on facility location and related clustering problems [26, 4, 56, 29, 17, 16] very few theoretical results are known on projective clustering, partly because the latter is a much harder problem. ....

P. K. Agarwal and C. M. Procopiuc, Exact and approximation algorithms for clustering, Proc. 9th ACM-SIAM Sympos. Discrete Algorithms, 1998, pp. 658--667.

.... with applications in many other problems global illumination [6] shadow 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 ....

P. K. Agarwal and S. Suri, Surface approximation and geometric partitions, Proc. 5th ACM-SIAM Sympos. Discrete Algorithms, 1994, pp. 24--33.

.... with applications in many other problems global illumination [5] shadow generation [7, 8, 9] visibility problems [3, 25] solid geometry [19, 20, 26] geometric data repair [16] ray tracing [18] robotics [4] and approximation algorithms for network design [15] and surface simpli cation [2]. Informally, a BSP B for a set of polygons in R 3 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 plane. The regions associated with the leaves of the tree form a convex decomposition ....

P. K. Agarwal and S. Suri, Surface approximation and geometric partitions, Proc. 5th ACM-SIAM Sympos. Discrete Algorithms, 1994, pp. 2433.

....One way of solving the problem is to run a binary search on the number of vertices of the approximating surface. We then need to solve the decision problem of determining whether there exists an approximation with at most k vertices, for some given k. Unfortunately, this problem is NP Hard [21], so one seeks efficient techniques for computing an approximation of size (number of vertices) close to kOPT , where kOPT is the minimum size of an approximation. Although several ad hoc algorithms have been developed for computing an approximation [88, 89, 159, 160, 197] none of them ....

....function) In most of the applications, P is represented as a finite set of n points, sampled from the input surface, and the goal is to compute a polyhedral terrain Q with the minimum number of vertices, such that the vertical distance between any point of P and Q is at most . Agarwal and Suri [21] showed that this problem is NP Hard. They also gave a polynomial time algorithm for computing an approximation of size O(kOPT log kOPT ) by reducing the problem to a geometric set cover problem, but the running time of their algorithm is O(n 8 ) which is rather high. Agarwal and Desiken [6] ....

P. K. Agarwal and S. Suri, Surface approximation and geometric partitions, Proc. 5th ACMSIAM Sympos. Discrete Algorithms, 1994, pp. 24--33.

....covers D. For the planar case, Drezner [93] gave an improved O(n 2p 1 ) time algorithm, which was subsequently improved by Hwang et al. 167] to n O( p p) Hwang et al. 166] have given another n O( p p) time algorithm for computing a discrete p center. Recently, Agarwal and Procopiuc [12] extended and simplified the technique by Hwang et al. 167] to obtain an n O(p 1 Gamma1=d ) time algorithm for computing a p center of n points in R d . Therefore, for a fixed value of p, the Euclidean p center (and also the Euclidean discrete p center) problem can be solved in polynomial ....

....minfNN Dist(j) ffi(s i ; d j )g; return fs 1 ; s p g; Figure 3: Greedy algorithm for approximate p center. This algorithm works equally well for any metric and for the discrete p center problem. The running time was improved to O(n log p) by Feder and Green [119] Agarwal and Procopiuc [12] showed that, for any 0, a set S of p supply points with c(D; S) 1 )r can be computed in time O(n log p) p= d ) O(p 1 Gamma1=d ) Dyer and Freeze [104] modified the greedy algorithm to obtain an approximation algorithm for the weighted p center problem, in which a weight w(p) ....

P. K. Agarwal and C. M. Procopiuc, Exact and approximation algortihms for clustering, Proc. 9th ACM-SIAM Sympos. Discrete Algorithms, 1998, pp. 658--667.

.... of space widely used in several areas, including computer graphics (global illumination [7] shadow generation [10, 11] visibility determination [4, 28] and ray tracing [21] solid modeling [22, 20, 29] geometric data repair [19] robotics [5] network design [18] and surface simplification [3]. Key to Support was provided by National Science Foundation research grant CCR 93 01259, by Army Research Office MURI grant DAAH04 96 1 0013, by a Sloan fellowship, by a National Science Foundation NYI award and matching funds from Xerox Corp, and by a grant from the U.S. Israeli ....

P. K. Agarwal and S. Suri, Surface approximation and geometric partitions, Proc. 5th ACM-SIAM Sympos. Discrete Algorithms, 1994, pp. 24--33.

.... with applications in many other problems global illumination [7] shadow 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 ....

P. K. Agarwal and S. Suri, Surface approximation and geometric partitions, Proc. 5th ACM-SIAM Sympos. Discrete Algorithms, 1994, pp. 24--33.

.... et al. 10] The BSP has been widely used in several areas, including computer graphics (global illumination [4] shadow generation [6, 7] visibility determination [3, 21] and ray tracing [15] solid modeling [16, 22] geometric data repair [12] network design [11] and surface simplification [2]. The BSP has A preliminary version of this paper appeared as a communication in the Proceedings of the 13th Annual ACM Symposium on Computational Geometry, 1997, pages 382 384. This author is affiliated with Brown University. Support was provided in part by National Science Foundation ....

P. K. Agarwal and S. Suri, Surface approximation and geometric partitions, Proc. 5th ACM-SIAM Sympos. Discrete Algorithms, 1994, pp. 24--33.

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