Gary L. Miller and Stephen A. Vavasis. Density graphs and separators. In Proceedings of the ACM-SIAM Symposium on Discrete Algorithms, pages 331--336, 1991. 32  Home/Search   Document Not in Database   Summary   Related Articles   Check

....based on the topology of the graph underlying the sparse matrix. It requires no geometric information. However, it is computationally quite demanding. The geometric partitioning technique by Miller et al. holds promise to be computationally less demanding than the spectral decomposition technique [24, 25]. Geometric information is typically available for meshes generated for the solution of partial differential equations, but may not be present in other applications. The results of applying the spectral bisection technique to two model problems are reported in [14, 15] and shown in Tables 2 and ....

Gary L. Miller and Stephen A. Vavasis. Density graphs and separators. In Second Annual ACMSIAM Symposium on Discrete Algorithms, pages 331--336. ACM Press, 1991.

....was obtained by Alon, Seymour, and Thomas [AST90] who showed that graphs that do not have an h clique minor have separators of O(h 3=2 p n) nodes. Plotkin, Rao, and Smith [PRS94] reduced the dependency on h from h 3=2 to h. Using geometric techniques, Miller, Teng, Thurston, and Vavasis [MT90, MTTV96a, MTTV96b, MTV91, MV91, Ten91] extended the planar separator theorem to graphs embedded in higher dimensions and showed that every wellshaped mesh in R d has a 1= d 2) separator of size O(n 1 Gamma1=d ) Using multicommodity flow, Leighton and Rao [LR88] designed a partitioning method guaranteed to return a cut whose ....

....the center of B and whose radius is larger by a multiplicative factor of ff. Overlap graphs are good models for well shaped meshes because each well shaped mesh in two, three, or higher dimensions is a bounded degree subgraph of some overlap graph (for suitable choices of the parameters ff and k) [MTTV96a, MTTW95, Ten96, MV91]. 5.2. Spherical Embeddings of Overlap Graphs In this section, we show that an ff overlap graph is a subgraph of the intersection graph obtained by projecting its neighborhoods onto the sphere and then dilating each by an O(ff) factor. By choosing the proper projection, we are able to use this ....

G. L. Miller and S. A. Vavasis. Density graphs and separators. In Proceedings of the ACM-SIAM Symposium on Discrete Algorithms, pages 331--336, 1991.

....information. However, it is computationally quite demanding. The geometric partitioning technique by Miller et al. holds promise to be computationally less demanding than the spectral decomposition technique, but relies on geometric information and geometric properties of the graph [MT90, MV91] Geometric information is typically available for meshes generated for the solution of partial differential equations, but may not be present in other applications. The geometric approach is currently being implemented as part of the CMSSL. Since we do not yet have much experience with the ....

Gary L. Miller and Stephen A. Vavasis. Density graphs and separators. In Second Annual ACM-SIAM Symposium on Discrete Algorithms, pages 331--336. ACM Press, 1991.

....CONTENTS 8 (c) a) b) Figure 9: The nodes of a finite element mesh (a) A 3 ply neighborhood systems for the nodes (b) The (1, 3) overlap graph for the mesh (c) these graphs have O(n (d;1) d )vertex separators 2 . In doing so, they extended results by Lipton and Tarjan [56] and others [59]. Meshes are considered well shaped if the angles and or aspect ratios of their elements are bounded within some values. Most of the meshes that are used in scientific simulations are well shaped according to this definition. Miller et al. used the concept of a neighborhood system to define an ....

G. Miller and S. Vavasis. Density graphs and separators. In Second Annual ACM-SIAM Symposium on Discrete Algorithms, pages 331--336, 1991.

....of the circumsphere to the shortest edge for and an element in the dual Delaunay triangulation must be bounded. The latter condition is called bounded radius aspect ratio. For our purposes, we will consider a mesh well shaped if it satisfies a density condition as defined by Miller and Vavasis [15]. Meshes with bounded element aspect ratio satisfy a density condition. Let G be a graph and let xyz be an embedding of its vertices in IR d . We say xyz is an embedding of density ff if the following inequality holds for all vertices v in G. Let w be the farthest vertex from v that is ....

G. L. Miller and S. A. Vavasis. Density graphs and separators. In Second Annual ACMSIAM Symposium on Discrete Algorithms, pages 331--336, San Francisco, January 1991. ACM-SIAM.

....no geometric information. However, it is computationally quite demanding. The geometric partitioning technique by Miller et al. holds promise to be computationally less demanding than the spectral decomposition technique, but relies on geometric information and geometric properties of the graph [34, 35]. Geometric information is typically available for meshes generated for the solution of partial differential equations, but may not be present in other applications. Since we do not yet have much experience with the geometric partitioning technique we only report the results from the spectral ....

Gary L. Miller and Stephen A. Vavasis. Density graphs and separators. In Second Annual ACMSIAM Symposium on Discrete Algorithms, pages 331--336. ACM Press, 1991.

....was obtained by Alon, Seymour, and Thomas [AST90] who showed that graphs that do not have an h clique minor have separators of O(h 3=2 p n) nodes. Plotkin, Rao, and Smith [PRS94] reduced the dependency on h from h 3=2 to h. Using geometric techniques, Miller, Teng, Thurston, and Vavasis [MT90, MTTV96a, MTTV96b, MTV91, MV91, Ten91] extended the planar separator theorem to graphs embedded in higher dimensions and showed that every well shaped mesh in R d has a 1= d 2) separator of size O(n 1 Gamma1=d ) Using multicommodity flow, Leighton and Rao [LR88] designed a partitioning method guaranteed to return a cut whose ....

....elements, so we can not analyze it as we do a triangulation. In general, the derivative of the conformal transformation must vary gradually with respect to the mesh size in order to produce good results (See, for example [TWM85] This means that the mesh will probably satisfy a density condition [BB87, MV91]. Let G be an undirected graph and let be an embedding of its nodes in R d . We say is an embedding of density ff if the following inequality holds for all vertices v in G: Let u be the node closest to v. Let w be the node farthest from v that is connected to v by an edge. Then jj (w) ....

[Article contains additional citation context not shown here]

Gary L. Miller and Stephen A. Vavasis. Density graphs and separators. In Proceedings of the ACM-SIAM Symposium on Discrete Algorithms, pages 331--336, 1991.

.... chordal graphs [6] grid graphs [16, 23] hypercubes [9] several interconnection graphs for parallel computation [10, 12, 13] graphs of bounded genus [2, 5] graphs that exclude a given minor [1] and some geometrically defined classes that include many three dimensional finite element graphs [19, 20, 21, 22]. In some applications it is useful to associate weights with the vertices of the graph. For example, weights can model different amounts of work in parts of a computation to be divided among parallel processors, or different sizes of circuit elements to be laid out. One may wish to divide a graph ....

Gary L. Miller and Steven A. Vavasis. Density graphs and separators. In Proceedings of the Second Annual ACM-SIAM Symposium on Discrete Algorithms, pages 331--336. SIAM, 1991.

....also given slower but more work efficient parallel algorithms for finding O( p n) planar separators. Recently Klein [66] has given a more work efficient (but not polylog time) algorithm for finding planar separators. Algorithms for finding small separators in more general graphs are given in [4,78,79,80,99]. 1.4.2 One way separators The separators we have examined so far separate the underlying undirected graph. All they guarantee is that any path from one side to the other must pass through the separator. They do not preclude a path from going back and forth many times across the separator. We now ....

G. L. Miller and S. A. Vavasis, "Density graphs and separators," Proc. 2nd Annual ACM-SIAM Symposium on Discrete Algorithms (1991).

....of the circumsphere to the shortest edge of any element in the dual Delaunay triangulation must be bounded. The latter condition is called bounded radius aspect ratio. For our purposes, we will consider a mesh well shaped if it satisfies a density condition as defined by Miller and Vavasis [16]. Meshes with bounded element aspect ratio satisfy a density condition. Let G be a graph and let xyz be an embedding of its vertices in IR d . We say xyz is an embedding of density ff if the following inequality holds for all vertices v in G. Let w be the farthest vertex from v that is connected ....

G. L. Miller and S. A. Vavasis. Density graphs and separators. In Second Annual ACMSIAM Symposium on Discrete Algorithms, pages 331--336, San Francisco, January 1991. ACM-SIAM.

....no geometric information. However, it is computationally quite demanding. The geometric partitioning technique by Miller et al. holds promise to be computationally less demanding than the spectral decomposition technique, but relies on geometric information and geometric properties of the graph [55, 56]. Geometric information is typically available for meshes generated for the solution of partial differential equations, but may not be present in other applications. The RSB technique has been used to partition the following five tetrahedral meshes on five CM 5 systems of different sizes [22] ....

Gary L. Miller and Stephen A. Vavasis. Density graphs and separators. In Second Annual ACM-SIAM Symposium on Discrete Algorithms, pages 331-- 336. ACM Press, 1991.

....in BE . 4) jBO j = O(ffk 1=d n 1 Gamma1=d ) S is called a sphere separator, and such a separator can be found in random linear time sequentially and in random constant time, using n processors. A special case of the overlap graph is the density graph (first introduced by Miller and Vavasis [28]) The density condition of an embedding is important for finite difference methods. Let G be an undirected graph and let be an embedding of its nodes in IR d . We say is an embedding of G of density ff if the following inequality holds for all vertices v in G. Let u be the closest node to v. ....

G. L. Miller and S. A. Vavasis. Density graphs and separators. In Second Annual ACM-SIAM Symposium on Discrete Algorithms, 331--336, 1991. ACMSIAM.

.... efficiently computing a small separator of the underlying graph [6, 7, 9, 10, 11, 18, 20, 28, 29, 30] By small separator we mean a relatively small subset of vertices whose removal divides the rest of the graph into two disconnected pieces of approximately equal size [19] In a series of papers [25, 27, 22, 26, 23, 30, 34], Miller, Teng, Thurston, and Vavasis have developed a geometric characterization of graphs (embedded in some fixed dimension) that have a small separator. Their characterization is based on a notion of a k ply neighborhood system, which is a collection of n balls in a fixed dimensional space ....

....the following theorem. Theorem 2.2 (Geometric Separators) Suppose Phi = fB 1 ; B n g is a k ply neighborhood system in IR d . Then there is an O(k 1=d n 1 Gamma1=d ) sphere separator S that (d 1) d 2) splits Phi. We now reviews some of the basic concepts and lemmas used in [25, 26, 23, 27, 30] for proving Theorem 2.2. Proofs to lemmas in this section can be found in [23, 30] A density function in IR d is a real valued nonnegative function f(x) defined on IR d such that f k is integrable for all k = 1; 2; 3; The f measure of a (d Gamma 1) dimensional sphere S, denoted by ....

Miller, G. L. and S. A. Vavasis. "Density graphs and separators". In Second Annual ACM-SIAM Symposium on Discrete Algorithms, pages 331--336, San Fransciso, January 1991. ACM-SIAM.

....2. Mesh partitioning. The main problem considered in this work is: given a mesh, cut it into pieces for the purpose of domain decomposition, nested dissection, and parallel computing. Vavasis initially considered this problem in a single author paper [113] and a series of joint papers followed [79 81], which steadily improved the method, making it more powerful and general. The basic method for solving the mesh partitioning problem is to re embed the mesh in a certain manner in R d and the cut it with a simple shape such as a hypersphere. It turns out that this simple, efficient approach ....

G. L. Miller and S. A. Vavasis. Density graphs and separators. In Proceedings of the SIAM-ACM Symposium on Discrete Algorithms, 1991.

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Gary L. Miller and Stephen A. Vavasis. Density graphs and separators. In Proceedings of the ACM-SIAM Symposium on Discrete Algorithms, pages 331--336, 1991. 32

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