## Towards a tighter coupling of bottom-up and top-down sparse matrix ordering methods (2001)

Venue: | BIT |

Citations: | 29 - 0 self |

### BibTeX

@ARTICLE{Schulze01towardsa,

author = {Jürgen Schulze},

title = {Towards a tighter coupling of bottom-up and top-down sparse matrix ordering methods},

journal = {BIT},

year = {2001},

volume = {41},

pages = {2001}

}

### Years of Citing Articles

### OpenURL

### Abstract

Most state-of-the-art ordering schemes for sparse matrices are a hybrid of a bottom-up method such as minimum degree and a top down scheme such as George's nested dissection. In this paper we present an ordering algorithm that achieves a tighter coupling of bottom-up and topdown methods. In our methodology vertex separators are interpreted as the boundaries of the remaining elements in an unfinished bottom-up ordering. As a consequence, we are using bottomup techniques such as quotient graphs and special node selection strategies for the construction of vertex separators. Once all separators have been found, we are using them as a skeleton for the computation of several bottom-up orderings. Experimental results show that the orderings obtained by our scheme are in general better than those obtained by other popular ordering codes.

### Citations

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Citation Context ...ioning phase). Finally, the edge separator is projected backwards to the next larger graph in the sequence until G is reached (uncoarsening phase). A local improvement heuristic such as Kernighan-Lin =-=[39]-=- or Fiduccia-Mattheyses [22] is used to refine the edge separator after each uncoarsening step. The software packages CHACO [32], METIS [38], and PARTY [52] provide a variety of methods for the coarse... |

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Citation Context ...ecomposition approach. The next section discusses improving a separator. Multilevel method In recent years, multilevel algorithms have been applied successfully to the construction of edge separators =-=[15, 33, 37]-=-. Roughly speaking, a multilevel algorithm consists of three phases. In the first phase the original graph G is approximated by a sequence of smaller graphs that maintain the essential properties of G... |

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Citation Context ...iciency of all vertices that are adjacent to a vertex in adj G k (v). Furthermore, it was thought that minimum deficiency would only marginally improve the ordering quality compared to minimum degree =-=[18]-=-. However, recent studies have shown that much better orderings can be obtained from minimum deficiency [47, 48, 56]. In order to reduce the runtime of the minimum deficiency algorithm, Rothberg and E... |

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Citation Context ...ecomposition approach. The next section discusses improving a separator. Multilevel method In recent years, multilevel algorithms have been applied successfully to the construction of edge separators =-=[15, 33, 37]-=-. Roughly speaking, a multilevel algorithm consists of three phases. In the first phase the original graph G is approximated by a sequence of smaller graphs that maintain the essential properties of G... |

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Citation Context ...edge separator is projected backwards to the next larger graph in the sequence until G is reached (uncoarsening phase). A local improvement heuristic such as Kernighan-Lin [39] or Fiduccia-Mattheyses =-=[22]-=- is used to refine the edge separator after each uncoarsening step. The software packages CHACO [32], METIS [38], and PARTY [52] provide a variety of methods for the coarsening, partitioning, and unco... |

302 | The University of Florida sparse matrix collection
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Citation Context ...uropean space shuttle. The last two matrices (CYL3 and DIME20) have been given to us by C. Walshaw at University of Southampton. All other matrices can be found in Tim Davis' sparse matrix collection =-=[17]-=-. Most of these matrices have been extracted from commercial structural analysis and computational fluid dynamics applications. Table 4.1 reports relevant statistics about our test matrices. The first... |

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Citation Context ...a variety of methods for the coarsening, partitioning, and uncoarsening phase. For the computation of a nested dissection ordering we need vertex separators. Several nested dissection implementations =-=[11, 15]-=- first find an edge separator via a multilevel algorithm and then derive a vertex separator from the edge separator using a matching technique that is described in section 2.3.2. However, the size of ... |

276 | MeTiS – A Software Package for Partitioning Unstructured Graphs, Partitioning Meshes, and Computing Fill-Reducing Orderings of Sparse Matrices – Version 4.0
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Citation Context ... It is well recognized that the quality of the orderings produced by bottom-up and top-down schemes is not uniformly good [9]. Therefore, most state-of-the-art ordering codes such as BEND [34], METIS =-=[38]-=-, SCOTCH [50], SPOOLES [4], and WGPP [30] are using a hybrid of both schemes. The main contribution of this paper is to present an ordering scheme that achieves a tighter coupling of bottom-up and top... |

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Citation Context ...amount of degree updates and leads to a significant acceleration of the minimum degree algorithm. The efficiency of the minimum degree algorithm can be further improved when using approximate degrees =-=[1, 29]-=- rather than exact degrees. Once a vertex v has been eliminated from G k 1 , the new degree of a vertex u 2 adj G k 1 (v) is estimated by an upper bound. The upper bound is much less expensive to comp... |

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Citation Context ...geneous. In figure 3.4 the scatter plot on the left shows that 83 % of the vertices have a degree of 26. The second matrix (BCSSTK25) belongs to the well-known Harwell-Boeing sparse matrix collection =-=[19]-=- and represents the finite element model of a tall skyscraper. In contrast to CFD1 the graph of BCSSTK25 is more heterogeneous. The scatter plot on the right of figure 3.4 shows that 14 % of the verti... |

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Citation Context ...he root. Note that any vertex v that is not adjacent to any element becomes a leaf of the tree. The tree is called elimination tree and plays an important role in the context of sparse direct solvers =-=[20, 21, 45, 57]-=-. In section 3 we consider quotient graphs that have a more general structure, i. e. there is no oneto -one relation between a variable of G and an uneliminated vertex of G. Each variable represents a... |

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Citation Context ...x L can be reduced significantly by reordering the columns and rows of A prior to factorization. Since the computation of a minimum fill ordering for general sparse matrices is an NP-complete problem =-=[59-=-], much effort has been devoted to the development of powerful heuristic algorithms. All heuristics are based on the observation that a symmetric n n matrix A can be interpreted as the adjacency matr... |

177 |
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Citation Context ...in the ordering and is eliminated from G k 1 to form the graph G k . The whole selection/elimination process is then repeated for G k . Another effective method for reducing fill is nested dissection =-=[23, 24]-=-. The method starts with computing a vertex separator S in G. All vertices in S are ordered after those in G(V S). The method is recursively applied to each component of G(V S) until a component This ... |

163 |
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Citation Context ...he root. Note that any vertex v that is not adjacent to any element becomes a leaf of the tree. The tree is called elimination tree and plays an important role in the context of sparse direct solvers =-=[20, 21, 45, 57]-=-. In section 3 we consider quotient graphs that have a more general structure, i. e. there is no oneto -one relation between a variable of G and an uneliminated vertex of G. Each variable represents a... |

138 |
Modification of the minimum-degree algorithm by multiple elimination
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Citation Context ...weight jIj. This reduces the size of the elimination graphs and, therefore, the runtime of the minimum degree algorithm. Closely related to the concept of supernodes is the notion of external degrees =-=[42]-=-. With our notations, the external degree of a vertex v in supernode I is j adj G k (I)j. Instead of using true degrees, the vertex to be eliminated next is selected according to its external degree. ... |

129 | Sparse matrices in MATLAB: design and implementation
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(Show Context)
Citation Context ...amount of degree updates and leads to a significant acceleration of the minimum degree algorithm. The efficiency of the minimum degree algorithm can be further improved when using approximate degrees =-=[1, 29]-=- rather than exact degrees. Once a vertex v has been eliminated from G k 1 , the new degree of a vertex u 2 adj G k 1 (v) is estimated by an upper bound. The upper bound is much less expensive to comp... |

126 |
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(Show Context)
Citation Context ...s have been proposed to the basic algorithm that have greatly improved its efficiency. In the following we briefly describe some of these enhancements. For a detailed survey the reader is referred to =-=[27]-=-. 3 Perhaps one of the most important enhancements is the concept of supernodes. Two vertices u; v of an elimination graph G k belong to the same supernode, if adj G k (u)[fug = adj G k (v)[fvg. In th... |

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Citation Context ...e adjacency matrix of an undirected graph G = (V; E), E V V . In this graph theoretic context an ordering (or labeling) is a bijection : V 7! f1; : : : ; ng. As observed by Parter [49] and Rose [53], the column-wise factorization of A can be modeled by a sequence of elimination graphs G k , 1 k n. When column k is factorized, G k is obtained from G k 1 (G 0 = G) by applying the following mod... |

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Citation Context ...rs of v k are pairwise connected (the neighbors of v k form a clique in G k ). Each edge added in step (2) corresponds to a fill-entry in L. One of the most popular ordering schemes is minimum degree =-=[46, 58]-=-. At stage k the basic minimum degree algorithm selects a vertex with minimum degree in G k 1 . This vertex is numbered next in the ordering and is eliminated from G k 1 to form the graph G k . The wh... |

86 |
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(Show Context)
Citation Context ...rs of v k are pairwise connected (the neighbors of v k form a clique in G k ). Each edge added in step (2) corresponds to a fill-entry in L. One of the most popular ordering schemes is minimum degree =-=[46, 58]-=-. At stage k the basic minimum degree algorithm selects a vertex with minimum degree in G k 1 . This vertex is numbered next in the ordering and is eliminated from G k 1 to form the graph G k . The wh... |

82 |
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(Show Context)
Citation Context ...ncoarsening phase). A local improvement heuristic such as Kernighan-Lin [39] or Fiduccia-Mattheyses [22] is used to refine the edge separator after each uncoarsening step. The software packages CHACO =-=[32]-=-, METIS [38], and PARTY [52] provide a variety of methods for the coarsening, partitioning, and uncoarsening phase. For the computation of a nested dissection ordering we need vertex separators. Sever... |

75 |
Über Graphen und ihre Anwendung auf Determinantentheorie und Mengenlehre
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Citation Context ...ximum matching is determined in H(S;B S ). The maximum matching is then used to construct a vertex cover so that every matching edge is incident to exactly one vertex of the cover. According to Konig =-=[40]-=- the 7 S 0 S 1 S 2 S 3 S 4 Fig. 2.1: Topmost separators of a rectangular grid. S 0 is the first separator, followed by the separators S 1 , S 2 , and S 3 , S 4 . vertex cover must have minimum cardina... |

58 |
A heuristic for reducing fill-in in sparse matrix factorization
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(Show Context)
Citation Context ...ecomposition approach. The next section discusses improving a separator. Multilevel method In recent years, multilevel algorithms have been applied successfully to the construction of edge separators =-=[15, 33, 37]-=-. Roughly speaking, a multilevel algorithm consists of three phases. In the first phase the original graph G is approximated by a sequence of smaller graphs that maintain the essential properties of G... |

55 | Fast and effective algorithms for graph partitioning and sparse matrix ordering
- Gupta
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(Show Context)
Citation Context ...ction 2.3.2. However, the size of an edge separator is only indirectly related to the size of a vertex separator. Therefore, the multilevel scheme has been extended to find vertex separators directly =-=[31, 35]-=-. In the extended scheme a variant of the Fiduccia-Mattheyses heuristic is applied to the refinement of vertex separators. This vertex Fiduccia-Mattheyses method is summarized in section 2.3.2. Domain... |

50 |
The use of linear graphs in Gauss elimination
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(Show Context)
Citation Context ...erpreted as the adjacency matrix of an undirected graph G = (V; E), E V V . In this graph theoretic context an ordering (or labeling) is a bijection : V 7! f1; : : : ; ng. As observed by Parter [49] and Rose [53], the column-wise factorization of A can be modeled by a sequence of elimination graphs G k , 1 k n. When column k is factorized, G k is obtained from G k 1 (G 0 = G) by applying the... |

48 | Improving the runtime and quality of nested dissection ordering
- Hendrickson, Rothberg
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(Show Context)
Citation Context ...ique. In this way the elimination tree is built from the root down to the leaves. In contrast to the bottom-up methods introduced in section 2.2 the nested dissection algorithm is quite ill-specified =-=[35]-=-. For an effective implementation the following two important questions have to be answered: (1) How should the separators be determined, and (2) how important is a balanced partitioning. In the follo... |

45 | Robust ordering of sparse matrices using multisection
- Ashcraft, Liu
- 1998
(Show Context)
Citation Context ...uses global information of the original graph to build in a top-down manner. It is well recognized that the quality of the orderings produced by bottom-up and top-down schemes is not uniformly good [=-=9]-=-. Therefore, most state-of-the-art ordering codes such as BEND [34], METIS [38], SCOTCH [50], SPOOLES [4], and WGPP [30] are using a hybrid of both schemes. The main contribution of this paper is to p... |

43 |
Compressed graphs and the minimum degree algorithm
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(Show Context)
Citation Context ...essed graph. As a result one obtains an ordering c for G c that will be expanded to an ordering for G during the postprocessing step. More information concerning compressed graphs can be found in [2=-=, 16, 35]-=-. Although the main algorithmic components of our approach have been introduced in section 3, a number of details have been left out. The following list describes some important parameters of pord and... |

38 |
An automatic nested dissection algorithm for irregular finite element problems
- George, Liu
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(Show Context)
Citation Context ...in the ordering and is eliminated from G k 1 to form the graph G k . The whole selection/elimination process is then repeated for G k . Another effective method for reducing fill is nested dissection =-=[23, 24]-=-. The method starts with computing a vertex separator S in G. All vertices in S are ordered after those in G(V S). The method is recursively applied to each component of G(V S) until a component This ... |

35 | SPOOLES: An object-oriented sparse matrix library
- Ashcraft, Grimes
- 1999
(Show Context)
Citation Context ... the quality of the orderings produced by bottom-up and top-down schemes is not uniformly good [9]. Therefore, most state-of-the-art ordering codes such as BEND [34], METIS [38], SCOTCH [50], SPOOLES =-=[4]-=-, and WGPP [30] are using a hybrid of both schemes. The main contribution of this paper is to present an ordering scheme that achieves a tighter coupling of bottom-up and top-down methods. In our meth... |

32 |
Orderings for parallel sparse symmetric factorization
- Leiserson, Lewis
- 1987
(Show Context)
Citation Context .... One simply computes a minimum vertex cover for the bipartite graph induced by the cut edges. Note that it also possible to use heuristics for finding an approximate minimal cardinality vertex cover =-=[41]-=-. 2.4 Multisection orderings The shortcomings of a bottom-up method such as minimum degree are largely due to the local nature of the algorithm. In [13] Berman and Schnitger describe a minimum degree ... |

32 | Hybridizing nested dissection and halo approximate minimum degree for efficient sparse matrix ordering
- Pellegrini, Roman, et al.
- 1999
(Show Context)
Citation Context ...rograms differ in how the domains are numbered. METIS uses multiple minimum degree, while SCOTCH relies on a constrained version of the approximate minimum degree algorithm proposed by Amestoy et al. =-=[51]-=-. SPOOLES implements the two-level approach described in section 2.3. The vertices in the domains as well as the vertices in the multisector are numbered using multiple minimum degree. However, a cons... |

32 |
Node selection strategies for bottom-up sparse matrix ordering
- Rothberg, Eisenstat
- 1998
(Show Context)
Citation Context ...um deficiency would only marginally improve the ordering quality compared to minimum degree [18]. However, recent studies have shown that much better orderings can be obtained from minimum deficiency =-=[47, 48, 56]-=-. In order to reduce the runtime of the minimum deficiency algorithm, Rothberg and Eisenstat [56] have developed several node selection strategies that rely on an approximate computation of the defici... |

31 |
A graph partitioning algorithm by node separators
- Liu
- 1989
(Show Context)
Citation Context ...e, we cannot use a linear time algorithm such as bucket sort as proposed by Fiduccia and Mattheyses. Direct methods An alternate algorithm for refining a vertex separator S has been introduced by Liu =-=[44]-=-. Again, let (S; B; W ) denote the partition of G and let B be the component with larger weight. If we define B S to be the set of vertices in B that are adjacent to vertices in S, i. e. B S = adj G (... |

21 | Complexity bounds for regular finite difference and finite element grids - Hoffman, Martin, et al. - 1973 |

18 |
A partition improvement algorithm for generalized nested dissection
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- 1996
(Show Context)
Citation Context ...evaluation of priorities, and randomization are used to improve both runtime as well as quality of the solutions [32]. The Fiduccia-Mattheyses algorithm can easily be generalized to vertex separators =-=[5]-=-. In this vertex Fiduccia-Mattheyses algorithm each move selects a vertex v 2 S, transfers it to B (or W ) and locks it. Once a vertex v has been transferred to B, all neighbors of v in W are pulled i... |

17 | Using domain decomposition to find graph bisectors
- Ashcraft, Liu
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(Show Context)
Citation Context ...inement of vertex separators. This vertex Fiduccia-Mattheyses method is summarized in section 2.3.2. Domain decomposition method In contrast to the multilevel method described above, Ashcraft and Liu =-=[7]-=- propose a two-level approach to construct a vertex separator. Analogous to the domain 5 decomposition methods for solving PDEs, the vertex set V of G is partitioned into V = b V [D 1 [ : : : [ D r wi... |

17 | Applications of the Dulmage–Mendelsohn decomposition and network flow to graph bisection improvement - Ashcraft, Liu - 1998 |

16 |
On the performance of the minimum degree ordering for Gaussian elimination
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(Show Context)
Citation Context ...an approximate minimal cardinality vertex cover [41]. 2.4 Multisection orderings The shortcomings of a bottom-up method such as minimum degree are largely due to the local nature of the algorithm. In =-=[13-=-] Berman and Schnitger describe a minimum degree elimination sequence for the k k grid so that the number of factor entries and the number of factor operations is an order of magnitude higher than op... |

16 |
The minimum degree ordering with constraints
- Liu
- 1989
(Show Context)
Citation Context ...orithm the recursion terminates after a few levels and the vertices in the remaining subgraphs/domains are numbered using either multiple minimum degree (MMD) [42] or constrained minimum degree (CMD) =-=[4-=-3]. The vertices in the multisector are numbered according to the given nested dissection ordering. Local Nested Dissection [14] -- MS(ND;PROFILE) The most successful ordering scheme for h k grids w... |

14 |
A fast implementation of the minimum degree algorithm using quotient graphs
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- 1980
(Show Context)
Citation Context ...ient graphs Quotient graphs play a crucial role in our methodology. Originally, these graphs have been used in bottom-up ordering algorithms such as minimum degree to represent the elimination graphs =-=[25]-=-. In the following, we are using Roman letters for graphs (i. e., G; V; E) and calligraphic letters for quotient graphs (i. e., G; X ; E ). Let us assume that G k = (V k ; E k ) has been obtained from... |

14 |
Sparse matrix ordering with scotch
- Pellegrini, Roman
- 1997
(Show Context)
Citation Context ...ecognized that the quality of the orderings produced by bottom-up and top-down schemes is not uniformly good [9]. Therefore, most state-of-the-art ordering codes such as BEND [34], METIS [38], SCOTCH =-=[50]-=-, SPOOLES [4], and WGPP [30] are using a hybrid of both schemes. The main contribution of this paper is to present an ordering scheme that achieves a tighter coupling of bottom-up and top-down methods... |

11 |
Generalized nested dissection: some recent progress
- Ashcraft, Liu
- 1994
(Show Context)
Citation Context ...order the separator vertices instead of following the given nested dissection ordering. This multisection ordering is referred to as MS(CMD;MMD) and has been independently discovered by Ashcraft, Liu =-=[6, 9]-=-, and Rothberg [54]. 3 Our methodology Two levels of hybridizing bottom-up and top-down methods can be found in the literature: incomplete nested dissection (i. e. MS(MMD;ND) or MS(CMD;ND)) and minimu... |

11 | Effective sparse matrix ordering: just around the bend
- Hendrickson, Rothberg
- 1997
(Show Context)
Citation Context ...down manner. It is well recognized that the quality of the orderings produced by bottom-up and top-down schemes is not uniformly good [9]. Therefore, most state-of-the-art ordering codes such as BEND =-=[34]-=-, METIS [38], SCOTCH [50], SPOOLES [4], and WGPP [30] are using a hybrid of both schemes. The main contribution of this paper is to present an ordering scheme that achieves a tighter coupling of botto... |

11 |
Performance of greedy heuristics for sparse Cholesky factorization
- Ng, Raghavan
- 1999
(Show Context)
Citation Context ...um deficiency would only marginally improve the ordering quality compared to minimum degree [18]. However, recent studies have shown that much better orderings can be obtained from minimum deficiency =-=[47, 48, 56]-=-. In order to reduce the runtime of the minimum deficiency algorithm, Rothberg and Eisenstat [56] have developed several node selection strategies that rely on an approximate computation of the defici... |

10 |
WGPP: Watson graph partitioning (and sparse matrix ordering) package
- Gupta
- 1996
(Show Context)
Citation Context ...f the orderings produced by bottom-up and top-down schemes is not uniformly good [9]. Therefore, most state-of-the-art ordering codes such as BEND [34], METIS [38], SCOTCH [50], SPOOLES [4], and WGPP =-=[30]-=- are using a hybrid of both schemes. The main contribution of this paper is to present an ordering scheme that achieves a tighter coupling of bottom-up and top-down methods. In our methodology the ver... |

9 |
Applications of an element model for gaussian elimination
- Eisenstat, Schultz, et al.
- 1976
(Show Context)
Citation Context ...he root. Note that any vertex v that is not adjacent to any element becomes a leaf of the tree. The tree is called elimination tree and plays an important role in the context of sparse direct solvers =-=[20, 21, 45, 57]-=-. In section 3 we consider quotient graphs that have a more general structure, i. e. there is no oneto -one relation between a variable of G and an uneliminated vertex of G. Each variable represents a... |

8 |
Incomplete nested dissection for solving n by n grid problems
- George, Poole, et al.
- 1978
(Show Context)
Citation Context ...g method used to number the multisector. In the literature, there are a number of existing ordering schemes using the multisection approach. Two important examples are: Incomplete Nested Dissection [=-=28]-=- -- MS(MMD;ND) and MS(CMD;ND) In this scheme the multisector is constructed by the recursive bisection process of nested dissection. In contrast to the original nested dissection algorithm the recursi... |

7 |
Exploring the tradeoff between imbalance and separator size in nested dissection ordering
- Rothberg
- 1996
(Show Context)
Citation Context ...ments. Therefore, the minimization of partition imbalance is only a secondary objective in our nested dissection process. Empirically, our choice of is also supported by the experiments reported in [=-=55]-=-. Numerous experiments have shown that the default parameters described above are very effective on a wide range of matrix types. Note that we did not provide default parameters for ord 1 and ord 2 . ... |