## Permuting Sparse Rectangular Matrices into Block-Diagonal Form (2002)

Venue: | SIAM Journal on Scientific Computing |

Citations: | 57 - 19 self |

### BibTeX

@ARTICLE{Aykanat02permutingsparse,

author = {Cevdet Aykanat and Ali Pinar and Ümit V. Catalyurek},

title = {Permuting Sparse Rectangular Matrices into Block-Diagonal Form},

journal = {SIAM Journal on Scientific Computing},

year = {2002},

volume = {25},

pages = {1860--1879}

}

### Years of Citing Articles

### OpenURL

### Abstract

We investigate the problem of permuting a sparse rectangular matrix into block diagonal form. Block diagonal form of a matrix grants an inherent parallelism for solving the deriving problem, as recently investigated in the context of mathematical programming, LU factorization and QR factorization. We propose bipartite graph and hypergraph models to represent the nonzero structure of a matrix, which reduce the permutation problem to those of graph partitioning by vertex separator and hypergraph partitioning, respectively. Our experiments on a wide range of matrices, using state-of-the-art graph and hypergraph partitioning tools MeTiS and PaToH, revealed that the proposed methods yield very effective solutions both in terms of solution quality and runtime.

### Citations

1043 |
An efficient heuristic procedure for partitioning graphs
- Kernighan, Lin
- 1970
(Show Context)
Citation Context ... FM, which is an iterative refinement heuristic proposed for graph/hypergraph partitioning by Fiduccia and Mattheyses [15] as a faster implementation of the KL algorithm proposed by Kernighan and Lin =-=[43]-=-. In the third phase, the partition found in the second phase is successively projected back towards the original graph/hypergraph by refining the projected partitions on the intermediate level uncoar... |

794 | A fast and high quality multilevel scheme for partitioning irregular graphs
- Karypis, Kumar
- 1998
(Show Context)
Citation Context ...A ). Hence, \Pi GPV S induces the same SB form ASB of A as shown in Fig. 6.1(b). 18 Aykanat, Pinar and C atalyurek 8. Graph and Hypergraph Partitioning Algorithms and Tools. Recently, multilevel GPES =-=[7, 21, 25, 41, 42]-=- and HP [8, 9, 24, 39] approaches have been proposed leading to successful GPES tools Chaco [26], MeTiS [38] and WGPP [22], and HP tools hMeTiS [40] and PaToH [10]. These multilevel heuristics consist... |

601 |
Numerical methods for least squares problems
- Bjorck
- 1996
(Show Context)
Citation Context ...ng B k and the associated coupling columns in C k concurrently, so that [B k C k ] = Q k R k S k 0 C 0 k for k = 1; 2; : : : ; K: In a subsequent phase, we factor C 0 = \Theta C 0 1 ; : : : ; C 0 KsT =-=[4]-=-. So, in permuting a given matrix A into a dual SB form, minimizing the number of coupling columns minimizes the work on the second phase of the algorithm, and equal-sized blocks provide load balance ... |

500 |
Computer Solution of Large Sparse Positive Definite Systems
- George, Liu
- 1981
(Show Context)
Citation Context ...direct GPVS algorithms and our multilevel HP tool PaToH [10] in our one-phase A-toA SB transformation approach. One of the most important applications of GPVS is George's nested--dissection algorithm =-=[18, 19]-=-, which has been widely used for reordering of the rows/columns of a symmetric, sparse, and positive definite matrix to reduce fill in the factor matrices. Here, GPVS is defined on the standard graph ... |

486 |
Paritioning sparse matrices with eigenvectors of graphs
- Pothen, Simon, et al.
- 1990
(Show Context)
Citation Context ...ndary vertices as the wide separator to be refined to a narrow separator. The wide-to-narrow refinement problem is described as a minimum vertex cover problem on the subgraph induced by the cut edges =-=[58]-=-. A minimum vertex cover is taken as a narrow separator for the whole graph, since each cut edge will be adjacent to a vertex in the vertex cover. That is, let VBk `V k denote the set of boundary vert... |

478 | Multilevel k-way Partitioning Scheme for Irregular Graphs
- Karypis, Kumar
- 1998
(Show Context)
Citation Context ...A ). Hence, \Pi GPV S induces the same SB form ASB of A as shown in Fig. 6.1(b). 18 Aykanat, Pinar and C atalyurek 8. Graph and Hypergraph Partitioning Algorithms and Tools. Recently, multilevel GPES =-=[7, 21, 25, 41, 42]-=- and HP [8, 9, 24, 39] approaches have been proposed leading to successful GPES tools Chaco [26], MeTiS [38] and WGPP [22], and HP tools hMeTiS [40] and PaToH [10]. These multilevel heuristics consist... |

443 |
A multilevel algorithm for partitioning graphs
- Hendrickson, Leland
- 1995
(Show Context)
Citation Context ...A ). Hence, \Pi GPV S induces the same SB form ASB of A as shown in Fig. 6.1(b). 18 Aykanat, Pinar and C atalyurek 8. Graph and Hypergraph Partitioning Algorithms and Tools. Recently, multilevel GPES =-=[7, 21, 25, 41, 42]-=- and HP [8, 9, 24, 39] approaches have been proposed leading to successful GPES tools Chaco [26], MeTiS [38] and WGPP [22], and HP tools hMeTiS [40] and PaToH [10]. These multilevel heuristics consist... |

427 |
A linear-time heuristic for improving network partitions
- M, Mattheyses
- 1982
(Show Context)
Citation Context ...on is obtained on the coarsest graph/hypergraph using various heuristics including FM, which is an iterative refinement heuristic proposed for graph/hypergraph partitioning by Fiduccia and Mattheyses =-=[13]-=- as a faster implementation of the KL algorithm proposed by Kernighan and Lin [30]. In the third phase, the partition found in the second phase is successively projected back towards the original grap... |

418 |
Combinatorial Algorithms for Integrated Circuit Layout
- Lengauer
(Show Context)
Citation Context ...nsidered as a net separator whose removal gives K disconnected node parts U 1 ; : : : ; UK as well as K disconnected net parts N 1 ; : : : ; NK . Two cutsize definitions widely used in VLSI community =-=[48]-=- for representing the cost of a partition \Pi HP are: (a) cost(\Pi HP ) = jNS j and (b) cost(\Pi HP ) = X n i 2NS ( i \Gamma 1): (3.3) Hence, the K-way HP problem can be defined as the task of dividin... |

376 |
Some simplified NP-complete graph problems
- Garey, Johnson, et al.
- 1976
(Show Context)
Citation Context ...tor of BA will also be a bipartite graph. Permuting Sparse Rectangular Matrices into Block-Diagonal Form 11 The minimum vertex cover problem is known to be NP-hard on K-partite graphs at least for K5 =-=[16]-=-, thus we need to resort to heuristics. Leiserson and Lewis [46] proposed two greedy heuristics for this purpose, namely minimum recovery (MR) and maximum inclusion (MI). The MR heuristic is based on ... |

274 | A software package for partitioning unstructured graphs, partitioning meshes, and computing fill-reducing orderings of sparse matrices version
- Karypis, Kumar
(Show Context)
Citation Context ... and Hypergraph Partitioning Algorithms and Tools. Recently, multilevel GPES [7, 21, 25, 41, 42] and HP [8, 9, 24, 39] approaches have been proposed leading to successful GPES tools Chaco [26], MeTiS =-=[38]-=- and WGPP [22], and HP tools hMeTiS [40] and PaToH [10]. These multilevel heuristics consist of 3 phases: coarsening , initial partitioning , and uncoarsening. In the first phase, a multilevel cluster... |

274 |
Combinatorial Optimization: Networks and
- LAWLER
- 1976
(Show Context)
Citation Context ... Wide-to-Narrow Separator Refinement. A minimum vertex cover of a bipartite graph can be computed optimally in polynomial time by finding a maximum matching, by exploiting the duality between the two =-=[45, 57, 58]-=-. So, the wide-to-narrow separator refinement problem can be easily solved using this scheme for 2-way indirect GPVS, because the edge separator of a 2-way GPES induces a bipartite subgraph. This sche... |

231 |
The Decomposition Principle for Linear Programs
- Dantzig, Wolfe
- 1960
(Show Context)
Citation Context ...near algebra and optimization: linear programming, LU and QR factorizations. 2.1. Linear Programming. Exploiting the block-angular structure of linear programs (LPs) dates back to the work of Dantzig =-=[11]-=-, when the motivation was solving large LPs with limited memory. Later studies investigated parallelization techniques [15, 23, 34]. The proposed techniques [11, 31, 35] lead to iterative algorithms, ... |

198 | Recent directions in netlist partitioning: a survey, Integr
- Alpert, Kahng
- 1995
(Show Context)
Citation Context ...ly, we provide a discussion on the matrix theoretical view of the relation between HP and GPVS problems. 7.1. Alternative Models for Solving HP Problem. As indicated in the survey by Alpert and Kahng =-=[1]-=-, hypergraphs are commonly used to represent circuit netlist connections in solving the circuit partitioning and placement problems in VLSI layout design. The circuit partitioning problem is to divide... |

189 |
The Chaco user’s guide: Version 2.0
- Hendrickson, Leland
- 1995
(Show Context)
Citation Context ...rek 8. Graph and Hypergraph Partitioning Algorithms and Tools. Recently, multilevel GPES [7, 21, 25, 41, 42] and HP [8, 9, 24, 39] approaches have been proposed leading to successful GPES tools Chaco =-=[26]-=-, MeTiS [38] and WGPP [22], and HP tools hMeTiS [40] and PaToH [10]. These multilevel heuristics consist of 3 phases: coarsening , initial partitioning , and uncoarsening. In the first phase, a multil... |

178 |
Nested dissection of a regular finite element mesh
- George
- 1973
(Show Context)
Citation Context ...n phase of the FH method and our multilevel HP tool PaToH [9] in our one-phase A-toASB transformation approach. One of the most important applications of GPVS is George's nested--dissection algorithm =-=[14]-=-, which has been widely used in fill-reducing orderings for sparse matrix factorizations. The basic idea in the nested dissection algorithm is to reorder symmetric matrix into a 2-way DB form so that ... |

149 | Multilevel algorithms for multi-constraint graph partitioning - Karypis, Kumar - 1998 |

123 | Electronic Mail Distribution of Linear Programming Test Problems - Gay - 1985 |

87 |
Hypergraph-partitioning-based decomposition for parallel sparse-matrix vector multiplication
- Çatalyurek, Aykanat
- 1999
(Show Context)
Citation Context .... 6. Hypergraph Model for A-to-A SB Transformation. In this section, we show that A-to-A SB transformation can be described as an HP problem on a hypergraph representation of A. In our previous works =-=[8, 9, 55, 56]-=-, we proposed two hypergraph models, namely row-net and column-net models, for representing rectangular as well as symmetric and nonsymmetric square matrices. These two models are duals of each other.... |

83 |
Computing the block triangular form of a sparse matrix
- Pothen, Fan
- 1990
(Show Context)
Citation Context ... Wide-to-Narrow Separator Refinement. A minimum vertex cover of a bipartite graph can be computed optimally in polynomial time by finding a maximum matching, by exploiting the duality between the two =-=[45, 57, 58]-=-. So, the wide-to-narrow separator refinement problem can be easily solved using this scheme for 2-way indirect GPVS, because the edge separator of a 2-way GPES induces a bipartite subgraph. This sche... |

82 |
The chaco user’s guide
- Hendrickson, Leland
- 1993
(Show Context)
Citation Context ... SB form. 6. Graph and hypergraph partitioning algorithms and tools. Recently, multilevel GPES [6, 20] and HP [8, 17, 29] approaches have been proposed, leading to successful GPES tools such as Chaco =-=[21]-=-, MeTiS [27], and WGPP [16] and HP tools hMeTiS [29] and PaToH [9]. These multilevel heuristics consist of 3 phases: coarsening, initial partitioning, and uncoarsening. In the first phase, a multileve... |

75 |
Finding good approximate vertex and edge partitions is NP-hard
- Bui, Jones
- 1992
(Show Context)
Citation Context ...ted problems are called the problem of graph partitioning by edge separator (GPES) and graph partitioning by vertex separator (GPVS), respectively. Both GPES and GPVS problems are known to be NP-hard =-=[5]-=-. Balance among subgraphs is usually defined by cumulative effect of weights assigned to vertices. Some alternatives have been studied recently [40]. We proceed with formal definitions. \Pi ES =fV 1 ;... |

73 | Graph partitioning models for parallel computing
- Hendrickson, Kolda
(Show Context)
Citation Context ...) is exactly equal to the parallel communication volume. The proposed HP model overcomes the major flaws and limitations of the standard GPES models, which are also addressed by Hendrickson and Kolda =-=[27, 30]-=-. In this work, we show that the A-to-ASB transformation problem can be described as an HP problem in which the cutsize metric given in (3.3.a) is exactly equal to the number of coupling rows in A SB ... |

70 |
A heuristic for reducing fill in sparse matrix factorization
- Bui, Jones
- 1993
(Show Context)
Citation Context |

61 | C.: Hypergraph-partitioning based decomposition for parallel sparse-matrix vector multiplication
- Catalyürek, Aykanat
- 1999
(Show Context)
Citation Context ... referred to as the cutsize metric in VLSI community. The connectivity metric is defined as cost(ΠHP) = � (3.4) (λi − 1) ni∈NS and is frequently used in VLSI [32] and scientific computing communities =-=[8]-=-. 3.3. Column-splitting method for ADB-to-ASB transformation. In the second phase of the Ferris–Horn (FH) algorithm [12], ADB is transformed into an SB form through the column-splitting technique used... |

59 |
A heuristic for reducing fill-in in sparse matrix factorization
- Bui, Jones
- 1993
(Show Context)
Citation Context ...b shows a corresponding partial permutation which transforms matrix A in Figure 4.1 directly into a 3-way SB form. 6. Graph and hypergraph partitioning algorithms and tools. Recently, multilevel GPES =-=[6, 20]-=- and HP [8, 17, 29] approaches have been proposed, leading to successful GPES tools such as Chaco [21], MeTiS [27], and WGPP [16] and HP tools hMeTiS [29] and PaToH [9]. These multilevel heuristics co... |

55 | Fast and Effective Algorithms for Graph Partitioning and Sparse-matrix
- Gupta
- 1997
(Show Context)
Citation Context |

55 |
New variants of bundle methods
- Lemaréchal, Nemirovskii, et al.
- 1995
(Show Context)
Citation Context ... (LPs) dates back to the work of Dantzig [13], when the motivation was solving large LPs with limited memory. Later works investigated parallelization techniques [20, 31, 50]. The proposed techniques =-=[13, 47, 51, 60]-=- lead to iterative algorithms, where each 4 Aykanat, Pinar and C atalyurek iteration involves solving K independent LP subproblems corresponding to the block constraints followed by a coordination pha... |

52 | An Evaluation of Bipartitioning Techniques
- Hauck, Borriello
- 1997
(Show Context)
Citation Context ...responding partial permutation which transforms matrix A in Fig. 4.1 directly into a 3-way SB form. 6. Graph and Hypergraph Partitioning Algorithms and Tools. Recently, multilevel GPES [6, 20] and HP =-=[8, 17, 29]-=- approaches have been proposed leading to successful GPES tools such as Chaco [21], MeTiS [27] and WGPP [16], and HP tools hMeTiS [29] and PaToH [9]. These multilevel heuristics consist of 3 phases: c... |

46 |
PaToH: A Multilevel Hypergraph Partitioning Tool, Version 3.0
- Çatalyürek, Aykanat
- 1999
(Show Context)
Citation Context ...ntly, multilevel GPES [7, 21, 25, 41, 42] and HP [8, 9, 24, 39] approaches have been proposed leading to successful GPES tools Chaco [26], MeTiS [38] and WGPP [22], and HP tools hMeTiS [40] and PaToH =-=[10]-=-. These multilevel heuristics consist of 3 phases: coarsening , initial partitioning , and uncoarsening. In the first phase, a multilevel clustering is applied starting from the original graph/hypergr... |

45 |
hMeTiS: A Hypergraph Partitioning Package, Version 1.5.3
- Karypis, Kumar
- 1998
(Show Context)
Citation Context ...and Tools. Recently, multilevel GPES [7, 21, 25, 41, 42] and HP [8, 9, 24, 39] approaches have been proposed leading to successful GPES tools Chaco [26], MeTiS [38] and WGPP [22], and HP tools hMeTiS =-=[40]-=- and PaToH [10]. These multilevel heuristics consist of 3 phases: coarsening , initial partitioning , and uncoarsening. In the first phase, a multilevel clustering is applied starting from the origina... |

41 | Graph partitioning and parallel solvers: Has the emperor no clothes? In IRREGULAR’98: solving irregularly structured problems
- Hendrickson
- 1998
(Show Context)
Citation Context ... (3.2) is exactly equal to the parallel communication volume. The proposed HP model overcomes some flaws and limitations of the standard GPES models, which are also addressed by Hendrickson and Kolda =-=[18, 19]-=-. In this work, we show that the A-to-A SB transformation problem can be described as an HP problem in which the cutsize metric in (3.3) is exactly equal to the number of coupling rows in A SB . 10 Ay... |

31 |
A graph partitioning algorithm by node separators
- Liu
- 1989
(Show Context)
Citation Context ...tite graph K(E S ). However, there is a wide-to-narrow separator refinement V S =fv 1 ; v 3 g of size 2 as shown in Fig. 5.3(c). As mentioned in Section 8, Liu's narrow separator refinement algorithm =-=[49]-=- can also be considered as exploiting the vertex cover model on the bipartite graph induced by the edges between V 1 and V S (V 2 and V S ) of a GPVS \Pi GPV S =fV 1 ; V 2 ; V S g. So, the discussion ... |

30 |
Diagonal quadratic approximation method for large scale linear programs
- Mulvey, Ruszczynski
- 1990
(Show Context)
Citation Context ...ADB -to-A SB Transformation. In the second phase of FH algorithm [14], ADB is transformed into an SB form through the column-splitting technique used in stochastic programming to treat anticipativity =-=[53]-=-. In this technique, we consider the variables corresponding to the linking columns. Consider a linking column c j in submatrix C= (C T 1 \Delta \Delta \Delta C T k \Delta \Delta \Delta C T K D T ) T ... |

29 | Improving the run time and quality of nested dissection ordering
- Hendrickson, Rothberg
- 1998
(Show Context)
Citation Context ...g code of MeTiS [38] to compare against our edge-weighted indirect GPVS algorithms. Recently, direct 2-way GPVS approaches have been embedded into various multilevel nested-dissection implementations =-=[22, 28, 38]-=-. In these implementations, a 2-way GPVS obtained on the coarsest graph is refined during the multilevel framework of the uncoarsening phase. Two distinct vertex-separator refinement schemes were prop... |

28 | Decomposing irregularly sparse matrices for parallel matrix-vector multiplication
- CATALYUREK, AYKANAT
- 1996
(Show Context)
Citation Context .... 5. Hypergraph model for A-to-ASB transformation. In this section, we show that A-to-ASB transformation can be described as an HP problem on a hypergraph representation of A. In our previous studies =-=[7, 8, 38, 39]-=-, we proposed two hypergraph models, namely, row-net and column-net models, for representing rectangular as well as symmetric and nonsymmetric square matrices. These two models are duals: the row-net ... |

25 | Partitioning mathematical programs for parallel solution
- Ferris, Horn
- 1998
(Show Context)
Citation Context ... that the sum of the number of coupling rows and columns is minimized while a given balance criterion is satisfied. The literature that addresses this problem is very rare and recent. Ferris and Horn =-=[12]-=- proposed a two-phase approach for A-to-A SB transformation. In the first phase, matrix A is transformed into a DB form ADB as an intermediate form. In the second phase, ADB is transformed into an SB ... |

22 | Net Partitions Yield Better Module Partitions
- Cong, Hagen, et al.
- 1992
(Show Context)
Citation Context ...g Sparse Rectangular Matrices into Block-Diagonal Form 15 Both dual hypergraph and NIG models view the HP problem in terms of partitioning nets instead of nodes. Kahng [37] and Cong, Hagen, and Kahng =-=[11]-=- exploited this perspective of the NIG model to formulate the hypergraph bipartitioning problem as a two-stage process. In the first stage, nets of H are bipartitioned through 2-way GPES of its NIG G.... |

18 |
A partition improvement algorithm for generalized nested dissection
- Ashcraft, Liu
- 1996
(Show Context)
Citation Context ...nt schemes were proposed and used for the uncoarsening phase. The first one is the extension of the FM edgeseparator refinement approach to vertex-separator refinement as proposed by Ashcraft and Liu =-=[1]-=-. This scheme considers vertex moves from vertex separator V S to both V 1 and V 2 in \Pi V S =fV 1 ; V 2 ; V S g. This refinement scheme is adopted in the onmetis 12 Aykanat, Pinar and C atalyurek Ta... |

17 | Applications of the Dulmage-Mendelsohn decomposition and network flow to graph bisection improvement
- ASHCRAFT, H
- 1996
(Show Context)
Citation Context ...oundary vertices of V 1 (V 2 ) is refined as in the GPES-based wide-to-narrow separator refinement scheme. The network-flow based minimum weighted vertex cover algorithms proposed by Ashcraft and Liu =-=[3]-=-, and Hendrickson and Rothberg [28] enabled the use of Liu's refinement approach [49] on the coarse graphs within the multilevel framework. In this work, we use the publicly available onmetis ordering... |

17 | Partitioning rectangular and structurally nonsymmetric sparse matrices for parallel processing - Hendrickson, Kolda |

17 | A massively parallel algorithm for nonlinear stochastic network problems - Nielsen, Zenios - 1993 |

16 | Multi-way VLSI circuit partitioning based on dual net representation
- Cong, Labio, et al.
- 1994
(Show Context)
Citation Context ...hng [37] used a winner-loser heuristic [23], whereas Cong et al. [11] used a matching-based (IG-match) algorithm for solving the 2-way module contention problem optimally. Cong, Labio, and Shivakumar =-=[12]-=- extended this approach to K-way HP through using the dual hypergraph model. In the first stage, a K-way net partitioning is obtained through partitioning the dual hypergraph. For the second stage, th... |

16 |
The minimum degree ordering with constraints
- Liu
- 1989
(Show Context)
Citation Context ...cheme is adopted in the onmetis ordering code of MeTiS [27], the ordering code of WGPP [16], and the ordering code BEND [22]. The second scheme is based on Liu’s narrow separator refinement algorithm =-=[33]-=-, which considers moving a set of vertices simultaneously from VS, in contrast to the FM-based refinement scheme [1], which moves only one vertex at a time. Liu’s refinement algorithm [33] can be cons... |

14 |
An Algorithm for optimal PLA Folding
- Hachtel, Newton, et al.
(Show Context)
Citation Context ...assignment for the shared nodes such that the cutsize (3.3) is minimized. This problem is known as the module (node) contention problem in the VLSI community. Kahng [37] used a winner-loser heuristic =-=[23]-=-, whereas Cong et al. [11] used a matching-based (IG-match) algorithm for solving the 2-way module contention problem optimally. Cong, Labio, and Shivakumar [12] extended this approach to K-way HP thr... |

14 | Partitioning for complex objectives
- Pinar, Hendrickson
- 2001
(Show Context)
Citation Context ... Both GPES and GPVS problems are known to be NP-hard [5]. Balance among subgraphs is usually defined by cumulative effect of weights assigned to vertices. Some alternatives have been studied recently =-=[40]-=-. We proceed with formal definitions. ΠES = {V1, V2,...,VK} is a K-way vertex partition of G by edge separator ES ⊂Eif the following conditions hold: Vk ⊂Vand Vk �= ∅ for 1 ≤ k ≤ K; Vk ∩Vℓ = ∅ for 1 ≤... |

11 | Partitioning sparse rectangular matrices for parallel processing - Kolda - 1457 |

9 |
A multilevel hypergraph partitioning tool, version 3.0
- PaToH
- 1999
(Show Context)
Citation Context ...s. Recently, multilevel GPES [6, 20] and HP [8, 17, 29] approaches have been proposed leading to successful GPES tools such as Chaco [21], MeTiS [27] and WGPP [16], and HP tools hMeTiS [29] and PaToH =-=[9]-=-. These multilevel heuristics consist of 3 phases: coarsening , initial partitioning , and uncoarsening. In the first phase, a multilevel clustering is applied starting from the original graph/hypergr... |

9 |
Parallel bundle-based decomposition for large-scale structured mathematical programming problems
- Medhi
- 1990
(Show Context)
Citation Context ...k-angular structure of linear programs (LPs) dates back to the work of Dantzig [13], when the motivation was solving large LPs with limited memory. Later works investigated parallelization techniques =-=[20, 31, 50]-=-. The proposed techniques [13, 47, 51, 60] lead to iterative algorithms, where each 4 Aykanat, Pinar and C atalyurek iteration involves solving K independent LP subproblems corresponding to the block ... |

9 | Decomposing linear programs for parallel solution
- Pınar, Çatalyürek, et al.
- 1996
(Show Context)
Citation Context ...mediate form. In the second phase, ADB is transformed into an SB form through column-splitting as discussed in Section 3.3. Our initial results of this problem were presented in two conference papers =-=[55, 56]-=-. In [55], we proposed the basics of our hypergraph model and how to exploit this model to permute matrices to block-diagonal form. In our subsequent work [56] we proposed our graph models. Later Hu e... |