A. Gupta. Fast and effective algorithms for graph partitioning and sparse matrix reordering. IBM Journal of Research and Development, 41(1/2):171--183, 1996.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....recent. As a result, the linear programming community has been using these well established heuristics that were not originally developed for their applications. This paper shows that MD based heuristics can be quite unsuitable for many linear programming problems. Recent work by the author [14], Hendrickson and Rothberg [19] Ashcraft and Liu [1] and Karypis and Kumar [23, 22] suggests that GP based heuristics are capable of producing better quality orderings than MD based heuristics for finite element problems while staying within a small constant factor of the run time of MD based ....

....fail for LP problems [34] The algorithms presented in this paper are robust and generate good quality orderings for both LP and finite element problems. These algorithms are included in the graph partitioning and sparse matrix ordering package (WGPP) 15] developed by the author. In [14], we present experimental results of using WGPP for graph partitioning and for ordering sparse matrices arising in different applications. In this paper, we show that WGPP offers significant advantages over MD based heuristics for the matrices arising in IP computations. For such matrices, WGGP ....

[Article contains additional citation context not shown here]

Anshul Gupta. Fast and effective algorithms for graph partitioning and sparse matrix reordering. Technical Report (Number to be assigned), IBM T. J. Watson Research Center, Yorktown Heights, NY, 1996.

.... multifrontal algorithm [3, 10] for sparse Cholesky factorization and a highly scalable parallel sparse Cholesky factorization algorithm [8, 4] The package also uses scalable parallel sparse triangular solvers [9] and an improved and parallelized version of a multilevel nesteddissection algorithm [5] for computing fill reducing orderings. For details on the implementation and performance of WSMP for solving symmetric sparse systems, please refer to [7] In this document, the term node refers to a uniprocessor or a multiprocessor computing unit with shared memory. A node may consist of one or ....

....and quality is determined by the integer value (speed) in IPARM(16) Speed = 1 results in the slowest but best ordering, speed = 3 results in fastest but worst ordering, and speed = 2 results in an intermediate and quality or ordering. WSMP uses graph partitioning based ordering algorithms [5] to minimize fill during factorization. IPARM(17) specifies the maximum number of nodes that a subgraph must have before it is ordered by using a minimum local fill algorithm without further subpartition. The user can obtain a pure minimum local fill ordering by specifying IPARM(17) greater than ....

Anshul Gupta. Fast and effective algorithms for graph partitioning and sparse matrix ordering. IBM Journal of Research and Development, 41(1/2):171--183, January/March, 1997.

....strategy is to construct the series of graphs until the number of vertices in the coarsest graph is smaller than some threshold and then carry out an initial partition. In parallel, the graph is already distributed and so an initial partition already exists. Here, following the idea of Gupta [14], we continue coarsening until the number of vertices in the coarsest graph is the same as the number of subdomains, P, and this gives us automatically an initial partition with one vertex per subdomain. However, although contraction down to a single vertex per subdomain is rapid in serial (since ....

A. Gupta, Fast and effective algorithms for graph partitioning and sparse matrix reordering, IBM J. Res. Dev. 41 (1/2) (1996) 171 183.

....strategy is to construct the series of graphs until the number of vertices in the coarsest graph is smaller than some threshold and then carry out an initial partition. In parallel, the graph is already distributed and so an initial partition already exists. Here, following the idea of Gupta, [10], we continue coarsening until the number of vertices in the coarsest graph is the same as the number of subdomains, # , and this gives us automatically an initial partition with one vertex per subdomain. However, although contraction down to a single vertex per subdomain is rapid in serial (since ....

A. Gupta. Fast and effective algorithms for graph partitioning and sparse matrix reordering. IBM Journal of Research and Development, 41(1/2):171--183, 1996.

....algorithm uses this heuristic. The initial partition: Having constructed the series of graphs until the number of vertices in the coarsest graph is smaller than some threshold, the normal practice of the multilevel strategy is to carry out an initial partition. Here, following the idea of Gupta [11], we contract until the number of vertices in the coarsest graph is the same as the number of subdomains, P , and then simply assign vertex i to subdomain S i . Unlike Gupta, however, we do not carry out repeated expansion contraction cycles of the coarsest graphs to find a well balanced initial ....

A. Gupta, Fast and effective algorithms for graph partitioning and sparse matrix reordering, IBM J. Res. Dev. 41 (1/2) (1996) 171--183.

....k clustedng algorithm, for which state of the art favours a k way partitioning implementation on the complement graph. In practice, for pure acceleration purposes, we choose clusters in an 329 ad hoc way. Second, those clusters can serve as the basis for a multi level version of LCDO 2 or G CDO 3 [7, 8], which means that all nodes within a cluster are always reduced simultaneously. Because the number of clusters k can be controlled by the user, it is possible to reduce the G CDO(k) rantime to a minimum. For example, using a greedy coloring technique for k clustering (after isolating initially ....

A.Gupta, "Fast and effective algorithms for graph partitioning and sparse-matrix ordering ", IBM Journal of R&D, Vol. 41, No 1/2 - Optical lithography, 1997.

....S of GNIG (HA ) shown in Fig. 7.1(a) Recall that 3 way partitioning Pi HP shown in Fig. 6.1(a) is induced by Pi GPV S of GNIG (HA ) Hence, Pi GPV S induces the same SB form ASB of A as shown in Fig. 6.1(b) 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 ....

A. Gupta, "Fast and Effective Algorithms for Graph Partitioning and Sparse Matrix Ordering," Technical Report RC 2049.

....algorithm uses this heuristic. The initial partition: Having constructed the series of graphs until the number of vertices in the coarsest graph is smaller than some threshold, the normal practice of the multilevel strategy is to carry out an initial partition. Here, following the idea of Gupta [11], we contract until the number of vertices in the coarsest graph is the same as the number of subdomains, P, and then simply assign vertex i to subdomain Si. Unlike Gupta, however, we do not carry out repeated expansion contraction cycles of the coarsest graphs to find a well balanced initial ....

A. Gupta, Fast and effective algorithms for graph partitioning and sparse matrix reordering, IBM J. Res. Dev. 41 (1/2) (1996) 171 183.

....uses this heuristic. 2.2.2. The initial partition Having constructed the series of graphs until the number of vertices in the coarsest graph is smaller than some threshold, the normal practice of the multilevel strategy is to carry out an initial partition. Here, following the idea of Gupta [10], we contract until the number of vertices in the coarsest graph is the same as the number of subdomains, P, and then simply assign vertex i to subdomain . Unlike Gupta, however, we do not carry out repeated expansion contraction cycles of the coarsest graphs to find a well balanced initial ....

A. Gupta, Fast and effective algorithms for graph partitioning and sparse matrix reordering, IBM J. Res. Development 41 (1/2) (1996) 171 183.

....are removed from the list The Initial Partition Having constructed the series of graphs until the number of vertices in the coarsest graph is smaller than some threshold, the normal practice of the multilevel strategy is to carry out an initial partition. Here, following the idea of Gupta [4] we contract until the number of vertices in the coarsest graph is the same as the number of subdomains P, and then simply assign vertex i to subdomain then commence on the expansion optimisation sequence. Partition Expansion Having optimised the partition on a graph G, the partition must be ....

....recently been successfully applied to a diverse set of problems providing useful examples of crossover and mutation operators which provide a guide for developing such operators for new problems. The operators described in this paper extend an approach to the Travelling Salesman Problem (TSP) 14] and the Constrained Minimum Spanning Tree Problem (CMSTP) 15] both of which require a search for a set of links satisfying constraints (forming a tour for the TSP) and for which the sum of their costs is a minimum. Clearly the graph partitioning problem is of similar character: find a set of ....

[Article contains additional citation context not shown here]

A.Gupta. Fast and effective algorithms for graph partitioning and sparse matrix reordering. IBM Journal of Research and Development, 41(1/2):171- 183, 1996.

....weight jvj = ju 1 j ju 2 j. The initial partition. Having constructed the series of graphs until the number of vertices in the coarsest graph is smaller than some threshold, the normal practice of the multilevel strategy is to carry out an initial partition. Here, following the idea of Gupta, [10], we contract until the number of vertices in the coarsest graph is the same as the number of subdomains, P , and then simply assign vertex i to subdomain S i . Unlike Gupta, however, we do not carry out repeated expansion contractioncycles of the coarsest graphs to find a well balanced initial ....

A. Gupta. Fast and effective algorithms for graph partitioning and sparse matrix reordering. IBM Journal of Research and Development, 41(1/2):171--183, 1996.

....weight incident in v) For each u 2 H v , let W v Gammau be the sum of the weights of the edges connecting u to a vertex adjacent to v. Then v is matched with the vertex u 2 H v for which W v Gammau is maximum among all the vertices in H v . ffl Heaviest Edge Matching (HEAV) proposed by Gupta [10], considers the edges in order of decreasing weights (with ties broken randomly) It behaves better than HEM and HEM in the last coarsening steps, when the size of the graph has been considerably reduced and thus the sorting on the weights is less computationally costly. Moreover, it is more ....

A. Gupta, "Fast and effective algorithms graph partitioning and sparse-matrix ordering, " IBM Journal of Research and Development 41(1/2), pp. 171-184, 1997.

....into the context of reordering matrices to special forms. Special forms and methods to obtain them are well studied in the literature on computational linear algebra: They can be exploited by solution methods for linear equation systems, see, e.g. Duff, Erisman, and Reid [1986] or Kumar, Grama, Gupta, and Karypis [1994] The matrices considered there mainly arise from the discretization of partial differential equations, and, recently, from interior point algorithms for linear programs, see Gupta [1996] and Rothberg and Hendrickson [1996] But all these matrices are symmetric and to the best ....

.... methods for linear equation systems, see, e.g. Duff, Erisman, and Reid [1986] or Kumar, Grama, Gupta, and Karypis [1994] The matrices considered there mainly arise from the discretization of partial differential equations, and, recently, from interior point algorithms for linear programs, see Gupta [1996] and Rothberg and Hendrickson [1996] But all these matrices are symmetric and to the best of our knowledge the proposed ordering algorithms only apply to symmetric (or almost symmetric) matrices. This work has been supported jointly by the Fundac ao Coordenac ao de Aperfeicoamento de ....

Gupta, A. (1996). Fast and effective algorithms for graph partitioning and sparse matrix ordering.

....cyl3 1.47 10929 8.68 1.52 16382 10.05 1.52 22355 12.03 1. 51 29926 15.97 Table 2: Final results using surface matching and local template gain template cost optimisation e1 e2 e3 e1 e2 e3 (a) surface matching (b) cost matching Figure 2: Surface (a) and cost (b) matching idea of Gupta, [13], we contract until the number of vertices in the coarsest graph is the same as the number of subdomains, P , and then simply assign vertex i to subdomain S i . Unlike Gupta, however, we do not carry out repeated expansion contraction cycles of the coarsest graphs to find a well balanced initial ....

A. Gupta. Fast and effective algorithms for graph partitioning and sparse matrix reordering. IBM Journal of Research and Development, 41(1/2):171--183, 1996.

....strategy is to construct the series of graphs until the number of vertices in the coarsest graph is smaller than some threshold and then carry out an initial partition. In parallel, the graph is already distributed and so an initial partition already exists. Here, following the idea of Gupta, [10], we continue coarsening until the number of vertices in the coarsest graph is the same as the number of subdomains, P , and this gives us automatically an initial partition with one vertex per subdomain. However, although contraction down to a single vertex per subdomain is rapid in serial (since ....

A. Gupta. Fast and effective algorithms for graph partitioning and sparse matrix reordering. IBM Journal of Research and Development, 41(1/2):171--183, 1996.

....algorithm uses this heuristic. The initial partition. Having constructed the series of graphs until the number of vertices in the coarsest graph is smaller than some threshold, the normal practice of the multilevel strategy is to carry out an initial partition. Here, following the idea of Gupta, [11], we contract until the number of vertices in the coarsest graph is the same as the number of subdomains, P , and then simply assign vertex i to subdomain S i . Unlike Gupta, however, we do not carry out repeated expansion contraction cycles of the coarsest graphs to find a well balanced initial ....

A. Gupta. Fast and effective algorithms for graph partitioning and sparse matrix reordering. IBM Journal of Research and Development, 41(1/2):171--183, 1996.

....algorithm uses this heuristic. The initial partition. Having constructed the series of graphs until the number of vertices in the coarsest graph is smaller than some threshold, the normal practice of the multilevel strategy is to carry out an initial partition. Here, following the idea of Gupta, [9], we contract until the number of vertices in the coarsest graph is the same as the number of subdomains, P , and then simply assign vertex i to subdomain S i . Unlike Gupta, however, we do not carry out repeated expansion contraction cycles of the coarsest graphs to find a well balanced initial ....

A. Gupta. Fast and effective algorithms for graph partitioning and sparse matrix reordering. IBM Journal of Research and Development, 41(1/2):171--183, 1996.

....corresponding to neighbors of v in N i . We use f i (v) to denote the component of the vector corresponding to node v. Next, the vector f i is improved. This is done by Rayleigh quotient iteration; see [BS93] for further details. The multilevel KL algorithm [HL95c, BJ93, KK95b, KK95c, Gup97] is another example of how a multilevel approach can be used to obtain a fast algorithm. During the coarsening phase the algorithm creates a sequence of increasingly coarser approximations of the initial graph. When a sufficiently coarse graph has been found, we enter the partitioning phase ....

A. Gupta. Fast and effective algorithms for graph partitioning and sparse-matrix ordering. IBM J. Res. Develop. , 41(1):171--183, 1997.

....of the subdomains. Finally, the dashed and dotted bisections were computed. The centers of mass of the mesh elements are shown in (a) The mesh elements are shaded in (b) to indicated their subdomains. with the importance of the problem, has led to the developmentofseveral heuristic approaches [1, 2, 5,6,9, 13,16,20,21,22,24,25,26,27,29,30,36,37,44,48,54,58,60,61,66,68,73,74,76,83,94,104]. These can be classified as either geometric [6,22,30,58,61,66,68,76] combinatorial [1, 2, 20,21,24,27,54] spectral [36,37,73,74,83] combinatorial optimization techniques [5, 26,104] or multilevel methods [9, 13,16,25,29, 44,48,60,94] In this section, we discuss several of these classes and ....

.... 3,16,20,21,22,24,25,26,27,29,30,36,37,44,48,54,58,60,61,66,68,73,74,76,83,94,104] These can be classified as either geometric [6,22,30,58,61,66,68,76] combinatorial [1, 2, 20,21,24,27,54] spectral [36,37,73,74,83] combinatorial optimization techniques [5, 26,104] or multilevel methods [9, 13,16,25,29, 44,48,60,94]. In this section, we discuss several of these classes and describe the importantschemes from them. 0.3.1 Geometric Techniques Geometric techniques [6, 22,30,58,61,66,68,76] compute partitionings based solely on the coordinate information of the mesh nodes, and not on the connectivity of the ....

[Article contains additional citation context not shown here]

A. Gupta. Fast and effective algorithms for graph partitioning and sparse matrix reordering. IBM Journal of Research and Development, 41(1/2):171--183, 1996.

....algorithm uses this heuristic. The initial partition. Having constructed the series of graphs until the number of vertices in the coarsest graph is smaller than some threshold, the normal practice of the multilevel strategy is to carry out an initial partition. Here, following the idea of Gupta, [10], we contract until the number of vertices in the coarsest graph is the same as the number of subdomains, P , and then simply assign vertex i to subdomain S i . Unlike Gupta, however, we do not carry out repeated expansion contraction cycles of the coarsest graphs to find a well balanced initial ....

A. Gupta. Fast and effective algorithms for graph partitioning and sparse matrix reordering. IBM Journal of Research and Development, 41(1/2):171--183, 1996. 16

....algorithm uses this heuristic. The initial partition. Having constructed the series of graphs until the number of vertices in the coarsest graph is smaller than some threshold, the normal practice of the multilevel strategy is to carry out an initial partition. Here, following the idea of Gupta, [7], we contract until the number of vertices in the coarsest graph is the same as the number of subdomains, P , and then simply assign vertex i to subdomain S i . Unlike Gupta, however, we do not carry out repeated expansion contraction cycles of the coarsest graphs to find a well balanced initial ....

A. Gupta. Fast and effective algorithms for graph partitioning and sparse matrix reordering. IBM Journal of Research and Development, 41(1/2):171--183, 1996.

....strategy is to construct the series of graphs until the number of vertices in the coarsest graph is smaller than some threshold and then carry out an initial partition. In parallel, the graph is already distributed and so an initial partition already exists. Here, following the idea of Gupta, [13], we continue coarsening until the number of vertices in the coarsest graph is the same as the number of subdomains, P , and this gives us automatically an initial partition with one vertex per subdomain. Unlike Gupta, however, we do not carry out repeated interpolation contraction cycles of the ....

A. Gupta. Fast and effective algorithms for graph partitioning and sparse matrix reordering. IBM Journal of Research and Development, 41(1/2):171--183, 1996.

....weight jvj = ju 1 j ju 2 j. The initial partition. Having constructed the series of graphs until the number of vertices in the coarsest graph is smaller than some threshold, the normal practice of the multilevel strategy is to carry out an initial partition. Here, following the idea of Gupta, [10], we contract until the number of vertices in the coarsest graph is the same as the number of subdomains, P , and then simply assign vertex i to subdomain S i . Unlike Gupta, however, we do not carry out repeated expansion contraction cycles of the coarsest graphs to find a well balanced initial ....

A. Gupta. Fast and effective algorithms for graph partitioning and sparse matrix reordering. IBM Journal of Research and Development, 41(1/2):171--183, 1996.

No context found.

Anshul Gupta. Fast and effective algorithms for graph partitioning and sparse matrix reordering. Technical Report (Number to be assigned), IBM T. J. Watson Research Center, Yorktown Heights, NY, 1996.

No context found.

Anshul Gupta. Fast and effective algorithms for graph partitioning and sparse matrix ordering. IBM Journal of Research and Development, 41(1/2):171--183, January/March, 1997.

.... of the multifrontal algorithm [10] for sparse Cholesky factorization and a highly scalable parallel sparse Cholesky factorization algorithm [8, 3] The package also uses scalable parallel sparse triangular solvers [9] and an improved and parallelized version of the previously released package WGPP [4, 5] for computing fill reducing orderings. Sparse symmetric factorization in WSMP has been clocked at up to 3.6 GFLOPS on an RS6000 workstation with four 375 MHz Power3 CPUs and 90 GFLOPS on a 128 node SP with two way SMP 200 MHz Power3 nodes. For more details on the implementation and performance of ....

....by the integer value (speed) in IPARM(16) Speed = 1 results in the slowest but best ordering, speed = 3 results in fastest but worst ordering, and speed = 2 results in an intermediate and quality or ordering. ffl IPARM(17) type I: WSMP uses graph partitioning based ordering algorithms [4] to minimize fill during factorization. IPARM(17) specifies the maximum number of nodes that a subgraph must have before it is ordered by using a minimum local fill algorithm without further subpartition. The user can obtain a pure minimum local fill ordering by specifying IPARM(17) greater than ....

Anshul Gupta. Fast and effective algorithms for graph partitioning and sparse matrix ordering. IBM Journal of Research and Development, 41(1/2):171--183, January/March, 1997.

....is from a linear programming problem derived from a quadratic assignment problem obtained from AT T. In all of our experiments, we used spectral nested dissection [48] to order the matrices. The factorization algorithm described in this paper will work well with any type of nested dissection. In [21, 22, 20, 31, 30], we show that nested dissection orderings with proper selection of separators can yield better quality orderings that traditional heuristics, such as, the multiple minimum degree heuristic. 20 The performance obtained by this algorithm in some of these matrices is shown in Table 1. The ....

Anshul Gupta. Fast and effective algorithms for graph partitioning and sparse matrix ordering. IBM Journal of Research and Development, 41(1/2):171--183, January/March, 1997.

....heuristics. In [17] we presented an optimally scalable parallel algorithm for factoring a large class of sparse symmetric matrices. This algorithm works efficiently only with graph partitioning based ordering. Although, traditionally, local ordering heuristics have been preferred, recent research [24, 6, 20, 13] has shown the graph partitioning based ordering heuristics can match and often exceed the fill reduction of local heuristics. The serial ordering heuristics that WSMP uses have been described in detail in [17] Basically, the sparse matrix is regarded as the adjacency matrix of an undirected ....

Anshul Gupta. Fast and effective algorithms for graph partitioning and sparse matrix ordering. IBM Journal of Research and Development, 41(1/2):171--183, January/March, 1997.

....The best time is in boldface and the second best time is underlined. significant differences among the three softwares that affect their performance. First, they use different schemes for fill reducing ordering. By default, WSMP uses a symmetric permutation based on a nested dissection ordering [17] computed on the structure of A A 0 . MUMPS uses a symmetric permutation based on the approximate minimum degree (AMD) algorithm [7] applied to the structure of A A 0 . UMFPACK uses a column approximate minimum degree algorithm [9] to prepermute only the columns of A and computes a row ....

Anshul Gupta. Fast and effective algorithms for graph partitioning and sparse matrix ordering. IBM Journal of Research and Development, 41(1/2):171--183, January/March, 1997.

....is from a linear programming problem derived from a quadratic assignment problem obtained from AT T. In all of our experiments, we used spectral nested dissection [50] to order the matrices. The factorization algorithms described in this paper will work well with any type of nested dissection. In [21, 22, 20, 32, 31], we show that nested dissection orderings with proper selection of separators can yield better quality orderings that traditional heuristics, such as, the multiple minimum degree heuristic. The performance obtained by this algorithm in some of these matrices is shown in Table 2. The operation ....

Anshul Gupta. Fast and effective algorithms for graph partitioning and sparse matrix reordering. Technical Report (Number to be assigned), IBM T. J. Watson Research Center, Yorktown Heights, NY, 1996.

....heuristics. In [5] we presented an optimally scalable parallel algorithm for factoring a large class of sparse symmetric matrices. This algorithm works efficiently only with graph partitioning based ordering. Although, traditionally, local ordering heuristics have been preferred, recent research [9, 1, 6, 3] has shown the graph partitioning based ordering heuristics can match and often exceed the fill reduction of local heuristics. The serial ordering heuristics that WSSMP uses have been described in detail in [5] Basically, the sparse matrix is regarded as the adjacency matrix of an undirected ....

Anshul Gupta. Fast and effective algorithms for graph partitioning and sparse matrix ordering. IBM Journal of Research and Development, 41(1/2):171--183, January/March, 1997.

.... of the multifrontal algorithm [9] for sparse Cholesky factorization and a highly scalable parallel sparse Cholesky factorization algorithm [6, 3] The package also uses scalable parallel sparse triangular solvers [7] and an improved and parallelized version of the previously released package WGPP [4, 5] for computing fill reducing orderings. Sparse symmetric factorization in WSSMP has been clocked at up to 210 MFLOPS on an RS6000 590, 500 MFLOPS on an RS6000 397, 1200 MFLOPS on a Silver node (4 way SMP with 332 MHz 604e processors) and in excess of 20 GFLOPS on a 64 processor SP with RS6000 397 ....

Anshul Gupta. Fast and effective algorithms for graph partitioning and sparse matrix ordering. IBM Journal of Research and Development, 41(1/2):171--183, January/March, 1997.

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A. Gupta. Fast and effective algorithms for graph partitioning and sparse matrix reordering. IBM Journal of Research and Development, 41(1/2):171--183, 1996.

No context found.

A. Gupta. Fast and effective algorithms for graph partitioning and sparse matrix reordering. IBM Journal of Research and Development, 41(1/2):171--183, 1996.

No context found.

A. Gupta. Fast and effective algorithms for graph partitioning and sparse matrix reordering. IBM Journal of Research and Development, 41(1/2):171--183, 1996.

No context found.

A. Gupta. Fast and effective algorithms for graph partitioning and sparse matrix reordering. IBM Journal of Research and Development, 41(1/2):171--183, 1996.

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A. Gupta. Fast and effective algorithms for graph partitioning and sparse matrix reordering. IBM Journal of Research and Development, 41(1/2):171--183, 1996. 16

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Anshul Gupta. Fast and effective algorithms for graph partitioning and sparse matrix reordering. Technical Report (Number to be assigned), IBM T. J. Watson Research Center, Yorktown Heights, NY, 1996.

No context found.

A. Gupta. 1996. Fast and effective algorithms for graph partitioning and sparse matrix reordering. IBM Journal of Research and Development, 41(1/2):171--183.

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