## A Parallel Algorithm for Multilevel Graph Partitioning and Sparse Matrix Ordering (1996)

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Citations: | 81 - 7 self |

### BibTeX

@MISC{Karypis96aparallel,

author = {George Karypis and Vipin Kumar},

title = {A Parallel Algorithm for Multilevel Graph Partitioning and Sparse Matrix Ordering },

year = {1996}

}

### Years of Citing Articles

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### Abstract

### Citations

1120 |
An efficient heuristic procedure for partitioning graphs
- Kernighan, Lin
- 1970
(Show Context)
Citation Context ...balanced (i.e., each part has the same weight). A class of local refinement algorithms that tend to produce very good results are those that are based on the Kernighan-Lin (KL) partitioning algorithm =-=[18]-=- and their variants (FM) [6, 12, 16]. 3 Parallel Multilevel Graph Partitioning Algorithm There are two types of parallelism that can be exploited in the p-way graph partitioning algorithm based on the... |

534 |
Computer Solution of Large Sparse Positive Definite Matrices
- GEORGE, LIU
- 1981
(Show Context)
Citation Context ... are used to solve a sparse system of equations, then a graph partitioning algorithm can be used to compute a fill reducing ordering that lead to high degree of concurrency in the factorization phase =-=[19, 8]-=-. The multiple minimum degree ordering used almost exclusively in serial direct methods is not suitable for parallel direct methods, as it provides limited concurrency in the parallel factorization ph... |

511 | Parallel Multilevel k-way Partition Scheme for Irregular Graphs
- Karypis, Kumar
- 1999
(Show Context)
Citation Context ...multilevel graph partitioning and sparse matrix ordering algorithm. A key feature of our parallel formulation (that distinguishes it from other proposed parallel formulations of multilevel algorithms =-=[2, 1, 24, 14]-=-) is that it partitions the vertices of the graph into pp parts while distributing the overall adjacency matrix of the graph among all p processors. This mapping results in substantially smaller commu... |

505 |
Introduction to Parallel Computing: Design and Analysis of Algorithms
- Kumar, Grama, et al.
- 1994
(Show Context)
Citation Context ...ration of these methods is the multiplication of a sparse matrix and a (dense) vector. Partitioning the graph that corresponds to matrix A, is used to significantly reduce the amount of communication =-=[19]-=-. If parallel direct methods are used to solve a sparse system of equations, then a graph partitioning algorithm can be used to compute a fill reducing ordering that lead to high degree of concurrency... |

464 |
A multilevel algorithm for partitioning graphs
- Hendrickson, Leland
- 1993
(Show Context)
Citation Context ...many algorithms have been developed that find a reasonably good partition. Recently, a new class of multilevel graph partitioning techniques was introduced by Bui & Jones [4] and Hendrickson & Leland =-=[12]-=-, and further studied by Karypis & Kumar [16, 15, 13]. These multilevel schemes provide excellent graph partitionings and have moderate computational complexity. Even though these multilevel algorithm... |

381 | A simple parallel algorithm for the maximal independent set problem
- Luby
- 1986
(Show Context)
Citation Context ...llelize algorithms based on depth-first traversal of the graph. Another possibility is to adapt some of the algorithms that have been developed for the PRAM model. In particular the algorithm of Luby =-=[21]-=- for computing the maximal independent set can be used to find a matching. However, parallel formulations of this type of algorithms also have high communication overhead because each processor pi nee... |

325 |
R.M.Mattheyses, “A Linear Time Heuristic for Improve Network
- Fiduccia
- 1982
(Show Context)
Citation Context ... the same weight). A class of local refinement algorithms that tend to produce very good results are those that are based on the Kernighan-Lin (KL) partitioning algorithm [18] and their variants (FM) =-=[6, 12, 16]-=-. 3 Parallel Multilevel Graph Partitioning Algorithm There are two types of parallelism that can be exploited in the p-way graph partitioning algorithm based on the multilevel bisection described in S... |

282 | A fast multilevel implementation of recursive spectral bisection for partitioning unstructured problems
- Barnard, Simon
- 1993
(Show Context)
Citation Context ... partition. multilevel does slightly better than the serial multilevel. Figure 7 shows the size of the edge-cut of the parallel algorithm compared to the Multilevel Spectral Bisection algorithm (MSB) =-=[3]-=-. The MSB algorithm is a widely used algorithm that has been found to generate high quality partitions with small edge-cuts. We used the Chaco [11] graph partitioning package to produce the MSB partit... |

144 |
Modification of the minimum degree algorithm by multiple elimination
- LIU
- 1985
(Show Context)
Citation Context ...lesky factor L of the resulting matrix for 16, 32, and 64 processors. On p processors, the ordering is computed by using nested dissection for the first log p levels, and then multiple minimum degree =-=[20]-=- (MMD) is used to order the submatrices stored locally on each processor. Matrix T16 jLj T32 jLj T64 jLj BCSSTK31 1.7 5588914 1.3 5788587 1.0 6229749 BCSSTK32 2.2 7007711 1.7 7269703 1.3 7430756 BRACK... |

122 |
Parallel algorithms for sparse linear systems
- MT, Ng, et al.
- 1991
(Show Context)
Citation Context ...duced to that of performing a sequence of 2-way partitions or bisections. Even though this scheme does not necessarily lead to optimal partition [27, 15], it is used extensively due to its simplicity =-=[8, 10]-=-. The basic structure of the multilevel bisection algorithm is very simple. The graph G D .V; E/ is first coarsened down to a few thousand vertices (coarsening phase), a bisection of this much smaller... |

120 | A scalable parallel algorithm for sparse matrix factorization
- Gupta, Kumar
- 1994
(Show Context)
Citation Context ...dvantage of the significantly higher amount of memory available in parallel computers. Furthermore, with recent development of highly parallel formulations of sparse Cholesky factorization algorithms =-=[9, 17, 25]-=-, numeric factorization on parallel computers can take much less time than the step for computing a fill-reducing ordering on a serial computer. For example, on a 1024-processor Cray T3D, some matrice... |

96 | Parallel multilevel graph partitioning - Karypis, Kumar - 1995 |

90 | How good is recursive bisection
- Simon, Teng
- 1997
(Show Context)
Citation Context ...Thus, the problem of performing a p-way partition is reduced to that of performing a sequence of 2-way partitions or bisections. Even though this scheme does not necessarily lead to optimal partition =-=[27, 15]-=-, it is used extensively due to its simplicity [8, 10]. The basic structure of the multilevel bisection algorithm is very simple. The graph G D .V; E/ is first coarsened down to a few thousand vertice... |

87 |
Computing the block triangular form of a sparse matrix
- Pothen, Fan
- 1990
(Show Context)
Citation Context ... boundary induced separator can be easily constructed by simply choosing the smaller of A and B. However, a different separator can be constructed using a minimum cover algorithm for bipartite graphs =-=[23]-=- that contains subsets of vertices from both A and B. In many cases, this new separator S may have 20% to 40% fewer vertices than either A or B. Since, the size of the vertex separator directly affect... |

75 |
A heuristic for reducing fill in sparse matrix factorization
- Bui, Jones
- 1993
(Show Context)
Citation Context ...lem is NP-complete. However, many algorithms have been developed that find a reasonably good partition. Recently, a new class of multilevel graph partitioning techniques was introduced by Bui & Jones =-=[4]-=- and Hendrickson & Leland [12], and further studied by Karypis & Kumar [16, 15, 13]. These multilevel schemes provide excellent graph partitionings and have moderate computational complexity. Even tho... |

75 |
A fast and highly quality multilevel scheme for partitioning irregular graphs
- Karypis, Kumar
(Show Context)
Citation Context ... a reasonably good partition. Recently, a new class of multilevel graph partitioning techniques was introduced by Bui & Jones [4] and Hendrickson & Leland [12], and further studied by Karypis & Kumar =-=[16, 15, 13]-=-. These multilevel schemes provide excellent graph partitionings and have moderate computational complexity. Even though these multilevel algorithms are quite fast compared with spectral methods, para... |

73 |
The chaco user’s guide, version 1.0
- Hendrickson, Leland
- 1993
(Show Context)
Citation Context ... to the Multilevel Spectral Bisection algorithm (MSB) [3]. The MSB algorithm is a widely used algorithm that has been found to generate high quality partitions with small edge-cuts. We used the Chaco =-=[11]-=- graph partitioning package to produce the MSB partitions. As before, any bars above the baseline indicate that the parallel algorithm generates partitions with higher edge-cuts. From this figure we s... |

45 | Parallel algorithms for dynamically partitioning unstructured grids
- Diniz, Plimpton, et al.
- 1995
(Show Context)
Citation Context ... our parallel refinement algorithm is to select a group of vertices to swap from one part to the other instead of selecting a single vertex. Refinement schemes that use similar ideas are described in =-=[26, 5]-=-;. However, our algorithm differs in two important ways from the other schemes: (i) it uses a different method for selecting vertices; (ii) it uses a two-dimensional partition to minimize communicatio... |

41 |
Performance of panel and block approaches to sparse Cholesky factorization on the iPSC/860 and Paragon multicomputers
- Rothberg
- 1994
(Show Context)
Citation Context ...dvantage of the significantly higher amount of memory available in parallel computers. Furthermore, with recent development of highly parallel formulations of sparse Cholesky factorization algorithms =-=[9, 17, 25]-=-, numeric factorization on parallel computers can take much less time than the step for computing a fill-reducing ordering on a serial computer. For example, on a 1024-processor Cray T3D, some matrice... |

36 | Pmrsb: Parallel multilevel recursive spectral bisection
- Barnard
- 1995
(Show Context)
Citation Context ...multilevel graph partitioning and sparse matrix ordering algorithm. A key feature of our parallel formulation (that distinguishes it from other proposed parallel formulations of multilevel algorithms =-=[2, 1, 24, 14]-=-) is that it partitions the vertices of the graph into pp parts while distributing the overall adjacency matrix of the graph among all p processors. This mapping results in substantially smaller commu... |

30 |
Finding clusters in VLSI circuits
- Garbers, Promel, et al.
- 1990
(Show Context)
Citation Context ...he context of the serial multilevel graph partitioning algorithm presented in [16]. However, nearly all of the discussion in this paper is applicable to other multilevel graph partitioning algorithms =-=[4, 12, 7, 22]-=-. The rest of the paper is organized as follows. Section 2 surveys the different types of graph partitioning algorithms that are widely used today. Section 2 briefly describes the serial multilevel al... |

27 |
Parallelism in graph partitioning
- Savage, Wloka
- 1991
(Show Context)
Citation Context ... our parallel refinement algorithm is to select a group of vertices to swap from one part to the other instead of selecting a single vertex. Refinement schemes that use similar ideas are described in =-=[26, 5]-=-;. However, our algorithm differs in two important ways from the other schemes: (i) it uses a different method for selecting vertices; (ii) it uses a two-dimensional partition to minimize communicatio... |

23 |
A parallel implementation of multilevel recursive spectral bisection for application to adaptive unstructured meshes
- Barnard, Simon
- 1995
(Show Context)
Citation Context ...multilevel graph partitioning and sparse matrix ordering algorithm. A key feature of our parallel formulation (that distinguishes it from other proposed parallel formulations of multilevel algorithms =-=[2, 1, 24, 14]-=-) is that it partitions the vertices of the graph into pp parts while distributing the overall adjacency matrix of the graph among all p processors. This mapping results in substantially smaller commu... |

20 | Graph contraction and physical optimization methods: a quality-cost tradeoff for mapping data on parallel computers
- Ponnusamy, Mansour, et al.
- 1993
(Show Context)
Citation Context ...he context of the serial multilevel graph partitioning algorithm presented in [16]. However, nearly all of the discussion in this paper is applicable to other multilevel graph partitioning algorithms =-=[4, 12, 7, 22]-=-. The rest of the paper is organized as follows. Section 2 surveys the different types of graph partitioning algorithms that are widely used today. Section 2 briefly describes the serial multilevel al... |

15 | Parallel ordering using edge contraction
- Raghavan
- 1995
(Show Context)
Citation Context |

3 |
Fast sparse Cholesky factorization on scalable parallel computers
- Karypis, Kumar
- 1994
(Show Context)
Citation Context ... isoefficiency [19] for 2D finite element graphs is O.p1:5 log3 p/, and for 3D finite element graphs is O.p log2 p/. We have recently developed a highly parallel sparse direct factorization algorithm =-=[17, 9]-=-. the isoefficiency of this algorithm is O.p1:5/ for both 2D and 3D finite element graphs. Thus, for 3D problems, the parallel ordering does not affect the overall scalability of the ordering-factoriz... |