## Fast and Effective Algorithms for Graph Partitioning and Sparse Matrix Ordering (1996)

Venue: | IBM JOURNAL OF RESEARCH AND DEVELOPMENT |

Citations: | 55 - 11 self |

### BibTeX

@ARTICLE{Gupta96fastand,

author = {Anshul Gupta},

title = {Fast and Effective Algorithms for Graph Partitioning and Sparse Matrix Ordering},

journal = {IBM JOURNAL OF RESEARCH AND DEVELOPMENT},

year = {1996},

volume = {41},

pages = {171--183}

}

### Years of Citing Articles

### OpenURL

### Abstract

Graph partitioning is a fundamental problem in several scientific and engineering applications. In this paper, we describe heuristics that improve the state-of-the-art practical algorithms used in graph-partitioning software in terms of both partitioning speed and quality. An important use of graph-partitioning is in ordering sparse matrices for obtaining direct solutions to sparse systems of linear equations arising in engineering and optimization applications. The experiments reported in this paper show that the use of these heuristics results in a considerable improvement in the quality of sparse-matrix orderings over conventional ordering methods, especially for sparse matrices arising in linear programming problems. In addition, our graph-partitioning-based ordering algorithm is more parallelizable than minimum-degree-based ordering algorithms, and it renders the ordered matrix more amenable to parallel factorization.

### Citations

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Citation Context ...r, coarser q-node graph have little impact on the overall run time of the entire algorithm. Depending on the number of partitions k, we either use a variation of the popular Kernighan-- Lin heuristic =-=[20, 21, 19, 22]-=- or a greedy refinement scheme [10, 23] for refining the edge-separators. The linear time Fiduccia--Mattheyses variation [21] of the Kernighan--Lin heuristic is used for a small number of partitions a... |

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Citation Context ...ithm by Davis, Amestoy, and Duff [4] represent the state of the art in MD-based heuristics. Recent work by the author [5, 6], Hendrickson and Rothberg [7], Ashcraft and Liu [8], and Karypis and Kumar =-=[9, 10]-=- suggests that GP-based heuristics are capable of producing betterquality orderings than MD-based heuristics for finite-element problems while staying within a small constant factor of the run time of... |

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Citation Context ...istics prompted intense research [2] to improve their run time and quality, and they have been the methods of choice among practitioners. The multiple minimum degree (MMD) algorithm by George and Liu =-=[3, 2]-=- and the approximate minimum degree (AMD) algorithm by Davis, Amestoy, and Duff [4] represent the state of the art in MD-based heuristics. Recent work by the author [5, 6], Hendrickson and Rothberg [7... |

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Citation Context ...he overall run time of the entire algorithm. Depending on the number of partitions k, we either use a variation of the popular Kernighan-- Lin heuristic [20, 21, 19, 22] or a greedy refinement scheme =-=[10, 23]-=- for refining the edge-separators. The linear time Fiduccia--Mattheyses variation [21] of the Kernighan--Lin heuristic is used for a small number of partitions and the greedy algorithm is used if the ... |

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Citation Context ...he graph of a sparse-matrix to minimize the edge-cut and distributing different partitions to different processors minimizes the communication overhead in parallel sparse matrix-vector multiplication =-=[12]-=-. Sparse matrix-vector multiplication is an integral part of all iterative schemes for solving sparse linear systems. GP-based ordering methods are more suitable for solving sparse systems using direc... |

457 |
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Citation Context ...arallel factorization and triangular solutions, a part of the parallelism would be lost if an MD-based heuristic is used to preorder the sparse matrix. 3 Multilevel graph partitioning Recent research =-=[18, 19, 10]-=- has shown multilevel algorithms to be fast and effective in computing graph-partitions. A typical multilevel graph-partitioning algorithm has four components: coarsening,sinitial partitioning, uncoar... |

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Citation Context ...ors mean less fill and work during factorization. For finite-element graphs with certain nice properties, it can be proved that this technique yields orderings within a constant factor of the optimal =-=[25, 26, 24, 3]-=-. Such bounds cannot be proved for arbitrary matrices without a well-defined structure, such as the sparse matrices arising in LP computations. However, as we show in [27] and Section 5, with steps) 9... |

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Citation Context ...r, coarser q-node graph have little impact on the overall run time of the entire algorithm. Depending on the number of partitions k, we either use a variation of the popular Kernighan-- Lin heuristic =-=[20, 21, 19, 22]-=- or a greedy refinement scheme [10, 23] for refining the edge-separators. The linear time Fiduccia--Mattheyses variation [21] of the Kernighan--Lin heuristic is used for a small number of partitions a... |

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Citation Context ...ave been the methods of choice among practitioners. The multiple minimum degree (MMD) algorithm by George and Liu [3, 2] and the approximate minimum degree (AMD) algorithm by Davis, Amestoy, and Duff =-=[4]-=- represent the state of the art in MD-based heuristics. Recent work by the author [5, 6], Hendrickson and Rothberg [7], Ashcraft and Liu [8], and Karypis and Kumar [9, 10] suggests that GP-based heuri... |

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Citation Context ...ors mean less fill and work during factorization. For finite-element graphs with certain nice properties, it can be proved that this technique yields orderings within a constant factor of the optimal =-=[25, 26, 24, 3]-=-. Such bounds cannot be proved for arbitrary matrices without a well-defined structure, such as the sparse matrices arising in LP computations. However, as we show in [27] and Section 5, with steps) 9... |

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Citation Context ...pplication of graph-partitioning is in computing fill-reducing orderings of sparse matrices for solving large sparse systems of linear equations. Finding an optimal ordering is an NP-complete problem =-=[1]-=- and heuristics must be used to obtain an acceptable non-optimal solution. Improving the run time and quality of ordering heuristics has been a subject of research for almost three decades. Two main c... |

131 |
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Citation Context ...rix of a graph and follow a divide-and-conquer strategy to label the nodes of the graph by partitioning it into smaller subgraphs. The initial success of MD-based heuristics prompted intense research =-=[2]-=- to improve their run time and quality, and they have been the methods of choice among practitioners. The multiple minimum degree (MMD) algorithm by George and Liu [3, 2] and the approximate minimum d... |

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Citation Context ...ithm by Davis, Amestoy, and Duff [4] represent the state of the art in MD-based heuristics. Recent work by the author [5, 6], Hendrickson and Rothberg [7], Ashcraft and Liu [8], and Karypis and Kumar =-=[9, 10]-=- suggests that GP-based heuristics are capable of producing betterquality orderings than MD-based heuristics for finite-element problems while staying within a small constant factor of the run time of... |

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Citation Context ...tation. In addition to being parallelizable itself, a GP-based ordering also aids the parallelization of the factorization and triangular solution phases of a direct solver. Gupta, Karypis, and Kumar =-=[15, 16]-=- have proposed a highly scalable parallel formulation of sparse Cholesky factorization. This algorithm derives a significant part of its parallelism from the underlying partitioning of the graph of th... |

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Citation Context ...n MD-based methods in two respects. There is strong theoretical and experimental evidence that the process of graph-partitioning and sparse-matrix ordering based on it can be parallelized effectively =-=[13]-=-. On the other hand, the only attempt to perform a minimum-degree ordering in parallel that we are aware of [14] was not successful in reducing the ordering time over a serial implementation. In addit... |

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Citation Context ...ubgraphs after each step of uncoarsening so that few edges connect nodes of different subgraphs in the final partitioning of the original graph. Then they use an algorithm for finding a minimum cover =-=[29, 30]-=- to compute a node-separator from the edge-separator. This approach relies heavily on the assumption that the size of a node-separator is proportional to the size of the edge-separator containing it. ... |

74 |
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Citation Context ...arallel factorization and triangular solutions, a part of the parallelism would be lost if an MD-based heuristic is used to preorder the sparse matrix. 3 Multilevel graph partitioning Recent research =-=[18, 19, 10]-=- has shown multilevel algorithms to be fast and effective in computing graph-partitions. A typical multilevel graph-partitioning algorithm has four components: coarsening,sinitial partitioning, uncoar... |

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Citation Context ...2] and the approximate minimum degree (AMD) algorithm by Davis, Amestoy, and Duff [4] represent the state of the art in MD-based heuristics. Recent work by the author [5, 6], Hendrickson and Rothberg =-=[7]-=-, Ashcraft and Liu [8], and Karypis and Kumar [9, 10] suggests that GP-based heuristics are capable of producing betterquality orderings than MD-based heuristics for finite-element problems while stay... |

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Citation Context ... minimum degree (AMD) algorithm by Davis, Amestoy, and Duff [4] represent the state of the art in MD-based heuristics. Recent work by the author [5, 6], Hendrickson and Rothberg [7], Ashcraft and Liu =-=[8]-=-, and Karypis and Kumar [9, 10] suggests that GP-based heuristics are capable of producing betterquality orderings than MD-based heuristics for finite-element problems while staying within a small con... |

42 |
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Citation Context ...shes have many points with more than one degree of freedom. The sparse matrices resulting from such finite-element problems have many small groups of nodes that share the same adjacency structure. In =-=[32]-=-, Ashcraft describes a technique to compress such graphs into smaller graphs by coalescing the nodes with identical adjacency structures. As a result of compression, the High-degree nodes Figure 6: Il... |

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Citation Context ...r, coarser q-node graph have little impact on the overall run time of the entire algorithm. Depending on the number of partitions k, we either use a variation of the popular Kernighan-- Lin heuristic =-=[20, 21, 19, 22]-=- or a greedy refinement scheme [10, 23] for refining the edge-separators. The linear time Fiduccia--Mattheyses variation [21] of the Kernighan--Lin heuristic is used for a small number of partitions a... |

27 |
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Citation Context ...rix columns corresponding to the nodes are eliminated during numerical factorization. The overall approach of our ordering algorithm follows the fundamental technique of generalized nested dissection =-=[24]-=-. The graph is bisected by finding and removing a node-separator, labeling the nodes of the two resulting subgraphs by applying the same technique recursively, and labeling the nodes of the separator ... |

16 | Sparse matrix ordering methods for interior point linear programming - Rothberg, Hendrickson - 1998 |

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Citation Context ...tation. In addition to being parallelizable itself, a GP-based ordering also aids the parallelization of the factorization and triangular solution phases of a direct solver. Gupta, Karypis, and Kumar =-=[15, 16]-=- have proposed a highly scalable parallel formulation of sparse Cholesky factorization. This algorithm derives a significant part of its parallelism from the underlying partitioning of the graph of th... |

16 |
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Citation Context ..., graph bisection yields two submatrices of the original matrix that can be factored independently in parallel. This is the basis of many efficient parallel algorithms for sparse-matrix factorization =-=[15, 16, 28]-=-. 4.1 Graph bisection A key step in our ordering algorithm is finding a small node bisector of a graph. This can be accomplished by the heuristics described in Section 3 with some modifications to the... |

11 |
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(Show Context)
Citation Context ...or; i.e., in the final steps of uncoarsening, the subject of refinement is changed from the edge-separator to a node-separator. For refining the node-separator, we use Ashcraft and Liu's modification =-=[31]-=- of the Fiduccia--Mattheyses algorithm [21]. 4.2 Recursive bisection and ordering Once a node-separator of the original graph is found, it is removed and the entire procedure is repeated recursively o... |

10 |
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Citation Context ... algorithm by George and Liu [3, 2] and the approximate minimum degree (AMD) algorithm by Davis, Amestoy, and Duff [4] represent the state of the art in MD-based heuristics. Recent work by the author =-=[5, 6]-=-, Hendrickson and Rothberg [7], Ashcraft and Liu [8], and Karypis and Kumar [9, 10] suggests that GP-based heuristics are capable of producing betterquality orderings than MD-based heuristics for fini... |

9 |
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(Show Context)
Citation Context ... algorithm by George and Liu [3, 2] and the approximate minimum degree (AMD) algorithm by Davis, Amestoy, and Duff [4] represent the state of the art in MD-based heuristics. Recent work by the author =-=[5, 6]-=-, Hendrickson and Rothberg [7], Ashcraft and Liu [8], and Karypis and Kumar [9, 10] suggests that GP-based heuristics are capable of producing betterquality orderings than MD-based heuristics for fini... |

8 | Parallel algorithms for forward and back substitution in direct solution of sparse linear systems
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Citation Context ...hly scalable parallel formulation of sparse Cholesky factorization. This algorithm derives a significant part of its parallelism from the underlying partitioning of the graph of the sparse matrix. In =-=[17]-=-, Gupta and Kumar present efficient parallel algorithms for solving lower- and upper-triangular systems resulting from sparse factorization. In both parallel factorization and triangular solutions, a ... |

4 |
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Citation Context ...ubgraphs after each step of uncoarsening so that few edges connect nodes of different subgraphs in the final partitioning of the original graph. Then they use an algorithm for finding a minimum cover =-=[29, 30]-=- to compute a node-separator from the edge-separator. This approach relies heavily on the assumption that the size of a node-separator is proportional to the size of the edge-separator containing it. ... |

3 |
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(Show Context)
Citation Context ...aph-partitioning and sparse-matrix ordering based on it can be parallelized effectively [13]. On the other hand, the only attempt to perform a minimum-degree ordering in parallel that we are aware of =-=[14]-=- was not successful in reducing the ordering time over a serial implementation. In addition to being parallelizable itself, a GP-based ordering also aids the parallelization of the factorization and t... |