## Multilevel Graph Partitioning Schemes (1995)

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Venue: | Proc. 24th Intern. Conf. Par. Proc., III |

Citations: | 30 - 0 self |

### BibTeX

@INPROCEEDINGS{Karypis95multilevelgraph,

author = {George Karypis and Vipin Kumar},

title = {Multilevel Graph Partitioning Schemes},

booktitle = {Proc. 24th Intern. Conf. Par. Proc., III},

year = {1995},

pages = {113--122},

publisher = {CRC Press}

}

### Years of Citing Articles

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### Abstract

Abstract – In this paper we present experiments with a class of graph partitioning algorithms that reduce the size of the graph by collapsing vertices and edges, partition the smaller graph, and then uncoarsen it to construct a partition for the original graph. We investigate the effectiveness of many different choices for all three phases: coarsening, partition of the coarsest graph, and refinement. In particular, we present a new coarsening heuristic (called heavy-edge heuristic) for which the size of the partition of the coarse graph is within a small factor of the size of the final partition obtained after multilevel refinement. We also present a new scheme for refining during uncoarsening that is much faster than the Kernighan-Lin refinement. We test our scheme on a large number of graphs arising in various domains including finite element methods, linear programming, VLSI, and transportation. Our experiments show that our scheme consistently produces partitions that are better than those produced by spectral partitioning schemes in substantially smaller timer (10 to 35 times faster than multilevel spectral bisection). Also, when our scheme is used to compute fill reducing orderings for sparse matrices, it substantially outperforms the widely used multiple minimum degree algorithm. 1

### Citations

1128 |
An efficient heuristic procedure for partitioning graphs
- Kernighan, Lin
- 1970
(Show Context)
Citation Context ...aph. A partition of G m can be obtained using various algorithms such as (a) spectral bisection [33, 2, 18], (b) geometric bisection [28] (if coordinates are available), and (c) combinatorial methods =-=[25, 8, 9]-=-. Since the size of the coarser graph Gm is small (i.e., |Vm| < 100), this step takes a small amount. We implemented three different algorithms for partitioning the coarse graph. The first algorithm u... |

538 |
Computer Solution of Large Sparse Positive Definite Systems
- George, Liu
- 1981
(Show Context)
Citation Context ...acilities was provided by Cray Research Inc. Related papers are available via WWW at URL: http://www.cs.umn.edu/users/kumar/papers.html 1 lead to high degree of concurrency in the factorization phase =-=[26, 9]-=-. The multiple minimum degree orderingused almost exclusively in serial direct methods is not suitable for parallel direct methods, as it provides very little concurrency in the parallel factorization... |

520 |
Partitioning sparse matrices with eigenvectors of graphs
- Pothen, Simon, et al.
- 1990
(Show Context)
Citation Context ...ms have been developed that find a reasonably good partition. Spectral partitioning methods are known to produce excellent partitions for a wide class of problems, and they are used quite extensively =-=[33, 20]-=-. However, these methods are very expensive since they require the computation of the eigenvector corresponding to the second smallest eigenvalue (Fiedler vector). Execution of the spectral methods ca... |

506 |
Introdu to Parallel Computing - Design and Analysis of Algorithms. The Benjamin/Cummings Publishing Company
- Kumar
- 1994
(Show Context)
Citation Context ...is the multiplication of a sparse matrix and a (dense) vector. The problem of minimizing communication in this step is identical to the problem of partitioning the graph corresponding to the matrix A =-=[26]-=-. 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 ∗ This work is sponsored by the AH... |

466 |
A multilevel algorithm for partitioning graphs
- Hendrickson, Leland
- 1993
(Show Context)
Citation Context ...ms have been developed that find a reasonably good partition. Spectral partitioning methods are known to produce excellent partitions for a wide class of problems, and they are used quite extensively =-=[33, 20]-=-. However, these methods are very expensive since they require the computation of the eigenvector corresponding to the second smallest eigenvalue (Fiedler vector). Execution of the spectral methods ca... |

326 |
A Linear Time Heuristic for Improving Network partitions
- Fiduccia, Mattheyses
- 1982
(Show Context)
Citation Context ...A\A ′ ∪ B ′ and B\B ′ ∪ A ′ is a bisection with a smaller edge-cut. A class of algorithms that tend to produce very good results are those that are based on the Kernighan-Lin (KL) partition algorithm =-=[25, 6, 20]-=-. The KL algorithm is iterative in nature. It starts with an initial partition and in each iteration it finds subsets A ′ and B ′ with the above properties. If such subsets exist, then it moves them t... |

286 | A Fast Multilevel Implementation of Recursive Spectral Bisection for Partitioning Unstructured Problems”, Concurency: Practice and Experience
- Barnard, Simon
- 1994
(Show Context)
Citation Context ...ctor corresponding to the second smallest eigenvalue (Fiedler vector). Execution of the spectral methods can be speeded up if computation of the Fiedler vector is done by using a multilevel algorithm =-=[2]-=-. This multilevel spectral bisection algorithm (MSB) usually manages to speedup the spectral partitioning methods by an order of magnitude without any loss in the quality of the edge-cut. However, eve... |

188 |
An Improved Spectral Graph Partitioning Algorithm for Mapping Parallel Computations
- Hendrickson, Leland
- 1995
(Show Context)
Citation Context ...se graph Gm = (Vm, Em) such that each part contains roughly half of the vertex weight of the original graph. A partition of G m can be obtained using various algorithms such as (a) spectral bisection =-=[33, 2, 18]-=-, (b) geometric bisection [28] (if coordinates are available), and (c) combinatorial methods [25, 8, 9]. Since the size of the coarser graph Gm is small (i.e., |Vm| < 100), this step takes a small amo... |

146 | Modification of the minimum-degree algorithm by multiple elimination - Liu - 1985 |

137 | The evolution of the minimum degree ordering algorithm - George, Liu - 1989 |

123 |
Parallel algorithms for sparse linear systems
- HEATH, NG, et al.
- 1991
(Show Context)
Citation Context ...ecause it minimizes the edge-cut, the communication overhead is also minimized. Another importantapplication of recursive bisection is to find a fill reducing ordering for sparse matrix factorization =-=[9, 26, 16]-=-. This type of algorithms are generally referred to as nested dissection ordering algorithms. Nested dissection recursively splits a graph into almost equal halves by selecting a vertex separator unti... |

120 | Highly scalable parallel algorithms for sparse matrix factorization
- Gupta, Karypis, et al.
- 1994
(Show Context)
Citation Context ...ing ordering using MSB can be several orders of magnitude higher than the time taken by the parallel factorization algorithm, and thus ordering time can dominate the overall time to solve the problem =-=[14]-=-. The execution time of MSB can be further speeded up by computing the Fiedler vector in parallel. The algorithm for computing the Fiedler vector, is iterative and in each iteration it performs a matr... |

105 | Geometric mesh partitioning: Implementation and experiments
- Gilbert, Miller, et al.
- 1995
(Show Context)
Citation Context ...ever, due to the randomized nature of these algorithms,smultiple trials are often required to obtain solutions that are comparable in quality to spectral methods. Multiple trials do increase the time =-=[13]-=-, but the overall runtime is still substantially lower than the time required by the spectral methods. However, geometric graph partitioning algorithms have limited applicability because often the geo... |

96 | Parallel multilevel graph partitioning
- Karypis, Kumar
- 1995
(Show Context)
Citation Context ...t compared with spectral methods, they can still be the bottleneck if the sparse system of equations is being solved in parallel [26, 14]. The coarsening phase of these methods is easy to parallelize =-=[23]-=-, but the Kernighan-Lin heuristic used in the refinement phase is very difficult to speedup in parallel computers [12]. Since both the coarsening phase and the refinement phase with Kernighan-Lin heur... |

89 |
A unified geometric approach to graph separators
- Miller, Teng, et al.
- 1991
(Show Context)
Citation Context ...me is spent in performing communication [21, 1]. Another class of graph partitioning techniques uses the geometric information of the graph to find a good partition. Geometric partitioning algorithms =-=[17, 28, 29]-=- tend to be fast but often yield partitions that are worse than those obtained by spectral methods. Among the most prominent of these scheme is the algorithm described in [28]. This algorithm produces... |

89 |
Computing the block triangular form of a sparse matrix
- Pothen, Fan
- 1990
(Show Context)
Citation Context ...ered by recursively applying nested dissection ordering. In our multilevel nested dissection algorithm (MLND) a vertex separator is computed from an edge separator by finding the minimum vertex cover =-=[31]-=-. The minimum vertex cover has been found to produce very small vertex separators. Overall quality of a fill reducing ordering depends on whether or not the matrix is factored on a serial or parallel ... |

75 |
A heuristic for reducing fill in sparse matrix factorization
- Bui, Jones
- 1993
(Show Context)
Citation Context ...the graph) by collapsing vertices and edges, partition the smaller graph, and then uncoarsen it to construct a partition for the original graph. These are called multilevel graph partitioning schemes =-=[3, 5, 15, 20, 7, 30]-=-. Some researchers investigated multilevel schemes primarily to decrease the partitioning time, at the cost of somewhat worse partition quality [30]. Recently, a number of multilevel algorithms have b... |

72 |
The chaco user’s guide, version 1.0
- Hendrickson, Leland
- 1993
(Show Context)
Citation Context ...irectly. Note that MSB is a significantly different scheme than the multilevel scheme that uses spectral bisection to partition the graph at the coarsest level. We used the MSB algorithm in the Chaco =-=[19]-=- graph partitioning package to produce partitions for some of the matrices in Table 1 and compared them against the partitions produced by our multilevel algorithm that uses HEM during coarsening phas... |

64 |
An Improved Two-Way Partitioning Algorithm with Stable Performance
- Cheng, Wei
- 1991
(Show Context)
Citation Context ...the graph) by collapsing vertices and edges, partition the smaller graph, and then uncoarsen it to construct a partition for the original graph. These are called multilevel graph partitioning schemes =-=[3, 5, 15, 20, 7, 30]-=-. Some researchers investigated multilevel schemes primarily to decrease the partitioning time, at the cost of somewhat worse partition quality [30]. Recently, a number of multilevel algorithms have b... |

57 |
Solving finite element equations on concurrent computers
- Nour-Omid, Raefsky, et al.
- 1986
(Show Context)
Citation Context ...me is spent in performing communication [21, 1]. Another class of graph partitioning techniques uses the geometric information of the graph to find a good partition. Geometric partitioning algorithms =-=[17, 28, 29]-=- tend to be fast but often yield partitions that are worse than those obtained by spectral methods. Among the most prominent of these scheme is the algorithm described in [28]. This algorithm produces... |

46 | A new approach to effective circuit clustering
- Hagen, Kahng
- 1992
(Show Context)
Citation Context ...the graph) by collapsing vertices and edges, partition the smaller graph, and then uncoarsen it to construct a partition for the original graph. These are called multilevel graph partitioning schemes =-=[3, 5, 15, 20, 7, 30]-=-. Some researchers investigated multilevel schemes primarily to decrease the partitioning time, at the cost of somewhat worse partition quality [30]. Recently, a number of multilevel algorithms have b... |

34 |
A parallel graph partitioning algorithm for a message-passing multiprocessor
- Gilbert, Zmijewski
- 1987
(Show Context)
Citation Context ...n parallel [26, 14]. The coarsening phase of these methods is easy to parallelize [23], but the Kernighan-Lin heuristic used in the refinement phase is very difficult to speedup in parallel computers =-=[12]-=-. Since both the coarsening phase and the refinement phase with Kernighan-Lin heuristic take roughly the same amount of time, the overall scheme cannot be speeded up significantly. Our new faster meth... |

30 |
Finding clusters in VLSI circuits
- Garbers, Promel, et al.
- 1990
(Show Context)
Citation Context |

30 | Spectral nested dissection
- Pothen, Simon, et al.
- 1992
(Show Context)
Citation Context ...LND algorithm is amenable to parallelization. In [23] we present a parallel formulation of our MLND algorithm that achieves a speedup of 57 on 128-processor Cray T3D. Spectral nested dissection (SND) =-=[32]-=- is a widely used ordering algorithm for ordering matrices for parallel factorization. As in the case of MLND, the minimum vertex cover algorithm was used to compute a vertex separator from the edge s... |

27 |
Nested dissection of a regular finite-element mesh
- George
- 1973
(Show Context)
Citation Context ...aph. A partition of G m can be obtained using various algorithms such as (a) spectral bisection [33, 2, 18], (b) geometric bisection [28] (if coordinates are available), and (c) combinatorial methods =-=[25, 8, 9]-=-. Since the size of the coarser graph Gm is small (i.e., |Vm| < 100), this step takes a small amount. We implemented three different algorithms for partitioning the coarse graph. The first algorithm u... |

23 |
A parallel implementation of multilevel recursive spectral bisection for application to adaptive unstructured meshes
- Barnard, Simon
- 1995
(Show Context)
Citation Context ...—a problem that MSB is trying to solve in the first place. As a result, parallel implementation of spectral methods exhibit poor efficiency since most of the time is spent in performing communication =-=[21, 1]-=-. Another class of graph partitioning techniques uses the geometric information of the graph to find a good partition. Geometric partitioning algorithms [17, 28, 29] tend to be fast but often yield pa... |

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 |

16 | A parallel formulation of interior point algorithms
- Karypis, Gupta, et al.
- 1994
(Show Context)
Citation Context ...ral nested dissection algorithm. The matrices are displayed in increasing number of equations. Bars above the baseline indicate that the MLND algorithm performs better. concurrency and better balance =-=[24, 16]-=-. Therefore, when the factorization is performed in parallel, the better utilization of the processors can cause the ratio of the run time of parallel factorization algorithms running ordered using MM... |

12 |
A Cartesian nested dissection algorithm
- Heath, Raghavan
- 1992
(Show Context)
Citation Context ...me is spent in performing communication [21, 1]. Another class of graph partitioning techniques uses the geometric information of the graph to find a good partition. Geometric partitioning algorithms =-=[17, 28, 29]-=- tend to be fast but often yield partitions that are worse than those obtained by spectral methods. Among the most prominent of these scheme is the algorithm described in [28]. This algorithm produces... |

4 | Finite element methods on the connection machine cm-5 system - Johan, Mathur, et al. - 1993 |

2 | Geometric spectral partitioning (draft - Chan, Gilbert, et al. - 1994 |

2 |
A parallel implementtaion of the multiple minimum degree ordering heuristic
- Ghose, Rothberg
- 1994
(Show Context)
Citation Context ...nts. The MMD algorithmusually takes two to three times less time to order the matrices in Table 1 than the time required by MLND. However, efforts to parallelize the MMD algorithm have had no success =-=[11]-=-. In fact, the MMD algorithm appears to be inherently serial in nature. On the other hand, the MLND algorithm is amenable to parallelization. In [23] we present a parallel formulation of our MLND algo... |