## On The Optimality Of The Median Cut Spectral Bisection Graph Partitioning Method (1997)

Venue: | SIAM JOURNAL ON SCIENTIFIC COMPUTING |

Citations: | 18 - 3 self |

### BibTeX

@ARTICLE{Chan97onthe,

author = {Tony F. Chan and P. Ciarlet and Jr. and W. K. Szeto},

title = {On The Optimality Of The Median Cut Spectral Bisection Graph Partitioning Method},

journal = {SIAM JOURNAL ON SCIENTIFIC COMPUTING},

year = {1997},

volume = {18},

pages = {943--948}

}

### Years of Citing Articles

### OpenURL

### Abstract

Recursive Spectral Bisection is a heuristic technique for finding a minimum cut graph bisection. In it, the second eigenvector of the Laplacian of the graph is computed and from it a bisection is obtained. The most common method is to use the median of the components of the second eigenvector to induce a bisection. We prove here that this median cut method is optimal in the sense that the partition vector induced by it is the closest partition vector, in any l s norm, for s 1, to the second eigenvector. Moreover, we prove that the same result also holds for any m-partition, that is a partition into m and (n - m) vertices, when using the mth largest or smallest components of the second eigenvector.

### Citations

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An efficient heuristic procedure for partitioning graphs
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Citation Context ...as staying as close to the global minimum as possible while remaining feasible for the discrete problem. Seen in this light, it is also clear that a local search method (e.g. the Kernighan-Lin method =-=[6]-=-) following the partition (see e.g. [4, 5, 9]) could be much more effective than using it to find the global minimum directly. 6. Acknowledgements. We wish to thank an anonymous referee for making a r... |

486 |
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Citation Context ...lete problem. Among the many heuristics proposed for approximately solving this problem, one of the most successful is the Recursive Spectral Bisection method first proposed by Pothen, Simon and Liou =-=[9]-=-. In it, the NP-hard combinatorial minimization problem of finding a partition vector with components equal to +1 or \Gamma1 is approximated by minimizing a quadratic form involving the Laplacian of t... |

289 |
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(Show Context)
Citation Context ...n vector. The most common method is to use the median of the components of the Fiedler eigenvector to induce a bisection. We shall call this the median cut method. It has been widely used in practice =-=[11, 13]-=-, especially for unstructured finite element meshes, and further improvements have since then been proposed [1, 4, 5]. However, a complete theoretical justification for using the median of the Fiedler... |

277 | A fast multilevel implementation of recursive spectral bisection for partitioning unstructured problems. Concurrency: Practice and Experience
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(Show Context)
Citation Context ...tion. We shall call this the median cut method. It has been widely used in practice [11, 13], especially for unstructured finite element meshes, and further improvements have since then been proposed =-=[1, 4, 5]. How-=-ever, a complete theoretical justification for using the median of the Fiedler vector still seems to be lacking in the literature. Barnard and Simon [1] did mention that the median cut method "ch... |

222 |
A property of eigenvectors of nonnegative symmetric matrices and its application to graph theory
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(Show Context)
Citation Context ... 3. Optimality of Median Cut Partitioning. The median cut vector, p m , which is obviously feasible by construction (i.e. p m 2 P ), has some additional favourable properties, first proved by Fiedler =-=[3]-=-: 1. The median cut vector guarantees that at least one of the two partitions of a connected graph is also connected. 2. Nevertheless, if x 2 has n 2 positive components and n 2 negative components, b... |

179 |
R.: An improved spectral graph partitioning algorithm for mapping parallel computations
- Hendrickson, Leland
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(Show Context)
Citation Context ...tion. We shall call this the median cut method. It has been widely used in practice [11, 13], especially for unstructured finite element meshes, and further improvements have since then been proposed =-=[1, 4, 5]. How-=-ever, a complete theoretical justification for using the median of the Fiedler vector still seems to be lacking in the literature. Barnard and Simon [1] did mention that the median cut method "ch... |

157 | Performance of dynamic load balancing algorithms for unstructured mesh calculations,” Concurrency: Practice and Experience
- Williams
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(Show Context)
Citation Context ...n vector. The most common method is to use the median of the components of the Fiedler eigenvector to induce a bisection. We shall call this the median cut method. It has been widely used in practice =-=[11, 13]-=-, especially for unstructured finite element meshes, and further improvements have since then been proposed [1, 4, 5]. However, a complete theoretical justification for using the median of the Fiedler... |

148 | The Laplacian spectrum of graphs
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Citation Context ...d, then the same procedure can be applied to each of its connected components. It is well known that the multiplicity of zero as an eigenvalue of Q is equal to the number of connected components of G =-=[7]-=-. In this case, the Fiedler vector is the eigenvector which corresponds to the smallest positive eigenvalue. 3. Optimality of Median Cut Partitioning. The median cut vector, p m , which is obviously f... |

41 | A Projection Technique for Partitioning the Nodes of Graph
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(Show Context)
Citation Context ...a and Nakano [12]. However, it is not clear from the general algorithm in [12] that, for the bisection problem, the solution is indeed the median cut partition. On the other hand, Rendl and Wolkowicz =-=[10]-=- chose to formulate the bisection problem as a quadratic assignment problem in which the partition corresponds to an n by 2 matrix. A feasible solution is obtained by linearizing the objective functio... |

39 |
Multidimensional spectral load balancing
- Hendrickson, Leland
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(Show Context)
Citation Context ...tion. We shall call this the median cut method. It has been widely used in practice [11, 13], especially for unstructured finite element meshes, and further improvements have since then been proposed =-=[1, 4, 5]. How-=-ever, a complete theoretical justification for using the median of the Fiedler vector still seems to be lacking in the literature. Barnard and Simon [1] did mention that the median cut method "ch... |

13 |
Geometric Algorithms for the Minimum Cost Assignment Problem. Random Structures and Algorithms 6(4):393–406
- Tokuyama, Nakano
- 1995
(Show Context)
Citation Context ...idered a multi-dimensional extension of the RSB method, in which they define the partition as the solution to a minimum cost assignment problem and solve it by an algorithm due to Tokuyama and Nakano =-=[12]-=-. However, it is not clear from the general algorithm in [12] that, for the bisection problem, the solution is indeed the median cut partition. On the other hand, Rendl and Wolkowicz [10] chose to for... |

3 |
A sign cut version of the recursive spectral bisection graph partitioning algorithm
- Chan, Szeto
- 1994
(Show Context)
Citation Context ...nimum distance partition can easily be shown to be p s = sign(x 2 ), i.e. p s solves min p2L jjx 2 \Gamma pjj s , where jj:jj s denotes the l s norm. A partition strategy based on p s can be found in =-=[2]-=-. Theorem 3.1. Given any v 2 R n , n even, let p m 2 P be any median cut partition vector induced by v. Then p m = arg min p2P jjv \Gamma pjj s : Proof. We shall prove the theorem by showing that, for... |

2 |
An analysis of spectral graph partitioning via quadratic assignment problems
- Pothen
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(Show Context)
Citation Context ...m in which the partition corresponds to an n by 2 matrix. A feasible solution is obtained by linearizing the objective function at the solution X of a relaxed version of the problem. Recently, Pothen =-=[8]-=- showed that, up to this linearization, a closest partition matrix to X is obtained by using a technique similar to the median cut, and extended the result to any m-partition, that is a partition into... |