## Graph Partitioning Algorithms With Applications To Scientific Computing (1997)

Venue: | Parallel Numerical Algorithms |

Citations: | 40 - 0 self |

### BibTeX

@INPROCEEDINGS{Pothen97graphpartitioning,

author = {Alex Pothen},

title = {Graph Partitioning Algorithms With Applications To Scientific Computing},

booktitle = {Parallel Numerical Algorithms},

year = {1997},

pages = {323--368},

publisher = {Kluwer Academic Press}

}

### OpenURL

### Abstract

Identifying the parallelism in a problem by partitioning its data and tasks among the processors of a parallel computer is a fundamental issue in parallel computing. This problem can be modeled as a graph partitioning problem in which the vertices of a graph are divided into a specified number of subsets such that few edges join two vertices in different subsets. Several new graph partitioning algorithms have been developed in the past few years, and we survey some of this activity. We describe the terminology associated with graph partitioning, the complexity of computing good separators, and graphs that have good separators. We then discuss early algorithms for graph partitioning, followed by three new algorithms based on geometric, algebraic, and multilevel ideas. The algebraic algorithm relies on an eigenvector of a Laplacian matrix associated with the graph to compute the partition. The algebraic algorithm is justified by formulating graph partitioning as a quadratic assignment p...

### Citations

10928 | Computers and Intractability: A Guide to the Theory of NP-Completeness - Garey, Johnson - 1979 |

1048 |
An Efficient Heuristic Procedure for Partitioning Graphs
- Kernighan, Lin
- 1970
(Show Context)
Citation Context ...inertial algorithm. 3.1 The Kernighan-Lin algorithm This is one of the earliest algorithms proposed for partitioning graphs (published in 1970), and variants are still in use today. Kernighan and Lin =-=[66]-=- were motivated by the problem of partitioning electronic circuits onto cards: each card contains a subset of nodes, and the objective is to minimize the number of connections between nodes assigned t... |

764 | The Symmetric Eigenvalue Problem - Parlett - 1980 |

538 | Direct Methods for Sparse Matrices - DUFF, ERISMAN, et al. - 1986 |

529 | Algebraic Graph Theory - Biggs - 1993 |

510 |
Optimization by simulated annealing: an experimental evaluation
- Johnson, Aragon, et al.
- 1989
(Show Context)
Citation Context ...tion orderings, is discussed in detail. We now briefly consider some of the other partitioning algorithms not described in detail in this survey. Simulated annealing has been used to partition graphs =-=[61]-=-, but computational results in [32] show that it provides edge separators of inferior quality and requires much greater computational times when compared with the spectral partitioning algorithm. Gene... |

500 | Computer Solution of Large Sparse Positive Definite Systems - George, Liu - 1981 |

486 |
ªPartitioning Sparse Matrices with Eigenvectors of Graphs,º
- Pothen, Simon, et al.
- 1990
(Show Context)
Citation Context ... to further reduce the number of edges cut, after another "global" algorithm is used to compute a good initial partition. The use of the Kernighan-Lin algorithm for this purpose has been adv=-=ocated in [9, 89]-=-, and this is one of the post-processing options in CHACO [53]. Bui and Jones have implemented a multilevel Kernighan-Lin algorithm to improve the quality of the partitions computed [15, 62]. The Kern... |

461 | Domain decomposition: parallel multilevel methods for elliptic partial differential equations - Bjørstad - 1996 |

405 | Algebraic connectivity of graphs - Fiedler - 1973 |

389 | A Separator Theorem for Planar Graphs
- Lipton, Tarjan
- 1979
(Show Context)
Citation Context ...tor result is for trees, and is due to Jordan in 1869: Every tree has a single vertex that separates it into two parts, with no part containing more than two-thirds of the vertices. Lipton and Tarjan =-=[75]-=- showed that every planar graph has a vertex separator of size at most O( p n) separating it into two parts with no part having more than two-thirds the number of vertices. These results extend to two... |

305 |
A linear time heuristic for improving network partitions
- Fiduccia, Mattheyses
- 1982
(Show Context)
Citation Context ... are performed at each iteration of the algorithm. An important practical advance was made by Fiduccia and Mattheyses who implemented an iteration of the Kernighan-Lin algorithm to run in O(jEj) time =-=[35]-=- . Fiduccia and Mattheyses described how to compute and update the gains of the vertices that are candidates to be moved during an iteration of the algorithm. In their efficient implementation, the ga... |

289 |
Partitioning of unstructured problems for parallel processing
- Simon
- 1991
(Show Context)
Citation Context ...ce meshes embedded in space contain geometric information about the coordinates of the mesh points. Algorithms for partitioning meshes by bisecting along coordinate axes have been considered by Simon =-=[96]-=-, Williams [105], and many others. A parallel nested dissection algorithm based on this idea has been described by Heath and Raghavan [52]. These algorithms have the virtue of being fast, and are easy... |

283 | Sparse matrix test problems - Duff, Grimes, et al. - 1989 |

277 | A fast multilevel implementation of recursive spectral bisection for partitioning unstructured problems. Concurrency: Practice and Experience
- Barnard, Simon
- 1994
(Show Context)
Citation Context ...me of the ideas involved in the multilevel algorithm were used to obtain efficient implementations of the Kernighan-Lin algorithm by Bui et al. [15], who called it graph compaction. Barnard and Simon =-=[8]-=- described a spectral partitioning algorithm in which they used a multilevel algorithm to compute the Fiedler vector of a given graph, by coarsening, interpolating, and then refining. This algorithm w... |

239 | An Approximate Max-Flow Min-Cut Theorem For Uniform Multicommodity Flow Problems with Applications to Approximation Algorithms - Leighton, Rao - 1988 |

211 | Fast spectral methods for ratio cut partitioning and clustering - Hagen, Kahng - 1991 |

198 | Recent directions in netlist partitioning: A survey
- Alpert, Kahng
- 1995
(Show Context)
Citation Context ...Kernighan-Lin algorithm [66] and the work of Donath and Hoffman [29] was circuit layout. Graph and hypergraph partitioning algorithms for VLSI design and layout have been surveyed by Alpert and Kahng =-=[3]-=-. Mesh partitioning algorithms for solving discretizations of partial differential equations on parallel computers are described in [20, 105]. The parallel solution of Euler equations and other CFD pr... |

183 |
Generalized nested dissection
- Lipton, Rose, et al.
- 1979
(Show Context)
Citation Context ...olumns in the two parts.) Then the vertices in the two parts are numbered recursively by the same strategy. Nested dissection and generalizations have been described by George [40], Lipton and Tarjan =-=[74]-=-, and Gilbert [43]. A good vertex separator in this context should satisfy two criteria: First, the size (number of vertices) of the separator should be GRAPH PARTITIONING ALGORITHMS 29 small. Second,... |

182 |
Spectra of Graphs
- Cvetković, Doob, et al.
- 1980
(Show Context)
Citation Context ...computational results. Hendrickson and Leland [56] extended the spectral algorithm to compute quadrisections and octasections. Work on eigenvalues and eigenvectors of graphs can be found in the books =-=[25, 26]-=-, and in the surveys by Mohar and Poljak [80, 81]. 5 The geometric algorithm Finite element or finite difference meshes embedded in space contain geometric information about the coordinates of the mes... |

179 |
R.: An improved spectral graph partitioning algorithm for mapping parallel computations
- Hendrickson, Leland
- 1995
(Show Context)
Citation Context ...u [89] used the spectral partitioning algorithm to compute separators for parallel computing, proved additional lower bounds on separators, and described computational results. Hendrickson and Leland =-=[56]-=- extended the spectral algorithm to compute quadrisections and octasections. Work on eigenvalues and eigenvectors of graphs can be found in the books [25, 26], and in the surveys by Mohar and Poljak [... |

178 |
Nested dissection of a regular finite element mesh
- George
- 1973
(Show Context)
Citation Context ...ctored after factoring columns in the two parts.) Then the vertices in the two parts are numbered recursively by the same strategy. Nested dissection and generalizations have been described by George =-=[40]-=-, Lipton and Tarjan [74], and Gilbert [43]. A good vertex separator in this context should satisfy two criteria: First, the size (number of vertices) of the separator should be GRAPH PARTITIONING ALGO... |

159 |
An r-dimensional quadratic placement algorithm
- Hall
- 1970
(Show Context)
Citation Context ...inate information since they are not embedded in space. One can apply the geometric algorithm to problems that do not have coordinate information if it is possible to embed the problem in space. Hall =-=[49]-=- has shown that the eigenvectors of the Laplacian matrix associated with a graph can be used to embed the graph. Chan, Gilbert, and Teng [18] have reported results on partitioning graphs by the geomet... |

157 | Performance of dynamic load balancing algorithms for unstructured mesh calculations,” Concurrency: Practice and Experience
- Williams
- 1991
(Show Context)
Citation Context ...ded in space contain geometric information about the coordinates of the mesh points. Algorithms for partitioning meshes by bisecting along coordinate axes have been considered by Simon [96], Williams =-=[105]-=-, and many others. A parallel nested dissection algorithm based on this idea has been described by Heath and Raghavan [52]. These algorithms have the virtue of being fast, and are easy to implement in... |

148 | The Laplacian spectrum of graphs
- Mohar
- 1988
(Show Context)
Citation Context ...se spectral methods to partition graphs in the context of circuit layout. Spectral partitioning algorithms were considered and analyzed by Barnes [9], Boppana [12], Alon, Galil, and Milman [2], Mohar =-=[80]-=-, and many others. Aspvall and Gilbert [6] used eigenvectors of the adjacency matrix to color the vertices of a graph. Pothen, Simon, and Liou [89] used the spectral partitioning algorithm to compute ... |

143 |
Lower bounds for the partitioning of graphs
- Donath, Hoffman
- 1973
(Show Context)
Citation Context ...g problem, and relaxing the integrality constraints in the problem; such bounds are described in Section 7. Lower bounds on edge separators by this technique were first obtained by Donath and Hoffman =-=[29]-=-; the strongest current bounds are due to Rendl and Wolkowicz [92]. These bounds potentially could be used to state how close a computed edge separator is to an optimal separator by finding the differ... |

137 |
Modification of the minimum degree algorithm by multiple elimination
- Liu
- 1985
(Show Context)
Citation Context ...n obtained by Hendrickson and Rothberg (personal communication, 1994), Karypis and Kumar [64], and by Kumfert and Pothen [69]. We compare three algorithms: the Multiple-Minimum-Degree ordering of Liu =-=[76]-=-, an MLND algorithm due to Kumfert and Pothen [69], and MLNDW, a weighted variant of the MLND algorithm, due to Karypis and Kumar, included in MeTiS version 2.0. The multilevel nested dissection algor... |

116 |
A version of the bundle idea for minimizing a nonsmooth function: conceptual idea, convergence analysis, numerical results
- Schramm, Zowe
- 1992
(Show Context)
Citation Context ...z T b Q(d)z + (1=2n p n )(m 1 \Gamma m 2 ) p m 1 m 2 d T z o ; subject to z T z = 1: (15) GRAPH PARTITIONING ALGORITHMS 26 The lower bound can be computed by nondifferentiable optimization techniques =-=[94]-=-. The lower bounds in terms of the unperturbed Laplacian and the perturbed adjacency matrix, described by Falkner et al. [32], are still weak for finite-element meshes with bounded aspect ratios. 7.4 ... |

115 | Parallel algorithms for sparse linear systems - Heath, Ng, et al. - 1991 |

102 | Geometric mesh partitioning: Implementation and experiments
- Gilbert, Miller, et al.
(Show Context)
Citation Context ... other algorithms, especially for adapted meshes. We now describe a geometric partitioning algorithm designed by Miller, Teng, Thurston, and Vavasis [79], and implemented by Gilbert, Miller, and Teng =-=[44]-=-. This algorithm computes a separator by using a circle rather than a straight-line to cut the mesh. Given GRAPH PARTITIONING ALGORITHMS 15 a graph embedded in d-dimensional space we disregard the edg... |

94 | Implementation of a parallel unstructured Euler solver on the CM-5. Pres. at the 32nd AIAA
- Morano, Mavriplis
- 1994
(Show Context)
Citation Context ...rithms for solving discretizations of partial differential equations on parallel computers are described in [20, 105]. The parallel solution of Euler equations and other CFD problems are described in =-=[28, 59, 95, 102]-=-. A survey of parallel computing in CFD has been provided in [93]. Applications in structural mechanics have been considered in [58, 82, 100]. A comparative study of partitioners in domain decompositi... |

92 | Parallel multilevel graph partitioning
- Karypis, Kumar
- 1996
(Show Context)
Citation Context ...ar [64]; they have further refined the algorithm, and included an implementation in their METIS software. They have also described a parallel implementation of this algorithm and provided an analysis =-=[63, 65]-=-. A multilevel partitioning algorithm has been implemented by Kumfert and Pothen [69], and results from a nested dissection ordering computed with a multilevel algorithm are included in Section 8. Coa... |

83 | Computing the block triangular form of a sparse matrix - Pothen, Fan - 1990 |

82 |
The chaco user’s guide
- Hendrickson, Leland
- 1993
(Show Context)
Citation Context ..." algorithm is used to compute a good initial partition. The use of the Kernighan-Lin algorithm for this purpose has been advocated in [9, 89], and this is one of the post-processing options in C=-=HACO [53]-=-. Bui and Jones have implemented a multilevel Kernighan-Lin algorithm to improve the quality of the partitions computed [15, 62]. The Kernighan-Lin algorithm has been used to partition the coarsest le... |

75 |
Finding good approximate vertex and edge partitions is NP-hard
- Bui, Jones
- 1992
(Show Context)
Citation Context ...are NP-hard, i.e., it is unlikely that vertex separators or edge separators of minimum cardinality can be computed efficiently (in polynomial time) for general classes of graphs. Worse, Bui and Jones =-=[14]-=- have shown that it is NP-hard to find approximately optimal vertex and edge separators, even in graphs with maximum degree three. More precisely, it is NP-hard to compute an approximately optimal sep... |

70 |
An algorithm for partitioning the nodes of a graph
- Barnes
(Show Context)
Citation Context ... to further reduce the number of edges cut, after another "global" algorithm is used to compute a good initial partition. The use of the Kernighan-Lin algorithm for this purpose has been adv=-=ocated in [9, 89]-=-, and this is one of the post-processing options in CHACO [53]. Bui and Jones have implemented a multilevel Kernighan-Lin algorithm to improve the quality of the partitions computed [15, 62]. The Kern... |

69 | Implicit parallel processing in structural mechanics, in - Farhat, Roux - 1994 |

67 |
1, isoperimetric inequalities for graphs, and superconcentrators
- Alon, Milman
- 1985
(Show Context)
Citation Context ...rliest to use spectral methods to partition graphs in the context of circuit layout. Spectral partitioning algorithms were considered and analyzed by Barnes [9], Boppana [12], Alon, Galil, and Milman =-=[2]-=-, Mohar [80], and many others. Aspvall and Gilbert [6] used eigenvectors of the adjacency matrix to color the vertices of a graph. Pothen, Simon, and Liou [89] used the spectral partitioning algorithm... |

62 | Dynamic load balancing for PDE solvers an adaptive unstructured meshes
- Walshaw, Berzins
- 1992
(Show Context)
Citation Context ...n [58, 82, 100]. A comparative study of partitioners in domain decomposition has been provided in [24]. Dynamic load balancing for time-dependent partial differential equations has been considered in =-=[57, 101, 103]-=-. 8.2 Spectral Nested Dissection Nested dissection is a divide-and-conquer scheme for ordering sparse symmetric positive definite systems of equations for fast solution by direct methods. At each step... |

60 |
TOP/DOMDEC—A Software Tool for Mesh Partitioning and Parallel Processing
- Farhat, Simon
- 1993
(Show Context)
Citation Context ...a sparse matrix [90]. The greedy algorithm A variant of this algorithm to partition a graph into k parts (with a requested number of vertices in the parts) has also been considered by several authors =-=[23, 33]-=-. As before, a pseudo-peripheral vertex is computed. The first part initially includes the pseudo-peripheral vertex, and then we include vertices in as many levels as necessary until the first part co... |

60 | Domain decomposition methods in computational mechanics - Tallec - 1994 |

59 | A spectral algorithm for envelope reduction of sparse matrices - Barnard, Pothen, et al. - 1995 |

59 |
Eigenvalues and graph bisection: an average case analysis
- Boppana
- 1987
(Show Context)
Citation Context ...h and Hoffman [29] were the earliest to use spectral methods to partition graphs in the context of circuit layout. Spectral partitioning algorithms were considered and analyzed by Barnes [9], Boppana =-=[12]-=-, Alon, Galil, and Milman [2], Mohar [80], and many others. Aspvall and Gilbert [6] used eigenvectors of the adjacency matrix to color the vertices of a graph. Pothen, Simon, and Liou [89] used the sp... |

59 |
A heuristic for reducing fill-in in sparse matrix factorization
- Bui, Jones
- 1993
(Show Context)
Citation Context ...vocated in [9, 89], and this is one of the post-processing options in CHACO [53]. Bui and Jones have implemented a multilevel Kernighan-Lin algorithm to improve the quality of the partitions computed =-=[15, 62]-=-. The Kernighan-Lin algorithm has been used to partition the coarsest level and to refine initial partitions within multilevel partitioning algorithms, as described in Section 6. Gilbert and Zmijewski... |

57 |
Solving finite element equations on concurrent computers
- Nour-Omid, Raefsky, et al.
- 1986
(Show Context)
Citation Context ...nal projection of the coordinates of the vertices onto this eigenvector v 1 . The median value of these projections can be used to partition the vertices into two sets. This algorithm is described in =-=[82]-=-. An extension to the above algorithm is to use the projections of a point on to the second and third eigenvectors of I as secondary and tertiary keys to break ties [84]. Here we form the eigenvector ... |

50 | Domain decomposition algorithms. Acta Numerica - Chan, Mathew - 1994 |

46 | Mapping unstructured grid computations to massively parallel computers - Hammond - 1992 |

46 | A rnimd implementation of a parallel Euler solver for unstructured grids," The Journal of Supercomputing 6(2
- Venkatakrishnan, Simon, et al.
- 1992
(Show Context)
Citation Context ...rithms for solving discretizations of partial differential equations on parallel computers are described in [20, 105]. The parallel solution of Euler equations and other CFD problems are described in =-=[28, 59, 95, 102]-=-. A survey of parallel computing in CFD has been provided in [93]. Applications in structural mechanics have been considered in [58, 82, 100]. A comparative study of partitioners in domain decompositi... |

43 |
A fast algorithm for reordering sparse matrices for parallel factorization
- Lewis, Peyton, et al.
- 1989
(Show Context)
Citation Context ...HMS 30 for parallel factorization were based on improving the concurrency in an MMD ordering by computing a perfect elimination reordering of the filled graph from the MMD ordering; for instance, see =-=[73]-=-. The SND ordering had the drawback that it was slower than the MMD ordering by a factor of five to twelve on a vector computer such as the Cray-Y/MP. On workstations without the vector floating point... |

41 | Eigenvalues in combinatorial optimization
- MOHAR, POIAAK
- 1993
(Show Context)
Citation Context ...] extended the spectral algorithm to compute quadrisections and octasections. Work on eigenvalues and eigenvectors of graphs can be found in the books [25, 26], and in the surveys by Mohar and Poljak =-=[80, 81]-=-. 5 The geometric algorithm Finite element or finite difference meshes embedded in space contain geometric information about the coordinates of the mesh points. Algorithms for partitioning meshes by b... |