## A General Graph Model For Representing Exact Communication Volume in Parallel Sparse Matrix–Vector Multiplication (2006)

Citations: | 3 - 0 self |

### BibTeX

@MISC{Trifunović06ageneral,

author = {Aleksandar Trifunović and William Knottenbelt},

title = {A General Graph Model For Representing Exact Communication Volume in Parallel Sparse Matrix–Vector Multiplication},

year = {2006}

}

### OpenURL

### Abstract

In this paper, we present a new graph model of sparse matrix decomposition for parallel sparse matrix–vector multiplication. Our model differs from previous graph-based approaches in two main respects. Firstly, our model is based on edge colouring rather than vertex partitioning. Secondly, our model is able to correctly quantify and minimise the total communication volume of the parallel sparse matrix– vector multiplication while maintaining the computational load balance across the processors. We show that our graph edge colouring model is equivalent to the fine-grained hypergraph partitioning-based sparse matrix decomposition model. We conjecture that the existence of such a graph model should lead to faster serial and parallel sparse matrix decomposition heuristics and associated tools.

### Citations

11502 |
Computers and Intractability: A Guide to the Theory of NPCompleteness
- Garey, Johnson
- 1979
(Show Context)
Citation Context ... subject to a load balancing constraint. The problem of sparse matrix decomposition can be reformulated in terms of a graph or hypergraph partitioning problem. These partitioning problems are NP-hard =-=[10]-=-, so (sub-optimal) heuristic algorithms are used in practice. The resulting graph or hypergraph partition is then used to direct the distribution of matrix elements across processors. The limits of th... |

864 | A fast and high quality multilevel scheme for partitioning irregular graphs
- Karypis, Kumar
- 1998
(Show Context)
Citation Context ...of every edge in a graph is two. Indeed, existing parallel multilevel graph partitioning algorithms have demonstrated more natural parallelism than has been hitherto shown for hypergraph partitioning =-=[13, 14, 20, 7, 15]-=-. 4 Conclusion and Future Work In this paper we have presented a new graph edge colouring-based model of sparse matrix decomposition for parallel sparse matrix–vector multiplication. Unlike previous g... |

325 |
R.M.Mattheyses, “A Linear Time Heuristic for Improve Network
- Fiduccia
- 1982
(Show Context)
Citation Context ...rsening algorithms developed for graph partitioning could be applied in the graph colouring context. We also expect that iterative improvement algorithms such as the Fiduccia and Mattheyses algorithm =-=[9]-=- may be applied to our graph colouring problem. Feasible moves are defined as colour changes of edges, such that the overall colouring imbalance satisfies the prescribed constraints (cf. Equation 2). ... |

131 |
ParMETIS: Parallel Graph Partitioning and Sparse Matrix Ordering
- Karypis, Schloegel, et al.
(Show Context)
Citation Context ...of every edge in a graph is two. Indeed, existing parallel multilevel graph partitioning algorithms have demonstrated more natural parallelism than has been hitherto shown for hypergraph partitioning =-=[13, 14, 20, 7, 15]-=-. 4 Conclusion and Future Work In this paper we have presented a new graph edge colouring-based model of sparse matrix decomposition for parallel sparse matrix–vector multiplication. Unlike previous g... |

92 |
Hypergraph-partitioning-based decomposition for parallel sparse-matrix vector multiplication
- Catalyurek, Aykanat
- 1999
(Show Context)
Citation Context ...ice. The resulting graph or hypergraph partition is then used to direct the distribution of matrix elements across processors. The limits of the existing graph partitioning approaches are outlined in =-=[11, 8, 4]-=-. For example, in the case of one-dimensional row-wise or column-wise partitioning of a sparse matrix for parallel sparse matrix–vector multiplication, existing graph models cannot optimise the exact ... |

87 | Parallel Dynamic Graph Partitioning for Adaptive Unstructured
- Walshaw, Cross, et al.
- 1997
(Show Context)
Citation Context ...of every edge in a graph is two. Indeed, existing parallel multilevel graph partitioning algorithms have demonstrated more natural parallelism than has been hitherto shown for hypergraph partitioning =-=[13, 14, 20, 7, 15]-=-. 4 Conclusion and Future Work In this paper we have presented a new graph edge colouring-based model of sparse matrix decomposition for parallel sparse matrix–vector multiplication. Unlike previous g... |

72 | A two dimensional data distribution method for parallel sparse matrix-vector multiplication
- VASTENHOUW, BISSELING
- 2005
(Show Context)
Citation Context ...1 Preliminaries Consider a sparse m × n matrix A. We require that the sparse matrix–vector product Ax = b is distributed across p processors, where x and b are dense n- and m-vectors respectively. In =-=[19]-=-, Vastenhouw and Bisseling note that the natural parallel algorithm, with an arbitrary non-overlapping distribution of the matrix and the vectors across the processors, has the following general form:... |

44 | Graph partitioning and parallel solvers: has the emperor no clothes
- Hendrickson
- 1457
(Show Context)
Citation Context ...ice. The resulting graph or hypergraph partition is then used to direct the distribution of matrix elements across processors. The limits of the existing graph partitioning approaches are outlined in =-=[11, 8, 4]-=-. For example, in the case of one-dimensional row-wise or column-wise partitioning of a sparse matrix for parallel sparse matrix–vector multiplication, existing graph models cannot optimise the exact ... |

41 | Parallel hypergraph partitioning for scientific computing
- Devine, Boman, et al.
- 2006
(Show Context)
Citation Context ...volume have been proposed and are thus preferred to graph modelsin practical applications. Moreover, recently two parallel hypergraph partitioning algorithms have also been developed and implemented =-=[17, 16, 15, 7]-=-. However, graph models do have the advantage that heuristic algorithms operating on graphs are faster and are significantly easier to parallelise than heuristic algorithms that operate on hypergraphs... |

35 | Encapsulating multiple communication-cost metrics in partitioning sparse rectangular matrices for parallel matrix-vector multiplies
- Uçar, Aykanat
- 2004
(Show Context)
Citation Context ...dimensional model provided the vector component xj is assigned to a processor allocated a non-zero in column j and the vector component bi is assigned to a processor allocated a non-zero in row i. In =-=[18, 2]-=-, the problem of vector partitioning is considered with the aim of improving the communication balance and reducing the number of messages sent between the processors, while maintaining the overall co... |

31 | A fine-grain hypergraph model for 2d decomposition of sparse matrices
- Çatalyürek, Aykanat
- 2001
(Show Context)
Citation Context ...he total interprocessor communication volume while maintaining the computational load balance. The graph model is derived from the fine-grained hypergraph model presented by Çatalyürek and Aykanat in =-=[5]-=-. The edges in the graph model the non-zeros in the matrix and thus instead of partitioning the set of vertices, as in existing graph and hypergraph sparse matrix decomposition models, our model requi... |

17 | Partitioning rectangular and structurally nonsymmetric sparse matrices for parallel processing
- Hendrickson, Kolda
(Show Context)
Citation Context ... w(e) the weight of, hyperedge e: fp(Π) = ∑ (λ(e) − 1)w(e) (3) e∈E Note that for graph partitioning this reduces to the edge-cut metric, since the cardinality of each edge is two. 2.3 Related Work In =-=[12]-=-, Hendrickson and Kolda outline a bipartite graph partitioning-based model for decomposition of a general rectangular non-symmetric sparse matrix. The non-zero structure of a sparse matrix A correspon... |

12 |
A parallel algorithm for multilevel k-way hypergraph partitioning
- Trifunovic, Knottenbelt
- 2004
(Show Context)
Citation Context ...volume have been proposed and are thus preferred to graph modelsin practical applications. Moreover, recently two parallel hypergraph partitioning algorithms have also been developed and implemented =-=[17, 16, 15, 7]-=-. However, graph models do have the advantage that heuristic algorithms operating on graphs are faster and are significantly easier to parallelise than heuristic algorithms that operate on hypergraphs... |

5 |
PaToH: Partitioning Tool for Hypergraphs, Version 3.0
- Catalyurek, Aykanat
(Show Context)
Citation Context ...idual processors. This has the effect of making the communication step 3 (or 1 in the column case) in the parallel sparse matrix–vector multiplication pipeline redundant. 2. Two-dimensional Cartesian =-=[6]-=-; each processor receives a submatrix defined by a partition of rows and columns of A. 3. Two-dimensional non-Cartesian with the Mondriaan structure [19]; obtained by recursively bipartitioning the ma... |

5 | Parallel Algorithms for Hyper graph Partitioning
- Trifunovic
- 2006
(Show Context)
Citation Context ...volume have been proposed and are thus preferred to graph modelsin practical applications. Moreover, recently two parallel hypergraph partitioning algorithms have also been developed and implemented =-=[17, 16, 15, 7]-=-. However, graph models do have the advantage that heuristic algorithms operating on graphs are faster and are significantly easier to parallelise than heuristic algorithms that operate on hypergraphs... |

1 |
2006. Personal communication. R.H. Bisseling and W. Meesen. Communication balancing in parallel sparse matrix–vector multiplication
- Bisseling
(Show Context)
Citation Context ...partitioning when describing our graph-based model, but note instead that incorporating vector partitioning heuristics within a model for the total communication volume is an area of ongoing research =-=[1]-=-. 3 The General Graph Model In this section, we derive our graph colouring-based model for sparse matrix decomposition. We show that our model yields a bipartite graph, which is in fact the same bipar... |

1 | Towards a Parallel Disk-Based Algorithm for Multilevel k-way Hypergraph Partitioning
- Trifunović, Knottenbelt
- 2004
(Show Context)
Citation Context |