## A Parallel Graph Coloring Heuristic (1992)

Venue: | SIAM J. SCI. COMPUT |

Citations: | 69 - 6 self |

### BibTeX

@ARTICLE{Jones92aparallel,

author = {Mark T. Jones and Paul E. Plassmann},

title = {A Parallel Graph Coloring Heuristic},

journal = {SIAM J. SCI. COMPUT},

year = {1992},

volume = {14},

pages = {654--669}

}

### Years of Citing Articles

### OpenURL

### Abstract

The problem of computing good graph colorings arises in many diverse applications, such as in the estimation of sparse Jacobians and in the development of efficient, parallel iterative methods for solving sparse linear systems. In this paper we present an asynchronous graph coloring heuristic well suited to distributed memory parallel computers. We present experimental results obtained on an Intel iPSC/860 which demonstrate that, for graphs arising from finite element applications, the heuristic exhibits scalable performance and generates colorings usually within three or four colors of the best-known linear time sequential heuristics. For bounded degree graphs, we show that the expected running time of the heuristic under the P-RAM computation model is bounded by EO(log(n)= log log(n)). This bound is an improvement over the previously known best upper bound for the expected running time of a random heuristic for the graph coloring problem.

### Citations

517 |
Partitioning sparse matrices with eigenvectors of graphs
- Pothen, Simon, et al.
- 1990
(Show Context)
Citation Context ... is not straightforward, a partitioning heuristic would have to be used. For example, recent advances in the automatic partitioning of three dimensional domains [11] or in spectral dissection methods =-=[9]-=- could be employed. We note that a partitioning that maintains locality is advantageous, although not essential, to the performance of the parallel heuristic. The heuristic requires only that the numb... |

378 | A simple parallel algorithm for the maximal independent set problem
- Luby
- 1986
(Show Context)
Citation Context ...es one of several good sequential coloring heuristics. The initial phase maintains the good expected running time bounds obtained for a Monte Carlo algorithm for determining a maximal independent set =-=[8]-=-. In fact, for bounded degree graphs, we show that an upper bound for the expected running time of our heuristic under the P-RAM computation model is EO(log(n)= log log(n)). This bound is an improveme... |

121 |
Estimation of sparse jacobian matrices and graph coloring problems
- Coleman, Moré
- 1983
(Show Context)
Citation Context ...atic number of a general graph is a well-known NP-hard problem [4]. However, a number of practical problems require the determination of nearly optimal graph colorings. For example, it has been shown =-=[3]-=- that the problem of directly estimating a sparse Jacobian by finite differences with a minimum number of function evaluations is equivalent to a graph coloring problem. Also, it has been shown [10] t... |

119 | Development of Parallel Methods for a 1024-Processor Hypercube
- Gustafson, Montry, et al.
- 1988
(Show Context)
Citation Context ... iPSC/860. In this section we present results obtained with this implementation. One of our main objectives is to demonstrate the scalability of this heuristic consistent with the definition given in =-=[5]-=-. Thus, we would like to show that, for a fixed number of vertices per processor, the running time of the heuristic is only a slowly increasing function of the number of processors used. 1 To achieve ... |

40 |
A Textbook of Graph Theory
- B, R
- 2000
(Show Context)
Citation Context ... V 0 / V 0 n fv i g; enddo Fig. 1. A sequential greedy coloring heuristic It is known that an optimal coloring can be obtained with a greedy heuristic if the vertices are visited in the correct order =-=[1]-=-. The basic structure of the greedy heuristic is shown in Figure 1. The only aspect of the sequential heuristic that must be specified is the rule for choosing the vertex v i . Many strategies for obt... |

36 | Scalable iterative solution of sparse linear systems
- Jones, Plassmann
- 1994
(Show Context)
Citation Context ...ective in practice. For graphs arising from a number of applications, it has been demonstrated that these heuristics are often able to find colorings that are within one or two of an optimal coloring =-=[3, 7]-=-. V 0 / V ; For i = 1; : : : ; n do Choose a vertex v i from V 0 ; oe(v i ) = smallest available consistent color; V 0 / V 0 n fv i g; enddo Fig. 1. A sequential greedy coloring heuristic It is known ... |

28 |
Worst case behavior of graph coloring algorithms", in
- Johnson
- 1974
(Show Context)
Citation Context ... NP-complete [4]. It is known that unless P = NP , there does not exist a polynomial approximation scheme for solving the graph coloring problem [4]. In fact, the best polynomial time heuristic known =-=[6] can-=- theoretically guarantee a coloring of only size c (n= log n) ��(G), where c is some constant. Given these pessimistic theoretical results, it is quite surprising that, for certain classes of grap... |

18 |
Automatic domain partitioning in three dimensions
- VAVASIS
- 1991
(Show Context)
Citation Context .... When the determination of a partition is not straightforward, a partitioning heuristic would have to be used. For example, recent advances in the automatic partitioning of three dimensional domains =-=[11]-=- or in spectral dissection methods [9] could be employed. We note that a partitioning that maintains locality is advantageous, although not essential, to the performance of the parallel heuristic. The... |

7 |
élaz. New methods to color the vertices of a graph
- Br
- 1979
(Show Context)
Citation Context ... for choosing the vertex v i . Many strategies for obtaining this vertex ordering have been proposed. One of the most effective heuristics is the saturation degree ordering (SDO) suggested by Br'elaz =-=[2]-=-. The SDO vertex ordering is defined as follows. Suppose that vertices v 1 ; : : : ; v i\Gamma1 have been chosen and colored. Vertex v i is chosen to be a vertex adjacent to the maximum number of diff... |

6 |
Vectorizing the conjugate gradient method. Unpublished manuscript
- Schreiber, Tang
- 1982
(Show Context)
Citation Context ...wn [3] that the problem of directly estimating a sparse Jacobian by finite differences with a minimum number of function evaluations is equivalent to a graph coloring problem. Also, it has been shown =-=[10]-=- that the minimum number of parallel steps in the solution of the triangular systems involving incomplete Cholesky factors can be obtained by a matrix reordering derived from the solution of a graph c... |