## Scalable Parallel Graph Coloring Algorithms (2000)

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Citations: | 23 - 7 self |

### BibTeX

@MISC{Gebremedhin00scalableparallel,

author = {Assefaw Hadish Gebremedhin and Fredrik Manne},

title = {Scalable Parallel Graph Coloring Algorithms},

year = {2000}

}

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### Abstract

Finding a good graph coloring quickly is often a crucial phase in the development of efficient, parallel algorithms for many scientific and engineering applications. In this paper we consider the problem of solving the graph coloring problem itself in parallel. We present a simple and fast parallel graph coloring heuristic that is well suited for shared memory programming and yields an almost linear speedup on the PRAM model. We also present a second heuristic that improves on the number of colors used. The heuristics have been implemented using OpenMP. Experiments conducted on an SGI Cray Origin 2000 super computer using very large graphs from finite element methods and eigenvalue computations validate the theoretical run-time analysis.

### Citations

376 | A simple parallel algorithm for the maximal independent set problem
- Luby
- 1986
(Show Context)
Citation Context ...red, Scheme 1 specializes into a number of variants. The Parallel Maximal Independent set (PMIS) coloring is one variant. This is a heuristic based on Luby’s maximal independent set finding algorithm =-=[15]-=-. Other variants are the asynchronous parallel heuristic by Jones and Plassmann (JP) [11], and the Largest-DegreeFirst(LDF) heuristic developed independently by Gjertsen Jr. et al. [12] and Allwright ... |

363 |
New methods to color the vertices of a graph
- Brélaz
- 1979
(Show Context)
Citation Context ...red. Some of the suggested coloring heuristics under this general framework include Largest-Degree-First-Ordering (LFO) [18], Incidence-Degree-Ordering (IDO) [4], and Saturation-Degree-Ordering (SDO) =-=[2]-=-. These heuristics choose at each step a vertex v from the set of uncolored vertices with the maximum “degree”. In LFO, the standard definition of degree of a vertex is used. In IDO, incidence degree ... |

199 |
allocation via coloring
- Chaitin, Auslander, et al.
- 1981
(Show Context)
Citation Context ... used. The GCP arises in a number of scientific computing and engineering applications. Examples include, among others, time tabling and scheduling [14], frequency assignment [6], register allocation =-=[3]-=-, printed circuit testing [8], parallel numerical computation [1], and optimization [4]. Coloring a general graph with the minimum number of colors is known to be an NP-hard problem [7], thus one ofte... |

121 |
Estimation of sparse jacobian matrices and graph coloring problems
- Coleman, Moré
- 1983
(Show Context)
Citation Context ... Examples include, among others, time tabling and scheduling [14], frequency assignment [6], register allocation [3], printed circuit testing [8], parallel numerical computation [1], and optimization =-=[4]-=-. Coloring a general graph with the minimum number of colors is known to be an NP-hard problem [7], thus one often relies on heuristics to obtain a usable solution. ∗Department of Informatics, Univers... |

94 |
Some lower bounds for a class of frequency assignment problems
- Gamst
- 1986
(Show Context)
Citation Context ...mize the number of colors used. The GCP arises in a number of scientific computing and engineering applications. Examples include, among others, time tabling and scheduling [14], frequency assignment =-=[6]-=-, register allocation [3], printed circuit testing [8], parallel numerical computation [1], and optimization [4]. Coloring a general graph with the minimum number of colors is known to be an NP-hard p... |

93 |
An upper bound on the chromatic number of a graph and its applications to timetabling problems
- Welsh, Powell
- 1967
(Show Context)
Citation Context ...he selection and coloring continues until all the vertices in the graph are colored. Some of the suggested coloring heuristics under this general framework include Largest-Degree-First-Ordering (LFO) =-=[18]-=-, Incidence-Degree-Ordering (IDO) [4], and Saturation-Degree-Ordering (SDO) [2]. These heuristics choose at each step a vertex v from the set of uncolored vertices with the maximum “degree”. In LFO, t... |

82 | On colouring random graphs - Grimmett, McDiarmid - 1975 |

69 | A parallel graph coloring heuristic
- Jones, Plassmann
- 1993
(Show Context)
Citation Context ...exist several linear time, or close to linear time, sequential greedy coloring heuristics. These heuristics have been found to be effective in coloring graphs that arise from a number of applications =-=[4, 11]-=-. Because of their inherent sequential nature, however, these heuristics are difficult to parallelize. This paper focuses mainly on the latter strategy where the goal is to develop scalable parallel c... |

55 |
An application of graph coloring to printed circuit testing
- Garey, Johnson, et al.
- 1976
(Show Context)
Citation Context ...mber of scientific computing and engineering applications. Examples include, among others, time tabling and scheduling [14], frequency assignment [6], register allocation [3], printed circuit testing =-=[8]-=-, parallel numerical computation [1], and optimization [4]. Coloring a general graph with the minimum number of colors is known to be an NP-hard problem [7], thus one often relies on heuristics to obt... |

34 | Iterated greedy graph coloring and the difficulty landscape
- Culberson
- 1992
(Show Context)
Citation Context ...s. 3.2 The Second Algorithm In this section we show how Algorithm 1 can be modified to use fewer colors. Our method is motivated by the idea behind Culberson’s Iterated Greedy coloring heuristic (IG) =-=[5]-=-. IG is based on the following result, stated here without proof. Lemma (Culberson) 3.3 Let C be a k-coloring of a graph G, and π a permutation of the vertices such that if C(vπ(i)) = C(vπ(l)) = c, th... |

27 | A comparison of parallel graph coloring algorithms
- Allwright, Bordawekar, et al.
- 1993
(Show Context)
Citation Context ...ineering applications. Examples include, among others, time tabling and scheduling [14], frequency assignment [6], register allocation [3], printed circuit testing [8], parallel numerical computation =-=[1]-=-, and optimization [4]. Coloring a general graph with the minimum number of colors is known to be an NP-hard problem [7], thus one often relies on heuristics to obtain a usable solution. ∗Department o... |

19 | Parallel heuristics for improved, balanced graph colorings
- Gjertsen, Jones, et al.
- 1996
(Show Context)
Citation Context ...ding algorithm [15]. Other variants are the asynchronous parallel heuristic by Jones and Plassmann (JP) [11], and the Largest-DegreeFirst(LDF) heuristic developed independently by Gjertsen Jr. et al. =-=[12]-=- and Allwright et al. [1]. Allwright et al. made an experimental, comparative study by implementing the PMIS, JP, and LDF coloring algorithms on both SIMD and MIMD parallel architectures [1]. They rep... |

14 | Graph coloring on a coarse grained multiprocessor
- Gebremedhin, Lassous, et al.
- 2000
(Show Context)
Citation Context ...ively straight forward task as all the communication is hidden from the programmer. We point out that in a recent development the algorithms presented in this paper have been adapted to the CGM model =-=[9]-=-. We believe that the general idea in these coloring heuristics of allowing inconsistency for the sake of concurrency can be applied to develop parallel algorithms for other graph problems and we are ... |

13 | A parallel algorithm for computing the extremal eigenvalues of very large sparse matrices (extended abstract
- Manne
- 1998
(Show Context)
Citation Context ...d with each vertex is fairly small and one repeatedly has to find new graph colorings then the overall time to perform the colorings might take up a significant portion of the entire computation. See =-=[16]-=- for an example of this case. In such a setting it is more important to compute a usable coloring fast than spending time on reducing the number of colors. For this purpose there exist several linear ... |

11 |
Practical implementations and applications of graph coloring
- Lewandowski
- 1994
(Show Context)
Citation Context ...rimary objective is to minimize the number of colors used. The GCP arises in a number of scientific computing and engineering applications. Examples include, among others, time tabling and scheduling =-=[14]-=-, frequency assignment [6], register allocation [3], printed circuit testing [8], parallel numerical computation [1], and optimization [4]. Coloring a general graph with the minimum number of colors i... |

10 |
A proposed industry standard api for shared memory programming. http://www.openmp.org
- OpenMP
(Show Context)
Citation Context ... now. The development of shared memory computers has been accompanied by the emergence of new shared memory programming paradigms of which OpenMP has become one of the most successful and widely used =-=[17]-=-. OpenMP 2is a directive-based, fork-join model for shared memory parallelism. In this paper we present a fast and scalable parallel graph coloring algorithm suitable for the shared memory programmin... |

6 |
Private Communication
- Karypis
(Show Context)
Citation Context ...graphs used in our experiments are divided into three categories as Problem Set I, II, and III (see Table 1). Problem Sets I and II consist of graphs (matrices) that arise from finite element methods =-=[13]-=-. Problem Set III consists of matrices that arise in eigenvalue computations [16]. In addition to providing some statistics about the structure of the test graphs, Table 1 also lists the number of col... |

4 | Iterated greedy graph coloring and the di#culty landscape - Culberson - 1992 |