Experiments with Parallel Graph Coloring

We report on experiments with a new hybrid graph coloring algorithm, which combines a parallel version of Morgenstern's S-Impasse algorithm 20], with exhaustive search. We contribute new test data arising in ve diierent application domains, including register allocation and class scheduling. We test our algorithms both on this test data and on several types of randomly generated graphs. We compare our parallel implementation, which is done on the CM-5, with two simple heuristics, the Saturation algorithm of Br elaz 4] and the Recursive Largest First (RLF) algorithm of Leighton 18]. We also compare our results with previous work reported by Morgenstern 20] and Johnson et al. 13]. Our main results are as follows. On the randomly generated graphs, the performance of Hybrid is consistently better than the sequential algorithms, both in terms of speed and number of colorings produced. However, on large random graphs, our algorithms do not come close to the best colorings found by other time-intensive algorithms such as the XRLF algorithm of Johnson et al. 13] and Morgenstern's tuned S-Impasse algorithm. Of the ve types of test data, three are easily colored even by the simple RLF and Saturation heuristics; one (the class scheduling data) is optimally colored by Hybrid but not by the simple heuristics, and one appears to be very hard. However, it should not be concluded that coloring is \easy" in most applications. In several cases, such as the class and exam scheduling graphs, nding an optimal coloring is not suucient to solve the problem at hand, but rather colorings satisfying additional restrictions are really needed. The Hybrid algorithm parallelizes well. This appears to be for three main reasons. First, the number of iterations needed by the S-impasse decreases as the number of processors increase. Second, in the exhaustive search algorithm the work involved in expanding the search tree is eeectively shared among the processors. Third, on some tests, the S-Impasse and Exhaustive-Search procedures progress in a symbiotic fashion, one using a good coloring obtained by the other as a basis for further improvement. Overall, we are satissed that our parallel algorithm eeectively exploits the processing power of the CM-5, and that further work on a hybrid algorithm can lead to even better results. Since the performance of all the implemented algorithms on random graphs does not correlate well with their performance on application data, we conclude that further eeort spent in collecting application …