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 e cient, 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 nite 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 PRAM 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.
Incomplete factorization has been shown to be a good preconditioner for the conjugate gradient method on a wide variety of problems. It is well known that allowing some fill-in during the incomplete factorization can significantly reduce the number of iterations needed for convergence. Allowing fill-in, however, increases the time for the factorization and for the triangular system solutions. Additionally, it is difficult to predict a priori how much fill-in to allow and how to allow it. The unpredictability of the required storage/work and the unknown benefits of the additional fill-in make such strategies impractical to use in many situations. In this article we motivate, and then present, two “black-box” strategies that significantly increase the effectiveness of incomplete Cholesky factorization as a preconditioner. These strategies require no parameters from the user and do not increase the cost of the triangular system solutions. Efficient implementations for these algorithms are described. These algorithms are shown to be successful for a variety of problems from the Harwell-Boeing sparse matrix collection.
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