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Scientific Programming
Volume 12, Issue 2, Pages 91-100

Graph Transformation and Designing Parallel Sparse Matrix Algorithms beyond Data Dependence Analysis

H.X. Lin

Delft Institute of Applied Mathematics (DIAM), Delft University of Technology, 2628 CD, Delft, The Netherlands

Received 19 July 2004; Accepted 19 July 2004

Copyright © 2004 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Algorithms are often parallelized based on data dependence analysis manually or by means of parallel compilers. Some vector/matrix computations such as the matrix-vector products with simple data dependence structures (data parallelism) can be easily parallelized. For problems with more complicated data dependence structures, parallelization is less straightforward. The data dependence graph is a powerful means for designing and analyzing parallel algorithms. However, for sparse matrix computations, parallelization based on solely exploiting the existing parallelism in an algorithm does not always give satisfactory results. For example, the conventional Gaussian elimination algorithm for the solution of a tri-diagonal system is inherently sequential, so algorithms specially for parallel computation has to be designed. After briefly reviewing different parallelization approaches, a powerful graph formalism for designing parallel algorithms is introduced. This formalism will be discussed using a tri-diagonal system as an example. Its application to general matrix computations is also discussed. Its power in designing parallel algorithms beyond the ability of data dependence analysis is shown by means of a new algorithm called ACER (Alternating Cyclic Elimination and Reduction algorithm).