We investigate the vectorization, on a Cray X-MP, of three graph algorithms, and introduce a data structure for representing graphs particularly amenable to this technique. The use of bit-wise instructions on the Cray, in conjunction with this data structure, leads to the efficient implementation of graph operations. In addition, this data structure makes the space requirement for the considered graph algorithms virtually optimal.
rnThe first investigated algorithm is the Breadth First Traversal, which is linear in execution time relative to the size of input. While this was found to be intrinsically non-vectorizable, execution times of its implementation using the introduced data structure did not increase.
rnThe second and third algorithms are both for calculating the chromatic polynomial of a graph, and whose execution time grows exponentially with the size of input. The second, by Read, is the fastest previously existing algorithm. However, since it uses levels of indirection in referencing edges, vectorization is usually inhibited. Only minor loops were vectorizable, and the speedup due to vectorization was less than two. The third algorithm uses the introduced data structure, and achieved a speedup of over three. Furthermore, this algorithm, when compared to the second one, was able to handle sparse graph (with average vertex degree of 4) approximately twice as large, while using similar time and space limits.
rnThis work was supported by NSF grant No. ASC-8900690.
我们在Cray X-MP上研究了三种图形算法的矢量化,并介绍了一种数据结构,用于表示特别适合该技术的图形。在Cray上使用按位指令以及此数据结构可导致图形操作的有效实现。此外,这种数据结构使所考虑的图算法的空间需求几乎最佳。 P> rn
首先研究的算法是广度优先遍历,其执行时间相对于输入大小呈线性关系。尽管发现它本质上是不可向量化的,但使用引入的数据结构实现该函数的执行时间并没有增加。 P> rn
第二和第三种算法都是用于计算图的色多项式,并且其执行时间随输入大小成指数增长。 Read表示的第二种是最快的现有算法。但是,由于它在引用边时使用间接级别,因此通常会禁止矢量化。只有较小的循环是可矢量化的,并且由于矢量化导致的加速小于2。第三种算法使用引入的数据结构,并实现了三倍以上的加速。此外,与第二种算法相比,该算法能够处理稀疏图(平均顶点度为4)约大两倍,而使用的时间和空间限制相似。 P> rn
由NSF授予ASC-8900690资助。 P>
Division of Computer and Information Sciences, University of South Alabama, Mobile, AL;
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