An Approach to Checking Link Conflicts in the Mapping of Uniform Dependence Algorithms into Lower Dimensional Processor Arrays

In this paper, we propose an enumeration method to check link conflicts in the mapping of $n$-dimensional uniform dependence algorithms with arbitrary convex index sets into $k$-dimensional processor arrays. Previous methods on checking the link conflicts had to examine either the whole index set or the I/O spaces whose size are $O(N^{2n})$ or $O(N^{n-1})$, respectively, where $N$ is the problem size of the $n$-dimensional uniform dependence algorithm. In our approach, checking the link conflicts is done by enumerating integer solutions of a mixed integer linear program. In order to enumerate integer solutions efficiently, a representation of the integer solutions is devised so that the size of the space enumerated is $O((2N)^{n-k})$. Thus, our approach to checking link conflicts has better performance than previous methods, especially for larger $k$. For the special case $k = n-2$, we show that link conflicts can be checked by solving two linear programs in one variable.