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<p><b>Abstract</b>—In this paper, we propose an enumeration method to check link conflicts in the mapping of <tmath>$n$</tmath>-dimensional uniform dependence algorithms with arbitrary convex index sets into <tmath>$k$</tmath>-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 <tmath>$O(N^{2n})$</tmath> or <tmath>$O(N^{n-1})$</tmath>, respectively, where <tmath>$N$</tmath> is the problem size of the <tmath>$n$</tmath>-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 <tmath>$O((2N)^{n-k})$</tmath>. Thus, our approach to checking link conflicts has better performance than previous methods, especially for larger <tmath>$k$</tmath>. For the special case <tmath>$k = n-2$</tmath>, we show that link conflicts can be checked by solving two linear programs in one variable.</p>
Uniform dependence algorithms, lower dimensional arrays, space-time mapping, link conflict, mixed integer linear programming, Hermite normal form, Smith normal form.
Jong-Chuang Tsay, Jenn-Yang Ke, "An Approach to Checking Link Conflicts in the Mapping of Uniform Dependence Algorithms into Lower Dimensional Processor Arrays", IEEE Transactions on Computers, vol. 48, no. , pp. 732-737, July 1999, doi:10.1109/12.780880
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