| Summary: | 博士 === 國立交通大學 === 資訊工程學系 === 85 === A uniform dependence algorithm can be represented by an index
set of index points and a finite set of data dependence vectors.
Usually, the convex hull of the index set is a nondegenerated
convex polytope in n-dimensional real vector space . And, we
call such an algorithm an n-dimensional uniform dependence
algorithm. To synthesize a regular array from the uniform
dependence algorithm, there are two main issues, namely, the
time schedule problem and the processor assignment problem. In
this dissertation, a unified approach is proposed for the
problems to find an optimal linear schedule and a space-optimal
linear array for a uniform dependence algorithm with an
arbitrary bounded convex index set. Both problems are reduced to
the problem of finding a vector of smallest norm (length)
satisfying the constraints of the problems, where the vector
norm is defined on a symmetric convex set (which is derived from
the convex hull of the index set). The optimal linear schedule
problem of a uniform dependence algorithm is to find a linear
schedule vector such that the total execution time of the
algorithm is minimized. It is found that the total execution
time of a linear schedule is related to the norm of the linear
schedule vector. For this problem, a linear programming problem
is derived for finding a vector with smallest norm. Time
complexity analysis shows that the empirical average time
complexity of our method is better than those of the existing
methods. The space-optimal linear array problem of a uniform
dependence algorithm is to find a PE (Processing Element)
allocation vector such that the number of PEs used in the linear
array is minimized. It is founded that the number of PEs used is
related to the norm of the PE allocation vector. Since the
design constraints of the space-optimal linear array problem
usually do not have a closed and linear form, we proposed an
enumeration procedure to find a vector with the smallest norm.
The enumeration procedure finds a space-optimal PE allocation
vector, assuming that a valid linear schedule has been given a
prior. A tool called SODTLA (Space-Optimal Design Tool for
Linear Array) was developed to find a space-optimal PE
allocation vector without the user''s intervention. To be used in
the enumeration procedure, a method is also proposed to check
the link conflicts in the mapping of n-dimensional uniform
dependence algorithms into lower dimensional processor arrays,
i.e., a k-dimensional processor arrays with 0 < k < n-1. Time
complexity estimation shows that our method to check link
conflicts has better performance than previous methods. 1
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