A Unified Approach to Finding the Optimal Linear Schedules and

• Main Authors: Ke, Jenn-Yang, 柯振揚
• Other Authors: Tsay Jong-Chuang
• Format: Others
• Language: zh-TW
• Published: 1997
• Online Access: http://ndltd.ncl.edu.tw/handle/32994957656730149454

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