Abstract

this paper, we extend the existing numerical methods to overcome these difficulties. A geometrical analysis reveals that we can take advantage of the regular shape of the convex sets derived from multi-dimensional arrays in a data dependence test. The general methods proposed before assume very general convex sets; this assumption causes their inefficiency. We have implemented a new algorithm called the l-test and performed some measurements. Results were quite encouraging (see Section 4). As in earlier numerical methods, the proposed scheme uses Diophantine equations and bounds of real functions. The major difference lies in the way multiple dimensions are treated. In earlier numerical methods, data areas accessed by two array references are examined dimension by dimension. If the examination of any dimension shows that the two areas representing the subscript expressions are disjoint, there is no data dependence between the two references. However, if each pair of areas appears to overlap in each individual dimension, it is unclear whether there is an overlapped area - 3 - when all dimensions are considered simultaneously. In this case, a data dependence has to be assumed. Our algorithm treats all dimensions simultaneously. Based on the subscripts, it selects a few suitable "viewing angles" so that it gets an exact view of the data areas. Selection of the viewing angles is rather straightforward and only a few angles are needed in most cases. We present the rest of our paper as follows. In Section 2, we give some examples to illustrate the difficulties in data dependence analysis on multi-dimensional array references. Some measurement results on a large set of real programs are presented to show the actual frequency of such difficult cases. In Section 3, we describe...

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