Introducing correct and optimal OpenMP parallelization directives in applications is a challenge. To parallelize a loop in an input application code automatically, parallelizing compilers need to disprove dependences with respect to variables across iterations of the loop. Performing such dependence analysis in the presence of index arrays or subscripted subscripts - a[b[i]] - has long been a challenge for automatic parallelizers. Loops with subscripted subscripts can be parallelized if the subscript array is known to possess a property such as monotonicity. This paper presents a compiletime algorithm that can analyze complex recurrence relations and determine irregular or intermittent monotonicity of one-dimensional and monotonicity of multi-dimensional subscript arrays. The new algorithm builds on a prior approach that is capable of analyzing simple recurrence relations and determining monotonic one-dimensional subscript arrays. Experimental results show that automatic parallelizers equipped with our new analysis techniques can substantially improve the performance of ten out of twelve or 83.33% of the benchmarks evaluated, 25--33.33% more than possible with state-of-the-art compile-time automatic parallelization techniques.
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