Many computationally-intensive programs, such as those for differential equations, spatial interpolation, and dynamic programming, spend a large portion of their execution time in multiply-nested loops which have a regular stencil of data dependences. Tiling is a well-known optimization that improves performance on such loops, particularly for computers with a multi-levelled hierarchy of parallelism and memory. Most previous work on tiling restricts the tile shape to be rectangular. Our previous work and its extension by Desprez, Dongarra, Rastello and Robert Showed that for doubly nested loops, using parallelograms can improve parallel execution time by decreasing the idle time, the time that a processor spends waiting for data or synchronization. In this technical report, we extend that work to more deeply nested loops, as well as to more complex loop bounds. We introduce a model which allows us to demonstrate the equivalence in complexity of linear programming and determining the execution time of a tiling in the model. We then identify a sub-class of these loops that constitute rectilinear iteration spaces for which we derive a closed form formula for their execution time. This formula can be used by a compiler to predict the execution tome of a loop nest. We then derive the tile shape that minimizes this formula. Using the duality property of linear programming, we also study how the longest path of dependent tiles within a rectilinear iteration space depends on the slope of only four of the facets defining the iteration space, independent of its dimensionality.
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