Determining the Idle Time of Tiling: New Results

In the framework of perfect loop nests with uniform dependencies, tiling has been studied extensively as a source-to-source program transformation. We build upon recent results by HH ogsted, Carter, and Ferrante 10], who aim at determining the cumulated idle time spent by all processors while executing the partitioned (tiled) computation domain. We propose new, much shorter proofs of all their results and extend these in several important directions. More precisely, we provide an accurate solution for all values of the rise parameter that relates the shape of the iteration space to that of the tiles, and for all possible distributions of the tiles to processors. In contrast, the authors in 10] deal only with a limited number of cases and provide upper bounds rather than exact formulas.