A Geometric Programming Framework for Optimal Multi-Level Tiling

Determining the optimal tile size-one that minimizes the execution time-is a classical problem in compilation and performance tuning of loop kernels. Designing a model of the overall execution time of a tiled loop nest is an important subproblem. Both problems become harder when tiling is applied at multiple levels. We present a framework for determining the optimal tile sizes for a fully permutable, perfectly nested, rectangular loop with uniform dependences. Our framework supports multiple levels of tiling and uses a BSP style high level model for estimating the overall execution time of a loop program. In our framework, the problem of determining the optimal tile sizes, subject to memory capacity and bandwidth constraints, is modeled as a geometric program and transformed into a convex optimization problem, which can be solved efficiently. The model is validated through experimental results obtained by running twenty loop programs for different levels of tiling and different program and tile parameters. Our framework is very general and can also be used to solve the optimal tile size problem with many other models of execution time.