A Geometric Programming Framework for Optimal Multi-Level Tiling
Proceedings of the 2004 ACM/IEEE conference on Supercomputing
Proceedings of the 13th ACM SIGPLAN Symposium on Principles and practice of parallel programming
Multi-level tiling: M for the price of one
Proceedings of the 2007 ACM/IEEE conference on Supercomputing
Parametric multi-level tiling of imperfectly nested loops
Proceedings of the 23rd international conference on Supercomputing
Parameterized tiling revisited
Proceedings of the 8th annual IEEE/ACM international symposium on Code generation and optimization
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This paper presents a new cost-effective algorithm to compute exact loop bounds when multilevel tiling is applied to a loop nest having affine functions as bounds (nonrectangular loop nest). Traditionally, exact loop bounds computation has not been performed because its complexity is doubly exponential on the number of loops in the multilevel tiled code and, therefore, for certain classes of loops (i.e., nonrectangular loop nests), can be extremely time consuming. Although computation of exact loop bounds is not very important when tiling only for cache levels, it is critical when tiling includes the register level. This paper presents an efficient implementation of multilevel tiling that computes exact loop bounds and has a much lower complexity than conventional techniques. To achieve this lower complexity, our technique deals simultaneously with all levels to be tiled, rather than applying tiling level by level as is usually done. For loop nests having very simple affine functions as bounds, results show that our method is between 15 and 28 times faster than conventional techniques. For loop nests caving not so simple bounds, we have measured speedups as high as 2,300. Additionally, our technique allows eliminating redundant bounds efficiently. Results show that eliminating redundant bounds in our method is between 22 and 11 times faster than in conventional techniques for typical linear algebra programs.