A cost-effective implementation of multilevel tiling

Abstract

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 1.5 and 2.8 times faster than conventional techniques. For loop nests having 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 2.2 and 11 times faster than in conventional techniques for typical linear algebra programs.

Multi-level tiling

“…Also, note that actual tile sizes are a product of all inner tile sizes because tiling at level k is a tiling on the (k + 1) tiled space, not the original iteration space. Although this formulation is a direct extension of Xue's definition of single level tiling [33], to the best of our knowledge, this is first formalization and presentation of itother formulations [15] of multi-level tiling are based on the strip-mine and interchange view of tiling. Now given the fact that this set P m tiled is a polyhedron, the scanning loops can be easily generated by existing tools, such as omega test and cloog.…”

“…The experimental results in Section 6.1 show how this exponential growth with respect to number of levels renders the technique inapplicable beyond two levels of tiling. The multi-level tiled loop generation method proposed by Jiminéz et al [15] has an exponential time complexity at each level of tiling, and this grows linearly with the number of levels of tiling.…”

“…However, several important applications such as numerical linear algebra routines (e.g., lu decomposition, triangular matrix product, symmetric rank updates from blas) and stencil computations (after applying skewing to enable tiling) have non-rectangular iteration spaces. Multi-level tiling has been shown to be very useful for them [14,15,27].…”

“…We are aware of only one solution that can generate arbitrary levels of multi-level tiled code for general polyhedral iteration spaces [15]. Their technique is limited to the case when tile sizes are fixed at compile (tiled loop generation) time, which is a severe limitation in the situations discussed above.…”

Automatic data movement and computation mapping for multi-level parallel architectures with explicitly managed memories

“…Multi-level tiling has become a key technique for high-performance computation. There has been work on generating efficient multi-level tiled code for polyhedral iteration spaces that handle tile sizes at compile time [23] and that handle tile sizes as symbolic parameters [26].…”