Tiling with limited resources

Abstract

In the framework of perfect loop nests with uniform dependences, tiling has been extensively studied as a source-to-source program transformation. Little work has been devoted to the mapping and scheduling of the tiles on to physical processors. We present several new results in the context of limited computational resources, and assuming communication-computation overlap. In particular, under some reasonable assumptions, we derive the optimal mapping and scheduling of tiles to physical processors.

Static tiling for heterogeneous computing platforms

“…7 This small example shows that our target problem is intrinsically more complex than the instance with same-speed processors: as shown in Ref. [8], a columnwise allocation would be optimal for our two-column iteration space with two processors of equal speed.…”

“…Rather than providing a detailed motivation for tiling, we refer the reader to the papers by Calland, Dongarra and Robert [8] and by H ogsted, Carter and Ferrante [16], which provide a review of the existing literature. Brie¯y, most of the work amounts to partitioning the iteration space of a uniform loop nest into tiles whose shape and size are optimized according to some criterion (such as the communication-to-computation ratio).…”

“…Specifying a columnwise execution may lead to the simplest code generation. When all processors have equal speed, it turns out that a pure cyclic columnwise allocation provides the best solution among all possible distributions of tiles to processors [8] ± provided that the communication cost for a tile is not greater than the computation cost. Since the communication cost for a tile is proportional to its surface, while the computation cost is proportional to its volume, 2 this hypothesis will be satis®ed if the tile is large enough.…”

A Survey of Loop Parallelization: Models, Approaches, and Recent Developments

“…Tilling can be taken as both an important transformation and a code generation technique, which is concerned by a large amount of literature [28,30]. The classical definition of the "tile" should include three points [28]: 1) it is constrained by bounds; 2) it is identical by translation; and 3) it is an atomic unit.…”

“…The classical definition of the "tile" should include three points [28]: 1) it is constrained by bounds; 2) it is identical by translation; and 3) it is an atomic unit. The bound constraints are obvious.…”