Tiling Nested Loops into Maximal Rectangular Blocks

techniques are described as follows. Wolfe discusses the techniques of strip mining and iteration space tiling [25], which organize the computations in the original loops into chunks of equal size to take advantage of vector registers, caches, or local memory. Nicolau [13] proposes a method called loop quantization to partition nested loops. King and Li [9] discuss the grouping of loop iterations for parallel execution on multicomputers. The cycle shrinking method proposed by Polychronopoulos [16] uses the data dependence graphs of loops to determine which loops can be executed in parallel. Irigoin and Triolet [8] present a method called supernode partitioning, which formulates a general condition for determining the admissible tiles formed by multiple hyperplanes. Schreiber and Dongarra [20] discuss a heuristic method of choosing a subset of dependence vectors for tiling loops. Ramanujam and Sadayappan [18] formulate an approach to determine the hyperplanes and the tile sizes for reducing the communication cost in distributed memory systems. As for the code generation issue for loop tiling, Ancourt and Irigoin [1] discuss how to generate codes for the tiles defined by a set of linear equations, and the code generation techniques for nonunimodular loop transformations [12, 17] can also provide a solution to this problem. In this paper, we propose a new approach to tiling nested loops for exploiting parallelism. The proposed approach aims at aggregating as many independent computations as possible into a tile in order to maximize parallelism. At first, we describe a systematic procedure to find all the computations that are independent when the loops are to be executed. All these independent computations are collected together as a union of sets, which are called initially independent computation sets. Then, we show that all the initially independent computations can be aggregated into rectangular blocks. So, based on these, the original loops can be partitioned into rectangular blocks to maximize parallelism. Since each partitioned region is block-shaped, the code generation is very easy. Also, we show that if the wavefront transformation is combined with the proposed method, the original loops can always be tiled so that the tile size is greater than one. JOURNAL OF PARALLEL AND DISTRIBUTED COMPUTING 35, 123–132 (1996) ARTICLE NO. 0075