Loop Distribution and Fusion with Timing and Code Size Optimization

Abstract

In this paper, a technique that combines loop distribution with maximum direct loop fusion (LD_MDF) is proposed. The technique performs maximum loop distribution, followed by maximum direct loop fusion to optimize timing and code size simultaneously. The loop distribution theorems that state the conditions distributing any multi-level nested loop in the maximum way are proved. It is proved that the statements involved in the dependence cycle can be fully distributed if the summation of the edge weight of the dependence cycle satisfies a certain condition; otherwise, the statements should be put in the same loop after loop distribution. Based on the loop distribution theorems, algorithms are designed to conduct maximum loop distribution. The maximum direct loop fusion problem is mapped to the graph partitioning problem. A polynomial graph partitioning algorithm is developed to compute the fusion partitions. It is proved that the proposed maximum direct loop fusion algorithm produces the fewest number of resultant loop nests without violating dependence constraints. It is also shown that the resultant code size of the fused loops by the technique of loop distribution with maximum direct loop fusion is smaller than the code size of the original loops when the number of fused loops is less than the number of the original loops. The simulation results are presented to validate the proposed technique.

Execution time optimisation using delayed multidimensional retiming

“…These techniques can be classified in terms of granularity, which are the iteration level parallelism and the instruction level parallelism. The first type guarantees exploring the nested loops in order to identify and parallelize the independent iterations [Grosser et al 2013;Liu et al 2011;Qasem and Kennedy 2008;Liu et al 2009, Xue et al 2007Wang et al 2014]. The second one consists in applying the software pipelining in order to parallelize the independent instruction [Rong et al 2007;Khan 2011;Zhuge et al 2008].…”

“…Few techniques explore the instruction level parallelism across all the nested loops [Morvan et al 2011;Muthukumar and Doshi 2001] whose Multidimensional Retiming (MR) is distinguished as one of the most important technique Sheliga et al 1996;Zhuge et al 2008]. It models the loop body into a multidimensional Data Flow Graph (MDFG) [Xue et al 2007;Lee et al 2005;Liu et al 2011;Zhuge et al 2008] which allows explicitly featuring the granularity of instructions and their data dependencies. Then, it formulates the parallelism as a graph transformation theory.…”