Achieving full parallelism using multidimensional retiming

Abstract

Most scienti c and Digital Signal Processing (DSP) applications are recursive or iterative. Transformation techniques are usually applied to get optimal execution rates in parallel and/or pipeline systems. The retiming technique is a common and valuable transformation tool in one-dimensional problems, when loops are represented by data ow graphs (DFGs). In this paper, uniform nested loops are modeled as multidimensional data ow graphs (MDFGs). Full parallelism of the loop body, i.e., all nodes in the MDFG executed in parallel, substantially decreases the overall computation time. It is well known that, for one-dimensional DFGs, retiming can not always achieve full parallelism. Other existing optimization techniques for nested loops also can not always achieve full parallelism. This paper shows an important and counter-intuitive result, which proves that we can always obtain full-parallelism for MDFGs with more than one dimension. This result is obtained by transforming the MDFG into a new structure. The restructuring process is based on a multi-dimensional retiming technique. The theory and two algorithms to obtain full parallelism are presented in this paper. Examples of optimization of nested loops, and digital signal processing designs are shown to demonstrate the e ectiveness of the algorithms.

Analysis of Parallel Multidimensional Wave Digital Filtering Network on IBM Cell Broadband Engine

“…It is of importance to note that following the principle of MD retiming [14,16], ( ) can be obtained immediately without any difficulty by the following formula:…”

“…Given any scheduled element , a legal retiming function of a realizable MDFG for each node of a loop body can be obtained when it is orthogonal to according to [11,14]. As a consequence, the chained MD scheduling generates a retiming vector ( ) for each computing node in the MDFG such that the MD delay is pushed from incoming edges of to its outgoing edges, and new delay of each edge is given by…”

“…The objective of the chained MD retiming technique applied to the MDWDF network of Figure 1(b) is to legally change the delay of edges in the MD data flow graph (MDFG) such that nonzero delays on all edges can be obtained in order to achieve the full parallelism of the MDWDF network. Here the chained MD retiming is based on the push up scheduling technique [14,16] where a good example illustrated in Figure 2(b) represents a retimed MDFG from a valid MDFG depicted in Figure 2(a). We note that the form of MDFG is described as a cyclic data flow graph with the tuple ( , , , ) to represent a node-weighted and edge-weighted graph where is the set (1, −1) Journal of Computational Engineering of computation nodes in the loop body, denotes the set of directed edges representing the dependence between two nodes, is a function representing the MD delay between two nodes, and is the discrete time required for computing a certain node [14,16].…”

“…The technique of the chained MD retiming [14,16] for scheduling MD problems starts with finding a schedule vector ∈ + where + is a strictly positive scheduled subspace defined by…”

Optimal loop parallelization for maximizing iteration-level parallelism

“…Retiming is originally proposed to minimize the cycle period of a synchronous circuit by evenly distributing registers. It has been extended to schedule data flow graphs on parallel systems in [7,6,21]. For loop transformation, retiming is used under the name of "index shift method" for parallelizing nested loops [16].…”