Decomposition approach to the construction of parallel algorithms for processing of two-dimensional data

presented in this paper. The tool is intended to represent a variety of variants of computation organization for two-dimensional convolution and correlation, two-dimensional discrete Fourier transform (DFT), and other operations structurally similar to them. The creation way of such a tool, characterized by the use of the group-theoretical decomposition approach, is shown. The tool obtained is a parameterized, space-time description of parallel computation organization and provides a basis for combining algorithmic and architectural parameters in a single formal description. The tool is also architecturally independent and allows us to implement a stage of concurrent exploration of algorithms and architectures. This research stage is an integral part of the process of designing advanced computation systems and its implementation leads to improving the efficiency of parallel computing. The development of the formal tool presented in the article is based on the study of the internal space-time structure of algorithms using the proposed decomposition approach. Using this approach, which was originally developed for constructing compositional forms of the basic operations of digital signal processing, became possible due to its extension to the field of processing of two-dimensional data. The paper presents such an extension that made it possible to transfer the processing of two-dimensional data into a parameterized coordination-computing environment (CCE) intended to describe parallel computations. As a result of the extension realized, a system of an actions aimed at studying the internal structure of the computational operations of processing of two-dimensional data in order to detect their compositional forms is formed. These actions are developed and implemented within the scope of the approach proposed. The analysis of the structures of the obtained forms is carried out as a result their model nature is discovered. Thus, the transition to compositional forms of operations (CFO) made it possible to change the form of representation of computation organization in the processing of two-dimensional data from algorithmic to model one. A general model description of computation organization CFO ( A ijp ( t 11 , t 21 ), CCE i ( j , p )) for various operations of the class in question is given, the distinguishing feature of which is immersion of computations in a parametrized coor-dination-computing environment. The components of the description presented are the algorithms A ijp ( t 11 , t 21 ) obtained as a result of decomposition, immersion into the spatial environment, and time compression of the sequential algorithms A i ( t ), as well as the parameterized coordination-computational environments CCE i ( j , p ). The parameters describing the processing of two-dimensional data x ( t 1 , t 2 ) given on a group H of order N = N 1 × N 2 are presented as follows: t = t 2 + t 1 N 2 , N 1 = h 11 L 1 , N 2 = h 21 L 2 , t 1 = 0, …, N 1 – 1, t 11 = 0, …, h 11 – 1, j 1 = 0, …, L 1 – 1, t 2 = 0, …, N 2 –1, t 21 = 0, …, h 21 – 1, j 2 = 0, …, L 2 – 1. Having moved from the algorithmic form of representing two-dimensional operations to their model form, we not only saw the internal spatial structure of the operations, but also got the opportunity to configure it by parameters h 11 , L 1 and h 21 , L 2 synthesizing various variants of computation organization. Such an opportunity allows one to solve urgent problems aimed at increasing the efficiency of parallel processing, reducing its com-plexity, as well as increasing the flexibility of analysis of the data processed.