The organization of data is clearly of great importance in the design of high performance algorithms and architectures. Although there are several landmark papers on this subject, no comprehensive treatment has appeared. This monograph is intended to fill that gap. We introduce a model of computation for parallel computer architec­ tures, by which we are able to express the intrinsic complexity of data or­ ganization for specific architectures. We apply this model of computation to several existing parallel computer architectures, e.g., the CDC 205 and CRAY vector-computers, and the MPP binary array processor. The study of data organization in parallel computations was introduced as early as 1970. During the development of the ILLIAC IV system there was a need for a theory of possible data arrangements in interleaved mem­ ory systems. The resulting theory dealt primarily with storage schemes also called skewing schemes for 2-dimensional matrices, i.e., mappings from a- dimensional array to a number of memory banks. By means of the model of computation we are able to apply the theory of skewing schemes to var­ ious kinds of parallel computer architectures. This results in a number of consequences for both the design of parallel computer architectures and for applications of parallel processing.

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1 Data Communication and Data Organization in Parallel Computations: Classification and Overview.- 1.1 Introduction.- 1.2 Some Classification Schemes for Parallel Computer Architectures.- 1.3 Data Communication in Parallel Computer Architectures: a New Computational Viewpoint of Parallel Computations.- 1.4 Data Organization in Parallel Computer Architectures: the Theory of Skewing Schemes.- 2 Arbitrary Skewing Schemes for d-Dimensional Arrays.- 2.1 The General Case.- 2.2 The Validity of Skewing Schemes for Block Templates.- 2.3 The Validity of Skewing Scheines for [x1, x2,..., xd]-Lines.- 2.4 The Validity of Skewing Schemes for Polyominoes (Rookwise Connected Templates).- 3 Compactly Representable Skewing Schemes for d-Dimensional Arrays.- 3.1 Linear Skewing Schemes.- 3.2 Periodic Skewing Schemes.- 3.3 Multi-Periodic Skewing Schemes.- 4 Arbitrary Skewing Schemes for Trees.- 4.1 The Validity of Skewing Schemes for Trees.- 4.2 Skewing Schemes for Strips.- 4.3 An Exact Characterization of the Number µT({P1, P2,..., Pt}).- 4.4 Some Applications and Simplifications of Theorem 4.6.- 4.5 Applications of Theorem 4.6 (Theorem 4.7) to Certain Collections of Templates.- 4.6 Some Specific Results.- 5 Compactly Representable Skewing Schemes for Trees.- 5.1 Preliminaries.- 5.2 Semi-Regular Skewing Schemes.- 5.3 The Insufficiency of Semi-Regular Skewing Schemes.- 5.4 Regular Skewing Schemes.- 5.5 The Validity of Regular Skewing Schemes.- 5.6 Linear Skewing Schemes for Trees.