Derivation, extensions and parallel implementation of regular iterative algorithms

Regular Iterative Algorithms (RIAs) can be used to solve problems in a wide variety of areas including signal processing, matrix algebra, and combinatorics. Notably, RIAs include the class of algorithms implementable on systolic arrays. An attractive feature of RIAs is that these algorithms can be efficiently implemented on locally connected arrays of essentially identical processor modules, with register pipelines of various lengths and/or LIFO buffers in some of the links. Although efficient procedures for analyzing and implementing RIAs on regular processor arrays have recently been developed, issues such as optimal scheduling and parallel implementation of any general RIA have not been fully resolved. Moreover, algorithms are seldom available as RIAs and there are no systematic procedures for deriving RIAs from higher level descriptions of algorithms. In this thesis, some previous work by Karp, Miller, and Winograd (1967) and more recently by Rao, Jagadish and Kailath (1985) is extended to solve the problem of optimal scheduling and parallel implementation of RIAs. It is demonstrated that any RIA defined over a bounded or semi-infinite index space can be scheduled and mapped on to regular processor arrays by solving a set of integer programming problems with a small number of variables. An asymptotically optimal geometric scheduling scheme that works for any RIA is developed; in particular, we analyze the so-called computability tree to determine linear subspaces in the index space of a given RIA such that all variables lying on the same subspace can be scheduled at the same time. Next, procedures for converting algorithms described in fairly general terms (e.g., mathematical expressions) to RIAs are discussed. A class of algorithms called Assignment Codes (ACs) is precisely defined and formal procedures for converting linearly indexed ACs to RIAs are presented. Procedures to directly schedule linearly indexed codes are also discussed.