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This monograph is devoted to a new method of parallel computing which uses VLSI technology in an effcient manner. By this method, data are fed to the cells of a systolic processor and results are obtained instantly. Some theoretical and algorithmic questions which arise in the design of hardware and software for systolic processing are considered. Special attention is devoted to the complexity of VLSI, complexity of algorithms, parallel algorithms, relations between graphs of algorithms and graphs of processors, parallel programming languages, and the use of systolic algorithms for vector programming. The book is unique for its inclusion of a library of systolic algorithms for solving problems from twelve branches of computer science, and will be useful for designers of hardware and software for parallel processing.