Time and Space

Time and space are fundamental parameters for measuring the efficiency of algorithms, and the notion of trading off one for the other is a familiar one in the programmer's informal repertoire. Nonetheless, providing satisfactory mathematical definitions of computational time and space and establishing formal relationships between them remains a central problem in computing theory. In this chapter we examine the interplay between time and space determined by a variety of machine models and explore the connection between time and space complexity classes. We consider a number of possible inclusion relationships among these classes and discuss their consequences, along with recent results indicating that mechanical procedures may be available for reducing the space used by programs. This rosy picture is darkened somewhat by a counterexample due to Cobham, which states that minimum time and space cannot always be achieved by a single program.

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