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The power of computers comes from their ability to execute the same task, or different versions of the same task, repeatedly. In computing, the theme of iteration is met in a number of guises. Many concepts in data models, such as lists, are forms of repetition, as “A list either is empty or is one element followed by another, then another, and so on.” Programs and algorithms use iteration to perform repetitive jobs without requiring a large number of similar steps to be specified individually, as “Do the next step 1000 times.” Programming languages use looping constructs, like the while- and for-statements of C, to implement iterative algorithms. Closely related to repetition is recursion, a technique in which a concept is defined, directly or indirectly, in terms of itself. For example, we could have defined a list by saying “A list either is empty or is an element followed by a list.”

Recursion is supported by many programming languages. In C, a function F can call itself, either directly from within the body of F itself, or indirectly by calling some other function, which calls another, and another, and so on, until finally some function in the sequence calls F. Another important idea, induction, is closely related to “recursion” and is used in many mathematical proofs. Iteration, induction, and recursion are fundamental concepts that appear in many forms in data models, data structures, and algorithms.

The following list gives some examples of uses of these concepts; each will be covered in some detail in this book. 1. Iterative techniques. The simplest way to perform a sequence of operations repeatedly is to use an iterative construct such as the for-statement of C. 2. Recursive programming. C and many other languages permit recursive functions, which call themselves either directly or indirectly. Often, beginning programmers are more secure writing iterative programs than recursive ones, but an important goal of this book is to accustom the reader to thinking and programming recursively, when appropriate. Recursive programs can be simpler to write, analyze, and understand.

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