One of the most important factors for choosing an algorithm is the speed with which it is likely to complete. Characterizing the expected computation time of an algorithm is inherently a mathematical process. This chapter presents the mathematical tools behind this time prediction. After reading the chapter, you should understand the various mathematical terms used throughout this book—and in the rest of the literature that describes algorithms.

An instance of a problem is a particular input data set given to a program. In most problems, the execution time of a program increases with the size of this data set. At the same time, overly compact representations (possibly using compression techniques) may unnecessarily slow down the execution of a program. It is surprisingly difficult to define the optimal way to encode an instance because problems occur in the real world and must be translated into an appropriate representation to be solved by a program.

When evaluating an algorithm, we want as much as possible to assume the encoding of the problem instance is not the determining factor in whether the algorithm can be implemented efficiently. Your representation of a problem instance should depend just on the type and variety of operations that need to be performed. Designing efficient algorithms often starts by selecting the proper data structures in which to represent the problem.

Because we cannot formally define ...

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