In this section we give some mathematical background that is related to the topics of this book. We do not attempt to give a detailed description of any of the subjects that we mention. Rather, we mention some definitions and facts and refer the reader to appropriate books for further reading.

Asymptotics

Usually, when a solution to some problem P is analyzed, we are not interested in the amount of resources that are used on some particular input; rather, we want to know how the amount of resources grows with the size of the input. Hence, we usually think of P as a sequence of problems P1, P2, P3, … where each Pn is the restriction of P to inputs of size n. For example, to analyze the number of bits Alice and Bob need to exchange in order to tell whether two input strings are equal, we make the analysis as a function of n, the length of the input strings. Hence, a complexity function is a function f(n) that depends on n, the size of the input.

It is most important to be able to compare such complexity functions. The difficulty is that it is possible that some functions f and g on input size n1 satisfy f(n1) < g(n1), whereas on input size n2 they satisfy f(n2) > g(n2). Hence it is useful to consider the asymptotic behavior of the complexity functions. That is, to see which function is larger for sufficiently large values of n. The following relations between f and g are of interest.