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By Gilles Brassard

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In this case the advantage that fib3 enjoys over fib2 is less marked, but their joint advantage over fib l remains just as striking. 11). 2. 553 * All times are in seconds. 25 Sec. 8 When Is an Algorithm Specified ? 6 Fourier Transforms The Fast Fourier Transform algorithm is perhaps the one algorithmic discovery that had the greatest practical impact in history. We shall come back to this subject in Chapter 9. For the moment let us only mention that Fourier transforms are of fundamental importance in such disparate applications as optics, acoustics, quantum physics, telecommunications, systems theory, and signal processing including speech recognition.

We shall be happiest if, when we analyse the asymptotic behaviour of an algorithm, its execution time is bounded simultaneously both above and below by positive real multiples (possibly different) of the same function. For this reason we introduce a final notation O(f(n)) = O(f(n)) n Q(f(n)), called the exact order of f (n). 14. (3c, d e Rt)(3n0 EN)(Vn > no) [cg(n)

4, provides a typical example of such behaviour: it takes a time in U(n 2) in the worst case, despite the fact that a time in the order of n is sufficient to solve arbitrarily large instances in which the items are already sorted. We shall be happiest if, when we analyse the asymptotic behaviour of an algorithm, its execution time is bounded simultaneously both above and below by positive real multiples (possibly different) of the same function. For this reason we introduce a final notation O(f(n)) = O(f(n)) n Q(f(n)), called the exact order of f (n).

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