Algorithmics: Theory and Practice by Gilles Brassard

Algorithmics: Theory and Practice by Gilles Brassard

By Gilles Brassard

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Let X be a set of functions from N into R*, possibly defined by some asymptotic notation. Now 0 (X) denotes UJ0(f(n)) = {t : N -I R* I (3f EX)[teO(f(n))] }. f x Q(X) and 0(X) are defined similarly. 1. 3) is 0 (e ° (5Ifn n Inln n where n is the value of the integer to be factorized. 5 Conditional Asymptotic Notation Many algorithms are easier to analyse if initially we only consider instances whose size satisfies a certain condition, such as being a power of 2. Conditional asymptotic notation handles this situation.

Lim f (n)/g(n) E R+ => 0 (f (n)) = 0 (g(n)), and n-e ii. limf(n)/g(n) fl = 0 0(f(n))C O(g(n)) = O(g(n) ± f(n)), but = -40- iii. it can happen that 0 (f(n)) = 0 (g(n)) although the limit of f(n)/g(n) does not exist as n tends to infinity, and iv. it can happen that 0 (f (n)) C 0 (g(n)) when the limit of f (n)/g(n) does not exist as n tends to infinity and when it is also not true that 0 (g(n)) = 0 (g(n) -f (n)). 0 De l'Hopital's rule is often useful in order to apply the preceding problem. Recall that if lim f (n) = lim g(n) = 0, or if both these limits are infinite, then provided n - n - that the domains of f and g can be extended to some real interval [no, +oo) in such a way that the corresponding new functions f and g are differentiable on this interval and also that k'(x), the derivative of g(x), is never zero for x E [no, +oo), then lim f (n)/g(n) = lim f '(x)/g'(x), nl-*- X-400 provided that this last limit exists.

If an algorithm takes a time in 0 (f(n)) in the worst case, there exists a real positive constant c such that a time of cf (n) is sufficient for the algorithm to solve the worst instance of size n, for each sufficiently large n. This time is obviously also sufficient to solve all the other instances of size n, since they cannot take more time than the worst case. Assuming only a finite number of instances of each given size exists, there can thus be only a finite number of instances, all of size less than the threshold, on which the algorithm takes a time greater than cf(n).

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