By Gilles Brassard
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Extra info for Algorithmics: Theory and Practice
For a more popular account of algorithms, see Knuth (1977) and Lewis and Papadimitriou (1978). Multiplication d la russe is described in Warusfel (1961), a remarkable little French book of popular mathematics. Although we do not use any specific programming language in the present book, we suggest that a reader unfamiliar with Pascal would do well to look at one of the numerous books on this language, such as Jensen and Wirth (1985) or Lecarme and Nebut (1985). Knuth (1973) contains a large number of sorting algorithms.
A heap is an essentially complete binary tree, each of whose nodes includes an element of information called the value of the node. The heap property is that the value of each internal node is greater than or equal to the values of its children. 8 gives an example of a heap. This same heap can be represented by the following array: l 10 l 7 l 9 l 4 l 7 | 5 l 2 l 2 1 l 6 l The fundamental characteristic of this data structure is that the heap property can be restored efficiently after modification of the value of a node.
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.