By Gilles Brassard
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Additional resources for Algorithmics: Theory and Practice
For instance, [O (f (n))]2 does not denote the set of pairs of functions chosen from the set 0 (f(n)). Similarly, 0 (f(n)) x 0 (g(n)) does not denote the Cartesian product of O (f(n)) and 0 (g(n)). On the other hand, if N is the set of vertices of a directed graph, then NxN denotes as usual the set of possible edges between these vertices. In every case but one the context removes any potential ambiguity. Still, there is one case that must be treated cautiously. If the symbol "-" were used to denote the difference of two sets, a genuinely ambiguous situation would arise: what would o (n 3) - 0 (n 2) mean, for example?
We say that the modified value has been percolated up to its new position (one often encounters the rather strange term sift-up for this process). 8. A heap. Sec. 9 Data Structures 27 the larger of the values in the children, and then to continue this process downwards in the tree until the heap property is restored. We say that the modified value has been sifted down to its new position. The following procedures describe more formally the basic heap manipulation process. For the purpose of clarity, they are written so as to reflect as closely as possible the preceding discussion.
7. For arbitrary functions f and g: IN -4 R, prove that + g(n)) = 0 (max(f (n), g(n))), where the sum and the maximum are to be taken pointwise. 0 O (f (n) The result of the preceding problem is useful for simplifying asymptotic calculations. For instance, n 3 +3n2 +n +8 E O(n 3 +(3n2 +n +8)) = 0 (max(n 3 , 3n2 + n + 8)) = 0 (n 3 ). The last equality holds despite the fact that max(n 3 , 3n2 + n + 8) • n3 when 0 < n < 3, because the asymptotic notation only applies when n is sufficiently large.
Algorithmics: Theory and Practice by Gilles Brassard