By Gilles Brassard

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The notation 0 (f(n)) defined previously is thus equivalent to 0 (f(n) | P(n)) where P(n) is the predicate whose value is always true. The notation Q(f(n) | P(n)) and t9(f(n) I P(n)) is defined similarly, as is the notation with several parameters. The principal reason for using this conditional notation is that it can generally be eliminated once it has been used to make the analysis of an algorithm easier. 12. A function f :N - R* is Analysing the Efficiency of Algorithms 46 Chap. 2 eventually nondecreasing if (3n 0 EIN)(Vn n0) [f(n)

1 Williams (1964). The improvements suggested at the end of the sub-section on heaps are described in Johnson (1975), Fredman and Tarjan (1984), Gonnet and Munro (1986), and Carlsson (1986, 1987). Carlsson (1986) also describes a data structure, which he calls the double-ended heap, or deap, that allows finding efficiently the largest and the smallest elements of a set. For ideas on building heaps faster, consult McDiarmid and Reed (1987). In this book, we give only some of the possible uses of disjoint set structures; for more applications see Hopcroft and Karp (1971) and Aho, Hopcroft, and Ullman (1974, 1976).

This algorithm can be described formally as follows. procedure make-heap (T [I .. n ]) I this procedure makes the array T [ I . 4 that this algorithm allows the creation of a heap in linear time. (a) The starting situation. Q Ta0 (b) The level I subtrees are made into heaps. (c) One level 2 subtree is made into a heap (the other already is a heap). 9. Making a heap. Preliminaries 30 Chap. 2. Let T[l .. 12] be an array such that T[i] =i for each i < 12. Exhibit the state of the array after each of the following procedure calls: make-heap (T) alter-heap(T, 12, 10) alter-heap (T.