# Download Algorithms and Complexity (Internet edition, 1994) by Herbert S. Wilf PDF

By Herbert S. Wilf

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Sample text

Module is as follows. function fact(n); if n = 1 then fact := 1 else fact := n · fact(n − 1); end. The hallmark of a recursive procedure is that it calls itself, with arguments that are in some sense smaller than before. Notice that there are no visible loops in the recursive routine. ). Another advantage of recursiveness is that the thought processes are helpful. Mathematicians have known for years that induction is a marvellous method for proving theorems, making constructions, etc. Now computer scientists and programmers can profitably think recursively too, because recursive compilers allow them to express such thoughts in a natural way, and as a result many methods of great power are being formulated recursively, methods which, in many cases, might not have been developed if recursion were not readily available as a practical programming tool.

Not only that, but suppose this kind of unlucky choice is repeated on each and every recursive call. If the splitter element is the smallest array entry, then it won’t do a whole lot of splitting. In fact, if the original array had n entries, then one of the two recursive calls will be to an array with no entries at all, and the other recursive call will be to an array of n − 1 entries. If L(n) is the number of paired comparisons that are required in this extreme scenario, then, since the number of comparisons that are needed to carry out the call to split an array of length n is n − 1, it follows that L(n) = L(n − 1) + n − 1 (n ≥ 1; L(0) = 0).

Let’s state a preliminary version of the recursive procedure as follows (look carefully for how the procedure handles the trivial case where n=1). procedure quicksortprelim(x : an array of n numbers); {sorts the array x into nondecreasing order} if n ≥ 2 then permute the array elements so as to create a splitter; let x[i] be the splitter that was just created; quicksortprelim(the subarray x1, . . , xi−1) in place; quicksortprelim(the subarray xi+1, . . {quicksortprelim} * C. A. R. Hoare, Comp.