By Herbert S. Wilf
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Additional info for Algorithms and Complexity (Second edition)
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 marvelous 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.
We will see how the recursive structure helps us to analyze the running time, or complexity, of the algorithms. We will also find that there is a bit of art involved in choosing the list of variables on which a recursive procedure operates. 2. Quicksort 51 detailed information than we have provided for it. , but more of this later. 1 Exercise 1. Write a recursive routine that will find the digits of a given integer n in the base b. There should be no visible loops in your program. 2 Quicksort Suppose that we are given an array x, .
In an easy case like this, we can write out the first few xs and then guess the answer. We find, successively, that x1 = b1 x0 , then x2 = b2 x1 = b2 b1 x0 and x3 = b3 x2 = b3 b2 b1 x0 etc. At this point, we can guess that the solution is: xn = x0 n Y bi (n = 0, 1, 2, . ). 27) i=1 Since that wasn’t hard enough, we’ll raise the ante a step further. Suppose we want to solve the first-order inhomogeneous (because xn = 0 for all n is not a solution) recurrence relation: xn+1 = bn+1 xn + cn+1 (n ≥ 0; x0 given).
Algorithms and Complexity (Second edition) by Herbert S. Wilf