Algorithms and Data Structures

N.Wirth. Algorithms and Data Structures. Oberon version 102
fib n+1 = fib n + fib n-1 for n > 0
and fib 1 = 1, fib 0 = 0. A direct, naive transcription leads to the recursive program
PROCEDURE Fib (n: INTEGER): INTEGER;
VAR res: INTEGER;
BEGIN
IF n = 0 THEN res := 0
ELSIF n = 1 THEN res := 1
ELSE res := Fib(n-1) + Fib(n-2)
END;
RETURN res
END Fib
Computation of fib n by a call Fib(n) causes this function procedure to be activated recursively. How
often? We notice that each call with n > 1 leads to 2 further calls, i.e., the total number of calls grows
exponentially (see Fig. 3.2). Such a program is clearly impractical.
5
4
3
3
2
2
1
2
1
1
0
1
0
1
0
Fig. 3.2. The 15 activations of Fib(5)
But fortunately the Fibonacci numbers can be computed by an iterative scheme that avoids the
recomputation of the same values by use of auxiliary variables such that x = fib i and y = fib i-1 .
i := 1; x := 1; y := 0;
WHILE i < n DO z := x; x := x + y; y := z; i := i + 1 END
Note: The assignments to x, y, z may be expressed by two assignments only without a need for the
auxiliary variable z: x := x + y; y := x - y.
Thus, the lesson to be drawn is to avoid the use of recursion when there is an obvious solution by
iteration. This, however, should not lead to shying away from recursion at any price. There are many good
applications of recursion, as the following paragraphs and chapters will demonstrate. The fact that
implementations of recursive procedures on essentially non-recursive machines exist proves that for
practical purposes every recursive program can be transformed into a purely iterative one. This, however,
involves the explicit handling of a recursion stack, and these operations will often obscure the essence of a
program to such an extent that it becomes most difficult to comprehend. The lesson is that algorithms which
by their nature are recursive rather than iterative should be formulated as recursive procedures. In order to
appreciate this point, the reader is referred to the algorithms for QuickSort and NonRecursiveQuickSort
in Sect. 2.3.3 for a comparison.
The remaining part of this chapter is devoted to the development of some recursive programs in
situations in which recursion is justifiably appropriate. Also Chap. 4 makes extensive use of recursion in
cases in which the underlying data structures let the choice of recursive solutions appear obvious and
natural.