Algorithms and Data Structures

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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.
N.Wirth. Algorithms and Data Structures. Oberon version 103 3.3 Two Examples of Recursive Programs The attractive graphic pattern shown in Fig. 3.4 consists of a superposition of five curves. These curves follow a regular pattern and suggest that they might be drawn by a display or a plotter under control of a computer. Our goal is to discover the recursion schema, according to which the drawing program might be constructed. Inspection reveals that three of the superimposed curves have the shapes shown in Fig. 3.3; we denote them by H 1 , H 2 and H 3 . The figures show that H i+1 is obtained by the composition of four instances of H i of half size and appropriate rotation, and by tying together the four H i by three connecting lines. Notice that H 1 may be considered as consisting of four instances of an empty H 0 connected by three straight lines. H i is called the Hilbert curve of order i after its inventor, the mathematician D. Hilbert (1891). H 1 H 2 H 3 Fig. 3.3. Hilbert curves of order 1, 2, and 3 Since each curve H i consists of four half-sized copies of H i-1 , we express the procedure for drawing H i as a composition of four calls for drawing H i-1 in half size and appropriate rotation. For the purpose of illustration we denote the four parts by A, B, C and D, and the routines drawing the interconnecting lines by arrows pointing in the corresponding direction. Then the following recursion scheme emerges (see Fig. 3.3). A: D ← A ↓ A → B B: C ↑ B → B ↓ A C: B → C ↑ C ← D D: A ↓ D ← D ↑ C Let us assume that for drawing line segments we have at our disposal a procedure line which moves a drawing pen in a given direction by a given distance. For our convenience, we assume that the direction be indicated by an integer parameter i as 45 × i degrees. If the length of the unit line is denoted by u, the procedure corresponding to the scheme A is readily expressed by using recursive activations of analogously designed procedures B and D and of itself. PROCEDURE A (i: INTEGER); BEGIN IF i > 0 THEN D(i-1); line(4, u); A(i-1); line(6, u); A(i-1); line(0, u); B(i-1) END END A This procedure is initiated by the main program once for every Hilbert curve to be superimposed. The
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Algorithms and Data Structures

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