COP 4020 Fall 2005 Presentation Joshua Burkholder 2005 NOV 27. Recursion. What, Why, How, When, and What If?

Transcription

1 COP 4020 Fall 2005 Presentation Joshua Burkholder 2005 NOV 27 Recursion What, Why, How, When, and What If?

2 What is Recursion? Recursion is the definition of a function in terms of itself Recursion is a substitute for iteration int factorial (int n) { if (n == 0) return 1; else return n * factorial(n-1); Reference: Programming Language Pragmattics, Scott 2000

3 Why Use Recursion? Recursive code can be easier to write, understand, and maintain than its iterative counterpart at the cost of its efficiency. Recursive code can produce smaller code than its iterative counterpart at the cost of larger memory usage at run-time. Reference: Data Abstraction and Structures Using C++, Headington 1997

4 Recursion vs. Iteration // Recursion int factorial (int n) { if (n == 0) return 1; else return n * factorial(n-1); // Iteration int factorial (int n) { int i; int result = 1; for (i = 1; i <= n; ++i) result *= i; return result;

5 Recursion int factorial (int n) { if (n == 0) return 1; else return n * factorial(n-1) 1); Advantage: Easy to understand. 0! = 1 n! = n n 1! ( ) Disadvantage: Inefficient... n recursive calls results in n+1 frames on the stack. factorial(12);

6 Iteration int factorial (int n) { int i; int result = 1; for (i = 1; i <= n; ++i) result *= i; return result; Advantage: More efficient. There is only one frame on the stack regardless of the value of n. factorial(12); Disadvantage: Not as easy to understand.

7 How to use Recursion? Follow the Rules of Recursion

8 Rules of Recursion 1) Base Case Rule. 2) Making Progress Rule. 3) Design Rule. 4) Compound Interest Rule. Reference: Data Structures & Algorithm Analysis in Java, Weiss 1999

9 1) Base Case Rule. You must always have some base case(s), which can be solved without recursion. int factorial (int n) { if (n == 0) return 1; else return n * factorial(n-1); Reference: Data Structures & Algorithm Analysis in Java, Weiss 1999

10 2) Making Progress Rule. For the cases that are to be solved recursively, the recursive call must always be a case that makes progress toward the base case. int factorial (int n) { if (n == 0) return 1; else return n * factorial(n-1); Reference: Data Structures & Algorithm Analysis in Java, Weiss 1999

11 3) Design Rule. Assume that all recursive calls work. int factorial (int n) { if (n == 0) return 1; else return n * factorial(n-1) 1); Reference: Data Structures & Algorithm Analysis in Java, Weiss 1999

12 4) Compound Interest Rule. Never duplicate work by solving the same instance of a problem in separate recursive calls. int factorial (int n) { if (n == 0) return 1; else return n * factorial(n-1); Reference: Data Structures & Algorithm Analysis in Java, Weiss 1999

13 When to use Recursion? If the problem is stated recursively and a recursive algorithm is less complex than a nonrecursive version, then recursion is likely to be preferable. Reference: Data Abstraction and Structures Using C++, Headington 1997

14 When to use Recursion? If both recursive and nonrecursive algorithms are of similar complexity, the nonrecursive algorithm is likely to be more efficient and therefore preferred. Reference: Data Abstraction and Structures Using C++, Headington 1997

15 When to use Recursion? Don t forget to consider table-driven code as an option. If this technique fits the problem, a table lookup will outperform, by far, bothing looping and recursive algorithms. // Since 13! = 6,227,020,800 and results in integer overflow, // then a table going from 0! To 12! would be relevant. int factorial (int n) { int result[] = {1, 1, 2, 6, 24, 120, 720, 5040, 40320, , , , ; return result[n]; // Overrun error handling code NOT included. Reference: Data Abstraction and Structures Using C++, Headington 1997

16 Good vs. Bad Recursion Good Recursion: Recursion that improves understanding and maintainability without overly compromising efficiency. int factorial (int n) { if (n == 0) return 1; else return n * factorial(n-1);

17 Good vs. Bad Recursion Bad Recursion: Recursion that overly compromises efficiency. int fibonacci (int n) { if (n == 0 n == 1) return 1; else return fibonacci(n-1) + fibonacci(n-2); // fibonacci() doesn t follow the 4th Rule of // Recursion because it duplicates work by solving // the same instances of a problem in separate // recursive calls. Example: fibonacci(3) performs // fibonacci(1) twice.

18 What If There Wasn t Recursion? If a language only offered iteration, vice recursion, would there exist situations that couldn t be handled? No. For every recursive algorithm, there exists a comparable iterative algorithm. Reference: Data Abstraction and Structures Using C++, Headington 1997

19 What If There Wasn t Recursion? Moreover, languages that don t offer recursion usually offer the ability to create recursive-like constructs. For instance, in FORTRAN 77, dummy functions or GOTOs and arrays can be used to simulate recursive algorithms.

20 What If There Wasn t Recursion? C The use of dummy functions in FORTRAN 77 to C simulate Recursion. PROGRAM MAIN INTEGER N, RESULT EXTERNAL IFACT N = 12 RESULT = IFACT(N, IFACT) PRINT *, N, '! = ', RESULT STOP END INTEGER FUNCTION IFACT (N, IDUMMY) INTEGER N EXTERNAL IDUMMY END IF (N.EQ. 0) THEN IFACT = 1 ELSE IFACT = N * IDUMMY( N-1, IDUMMY ) ENDIF C Place the above code in the file: test.f C To compile: g77 W Wall pedantic test.f o test.exe Reference:

21 What If There Wasn t Recursion? C The use of GOTOs and arrays in FORTRAN 77 to simulate C Recursion by recreating the Stack (4*1024 bytes high). PROGRAM MAIN INTEGER NSTK(1024), INDEX, RTNSTK(1024), RESULT INDEX = 0 NSTK(INDEX+1) = 12 RTNSTK(INDEX+1) = 1 GOTO 2 1 PRINT *, NSTK(INDEX+1), '! = ', RESULT GOTO 5 2 INDEX = INDEX + 1 IF ( NSTK(INDEX).EQ. 0 ) THEN RESULT = 1 GOTO 4 ENDIF NSTK(INDEX+1) = NSTK(INDEX) - 1 RTNSTK(INDEX+1) = 3 GOTO 2 3 RESULT = NSTK(INDEX) * RESULT 4 CONTINUE INDEX = INDEX - 1 GOTO (1, 2, 3, 4, 5) RTNSTK(INDEX+1) 5 CONTINUE STOP END C Place the above code in the file: test.f C To compile: g77 W Wall pedantic test.f o test.exe

22 Summary of Recursion What? A function that calls itself. Why? To make the function easier to understand. How? Follow the four Rules of Recursion. When? When the increase in understanding outweighs the loss of efficiency. What If? Use iteration or (if they exist) recursive constructs.