Koch snowflake. Fractal Fern

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1 CSC 111: Recursive Methods

2 Fractals: Self Similar Shapes Koch snowflake Fractal Fern

3 Functions: Example Problem Factorial of a number: 0! = 1 Factorial(N)= 1! = 1 Product of 1..N 2! = 2 3! = 6 4! = 24 5! = 120

4 Functions: Example Problem For Loop Implementation of Factorial int factorial(int int inputvalue) int product = 1; for (int i= 1; i <= inputvalue; i++) product = product * i; return product;

5 Functions: Example Problem Factorials have the interesting property that N! = N * (N-1)! 5! = 120 4! = 24 =>5!= 5 * 4! So, if we have a very simple function to compute (N- 1)!, we could use it to solve N! factorial Really solving same problem, but a smaller version This technique has a name: recursion

6 Recursion 2 Requirements: Solution must be able to be found with the same algorithm, with a smaller input (recursive case) There must exist a base case input that prevents further recursion

7 Recursion: Factorials int factorial(intint inputvalue) // base case if f( (inputvalue == 0) return 1; else if (inputvalue == 1) return 1; // recursive case else return inputvalue * factorial(inputvalueinputvalue 1);

8 Recursion: Examples Fibonacci Sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, Simple definition: Fib[0] = 1 Fib[1] = 1 Fib[N] = Fib(N-1) + Fib(N-2)

9 Recursive Fibonacci int fibonacci(int int input) // base case if ((input == 0) (input == 1)) return 1; // recursive case else return (fibonacci(input-1) + fibonacci(input-2));

10 Approach to Writing Recursive Functions Using factorial as an example, 1. Write the function header so that you are sure what the function will do and how it will be called. Factorial: Send in the integer for which we want to compute the factorial. Receive the integer result back. int factorial(int n)

11 Writing Recursive Functions 2. Decompose the problem into subproblems (if necessary). For factorial, we must compute (n-1)! 3. Write recursive calls to solve those subproblems whose form is similar to that of the original problem. There is a recursive call to write: factorial(n-1) ;

12 Writing Recursive Functions 4. Write code to combine the results of the recursive call(s) () to construct the desired return value. When a factorial has been computed, the result must be multiplied by the current value of the number for which the factorial was taken: return (n * factorial(n-1));

13 Writing Recursive Functions 5. Write base case(s) () to handle any situations that are not handled properly p by the recursive portion of the program. The base case for factorial is that if n=0, the factorial is 1: if (n == 0) return(1);

14 Recursion: Factorials int factorial(int inputvalue) if (inputvalue == 0) return 1; else return inputvalue * factorial(inputvalue 1);

15 Recursion: Examples Recursion is useful when datastructure is inherently recursive Directory hierarchy is a tree datastructure \ \c: /d: /e: \c:\program Files \c:\userdata \c:\program Files\Word \c:\program Files\Matlab

16 Directory Traversal listdirectory(directory y basedirectory) file[] files = getfiles(); int filecount = getfilecount(); for (int i= 0; i < filecount; i++) if (files[i].type == dir ) listdirectory(file); else listfile(file);

17 Working With Arrays Recursively Arrays are a natural target for recursion Decompose into a head element (at the front) A smaller tail array (everything but front) array head tail (also an array, but 1 size smaller)

18 Working With Arrays Recursively The tail array is itself an array, with a head element and a tail element array head tail (also an array, but 1 size smaller) head (of tail) tail (of tail)

19 Working With Arrays Recursively Base Cases: Hit end of array (looked at all elements) Some particular feature of current position (head element) array head tail (also an array, but 1 size smaller)

20 Recursive Find In Array boolean findinarray(int[] array, int position, int value) // end of array y( (base case) if (position == (array.length)) return false; // match target value (base case) else if (array[position] == value) return true; // not a match keep searching (recursive) else return findinarray(array, position+1, value);

21 Iterative Find In Array boolean findinarray(int int[] array, int value) for (int position = 0; position < array.length; position++) if (array[position] == value) return true; return false;

22 Iteration vs Recursion While Loops and Recursion are essentially interchangeable Considerations: Efficiency Simplification of programming Readability/Understandability

23 Recursive Fibonacci int fibonacci(int input) if ((input == 0) (input == 1)) return 1; else return (fibonacci(input-1) + fibonacci(input-2));

24 Iterative Fibonacci int fibonacci(int input) int first = 1; int second = 1; int temp; for (int k = 0; k < input; k++) temp = first; first = second; second = temp + second; return first;

25 Recursion: Factorials int factorial(int inputvalue) if (inputvalue == 0) return 1; else return inputvalue * factorial(inputvalue 1);

26 Iterative Factorials Implementation of an iterative factorial method? int factorial(int int inputvalue) int product = 1; for (int i= 1; i <= inputvalue; i++) product = product * i; return product;

34. Recursion. Java. Summer 2008 Instructor: Dr. Masoud Yaghini

34. Recursion. Java. Summer 2008 Instructor: Dr. Masoud Yaghini 34. Recursion Java Summer 2008 Instructor: Dr. Masoud Yaghini Outline Introduction Example: Factorials Example: Fibonacci Numbers Recursion vs. Iteration References Introduction Introduction Recursion