1. CSE 1342 Programming Concepts Recursion

2. Overview of Recursion nRecursion is present when a function is defined in terms of itself. nThe factorial of an integer can be expressed using a recursive definition. For example, 5 factorial (5!) may be expressed as: 5! = 5 * 4 * 3 * 2 * 1 However, a more concise definition involving recursion would be 5! = 5 * 4! Now, in order to find the definition of 5! We must first find the definition of 4!, then 3!, then 2! And finally 1!.

3. 5! = 5 * 4! 4! = 4 * 3! 3! = 3 * 2! 2! = 2 * 1! 1! = 1 (basis case or stopping state) nA more generic recursive factorial definition would be: N! = N * (N-1)! n This assumes that N >= 0 and that the factorial of 0 is 1 and the factorial of 1 is also 1 (a non-recursive definition). Overview of Recursion

4. Factorial Example nConverting the mathematical definition to an equivalent software algorithm we have... unsigned int factorial(unsigned long number) { if (number <= 1) return 1; // basis case or stopping state else return number * factorial(number - 1); // direct recursive call } nNotice how the last statement conforms to the expression N * (N-1)!

5. Overview of Recursion nAs discussed earlier, control structures may be categorized as either sequential structures, selection structures or repetition structures. nRecursion is a technique by which to achieve repetition. n Iterative loops such as while, do while and for are the other technique. nAny algorithm that can be expressed iteratively can also be expressed recursively, and visa versa.

6. Overview of Recursion nIn many cases an algorithm expressed recursively is more concise than when expressed iteratively. nIn general, recursive algorithms run slower and require more memory than do iterative algorithms. n This often leads to a trade-off between efficiency and simplicity of expression. nThere are two types of recursion n Direct recursion is when a function contains a call to itself within its own body. The factorial function is an example of direct recursion.

7. n Indirect recursion is when a function calls a second function which in turn calls the first function. void g( ) { f ( ); // indirect recursive call } void f ( ) { g ( ); } void main ( ) { f ( ); } Overview of Recursion

8. Stacks and Function Calls nA stack is a last-in-first-out data structure used by program translators in the implementation of function calls, both recursive and non-recursive. nA stack is is operated on by two functions - push and pop. n Push adds a new data element to the top of the stack. n Pop removes an existing data element from the top of the stack. nWhenever a program translator encounters a function call it generates the code to push a return address and any function arguments that need to be passed to the called function onto the stack.

9. Stacks and Function Calls nWhenever a program translator encounters a return statement it generates the code to pop a return address from the top of the stack (and into the instruction pointer) as well as any parameters that might have been previously pushed. n The pops must be in the opposite order of the corresponding pushes. nIn this way a program can find its way back from any number of nested function calls. nUnderstanding the relationship between stacks and function calls can help in walking through complex recursive algorithms.

10. Recursion Examples

11. Non- Recursive Factorial Example unsigned int factorial (unsigned int x) { unsigned int fact=1; while(x>1) { fact=fact*x; x--; } return fact; }

12. Recursive Factorial Example unsigned int factorial(unsigned long number) { if (number <= 1) return 1; // basis case or stopping state else return number * factorial(number - 1); }

13. NON-RECURSIVE PALINDROME DETECTION FUNCTION bool pdrome(char a[ ], int lb, int ub) { bool pflag = true; while(lb < ub && pflag) { if(a[lb] != a[ub]) //string is not a palindrome pflag = false; else { //string is a palindrome so far lb++; ub--; } } return pflag; }

14. RECURSIVE PALINDROME DETECTION FUNCTION bool recPdrome(char a[ ], int lb, int ub) { if(lb >= ub) //string is a palindrome return true; else if(a[lb] != a[ub]) //string not a palindrome return false; else //string is palindrome so far return recPdrome(a, lb+1, ub-1); }

15. NON-RECURSIVE BINARY SEARCH int search(int a[], int lb, int ub, int value) { //Non-recursive binary search routine int half, found = 0; while (lb <= ub && found == 0) { half = (lb+ub) / 2; if(a[half] == value) found = 1; else if(a[half] > value) ub = half-1; //calculate new upper bound else lb = half+1; //calculate new lower bound } (found) ? return half : return -1; }

16. RECURSIVE BINARY SEARCH int recSearch(int a[], int lb, int ub, int value) { //Recursive binary search routine int half; if(lb > ub) return -1; //value is not in the array half = (lb+ub) / 2; if(a[half] == value) //value is in the array return half; //return value's location else if(a[half] > value) return recSearch(a, lb, half-1, value); //search lower half of array else return recSearch(a, half+1, ub, value); //search upper half of array }

17. TOWERS OF HANOI DRIVER void towers(int, int, int, int); main() { int nDisks; cout << “Enter the starting number of disks: “; cin >> nDisks; towers (nDisks, 1, 3, 2); return 0; }

18. TOWERS OF HANOI void towers(int disks, int start, int end, int temp) { if (disks == 1) { cout << start << “ TO ” << end << endl; return; } // move disks – 1 disks from start to temp towers (disks – 1, start, temp, end); // move last disk from start to end cout << start << “  ” << end << endl; // move disks – 1 disks from temp to end towers (disks – 1, temp, end, start); }