Two-week ISTE workshop on Effective teaching/learning of computer programming

Dr Deepak B Phatak Subrao Nilekani Chair Professor Department of CSE, Kanwal Rekhi Building IIT Bombay Lecture 6, Recursion Thursday 1 July 2010 Two-week ISTE workshop onEffective teaching/learning of computer programming

Overview • Recursion • Implementing a solution through recursive function calls • Normal execution of a function • Recursive execution • Computational cost of recursion • Need, definition and usage Lecture 6 Recursion

Iteration Vs Recursion • We have seen some problems which implement the algorithm using iteration • Newton-Raphson method • Finding maximum of a set of given values • Calculating factorial of a given number • Determining Nth term of Hemachandra numbers (Fibonacci numbers) • There are problems where the solution can also be defined as a recursive formula, e.g. f(n) = n! = 1 * 2 * 3 * … *(n-1) * n which can also be expressed as f(n) = n * f(n-1) Lecture 6 Recursion

Recursive definitions • Recursion is the process of defining something in terms of itself. • It is an elegant mathematical concept • permits very concise definition of operations • nth term f(n) of Hemachandra (Fibonacci) series is given by f(n) = f(n-1) + f(n-2) [nth term is defined in terms the (n-1)th and (n-2)th terms] Lecture 6 Recursion

Recursive definitions … • For such definitions to be meaningful and valid, some ‘initial conditions must be defined, otherwise the recursive process simply cannot begin. • Thus, n! is defined as f(0) = 1 f(n) = n * f(n-1), for n > 0 • Similarly, terms in Hemachandra (Fibonacci) series are defined as f(0) = 1 f(1) = 1 f(n) = f(n-1) + f(n-2), for n > 1 Lecture 6 Recursion

Execution of the iterative solution • To understand what happens internally when recursion is used, we should first look at the iterative solution int factorial (int n){ int f = 1; if (n == 0) return f; for (i =1; i <= n, i++){ f = f * i; }; return f; } Lecture 6 Recursion

function invocation • Assume that the above function is invoked in the main program in following manner … int n =3, answer; answer = factorial (n); cout << answer; --- • What happens when the function call is made ? Lecture 6 Recursion

function invocation • The control is handed over to factorial function with value of the parameter n = 3 • The computer allocates memory to the variables used in the function, i.e., to f, i , n • This n is different from the n in main program, hence it gets a separate memory location Lecture 6 Recursion

Function invocation Remember the instruction being executed, save context Collect all parameter values Hand over control to function Restore old context and resume execution Set up a logical block for factorial execution Allocate memory to variables n, i, f Calculate answer Release allocated memory and go back main function Lecture 6 Recursion

Function invocation • There are two type of ‘overheads’ which accompany a function call • Additional memory has to be allocated every time a function is invoked • For the data elements defined and used in the function • For temporarily storing the ‘context’ of the current program • Additional computational overhead • For transfer of control, parameter transfer, etc. Lecture 6 Recursion

Factorial function using recursion #include <iostream> using namespace std; // prog6-1.c // calculate factorial of a given integer n, int findfactorial(int n){ int f; if (n ==0) { f = 1;} else { f = findfactorial (n-1) * n; } return f; } Lecture 6 Recursion

Factorial … recursion int main(){ int n, fact; cout << “give an integer value”; n = 3; fact = findfactorial(n); cout << “factorial is “ << fact return 0; } Lecture 6 Recursion

Two-week ISTE workshop on Effective teaching/learning of computer programming