[ITP - Lecture 14] Recursion
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Recursion is a process where a function calls itself. In C, functions can call themselves recursively. As an example, calculating the factorial of a number recursively is described. Factorial of a number n is defined as n * (n-1) * (n-2) ... * 1. The recursive factorial function calls itself with decreasing arguments until it reaches 1, at which point it returns 1. It then multiplies the returned values together up the call stack to calculate the final factorial value. Visualizing the recursive calls is difficult, but it is drawn out step-by-step in the document as an example for calculating 3!. Though recursion may seem complex, it can often be the most direct way to code certain algorithms
[ITP - Lecture 14] Recursion
- 1. Recursion Intro to Programming MUHAMMAD HAMMAD WASEEM 1 Recursion In C, it is possible for the functions to call themselves. A function is called ‘recursive’ if a statement within the body of a function calls the same function. Sometimes called ‘circular definition’, recursion is thus the process of defining something in terms of itself. Let us now see a simple example of recursion. Suppose we want to calculate the factorial value of an integer. As we know, the factorial of a number is the product of all the integers between 1 and that number. For example, 4 factorial is 4 * 3 * 2 * 1. This can also be expressed as 4! = 4 * 3! where ‘!’ stands for factorial. Thus factorial of a number can be expressed in the form of itself. Hence this can be programmed using recursion. However, before we write a recursive function for calculating factorial let take a look at the non-recursive function for calculating the factorial value of an integer. main( ) { int a, fact ; printf ( "nEnter any number " ) ; scanf ( "%d", &a ) ; fact = factorial ( a ) ; printf ( "Factorial value = %d", fact ) ; } factorial ( int x ) { int f = 1, i ; for ( i = x ; i >= 1 ; i-- ) f = f * i ; return ( f ) ; } And here is the output... Enter any number 3 Factorial value = 6 Following is the recursive version of the function to calculate the factorial value. main( ) { int a, fact ; printf ( "nEnter any number " ) ; scanf ( "%d", &a ) ; fact = rec ( a ) ; printf ( "Factorial value = %d", fact ) ; } rec ( int x ) { int f ; if ( x == 1 ) return ( 1 ) ; else
- 2. Recursion Intro to Programming MUHAMMAD HAMMAD WASEEM 2 f = x * rec ( x - 1 ) ; return ( f ) ; } And here is the output for four runs of the program Enter any number 1 Factorial value = 1 Enter any number 2 Factorial value = 2 Enter any number 5 Factorial value = 120 Let us understand this recursive factorial function thoroughly. In the first run when the number entered through scanf( ) is 1, let us see what action does rec( ) take. The value of a (i.e. 1) is copied into x. Since x turns out to be 1 the condition if ( x == 1 ) is satisfied and hence 1 (which indeed is the value of 1 factorial) is returned through the return statement. When the number entered through scanf( ) is 2, the ( x == 1 ) test fails, so we reach the statement, f = x * rec ( x - 1 ) ; And here is where we meet recursion. How do we handle the expression x * rec ( x - 1 )? We multiply x by rec ( x - 1 ). Since the current value of x is 2, it is same as saying that we must calculate the value (2 * rec ( 1 )). We know that the value returned by rec ( 1 ) is 1, so the expression reduces to (2 * 1), or simply 2. Thus the statement, x * rec ( x - 1 ) ; evaluates to 2, which is stored in the variable f, and is returned to main( ), where it is duly printed as Factorial value = 2 In case the value of a is 5, main( ) would call rec( ) with 5 as its actual argument, and rec( ) will send back the computed value. But before sending the computed value, rec( ) calls rec( ) and waits for a value to be returned. It is possible for the rec( ) that has just been called to call yet another rec( ), the argument x being decreased in value by 1 for each of these recursive calls. We speak of this series of calls to rec( ) as being different invocations of rec( ). These successive invocations of the same function are possible because the C compiler keeps track of which invocation calls which. These recursive invocations end finally when the last invocation gets an argument value of 1, which the preceding invocation of rec( ) now uses to calculate its own f value and so on up the ladder. So we might say what happens is, rec ( 5 ) returns ( 5 times rec ( 4 ), which returns ( 4 times rec ( 3 ), which returns ( 3 times rec ( 2 ), which returns ( 2 times rec ( 1 ), which returns ( 1 ) ) ) ) ) Foxed? Well, that is recursion for you in its simplest garbs. I hope you agree that it’s difficult to visualize how the control flows from one function call to another. Possibly Figure 5.4 would make things a bit clearer. Assume that the number entered through scanf( ) is 3. Using Figure 5.4 let’s visualize what exactly happens when the recursive function rec( ) gets called. Go through the figure carefully. The first time when rec( ) is called from main( ), x collects 3. From here, since x is not equal to 1, the if block is
- 3. Recursion Intro to Programming MUHAMMAD HAMMAD WASEEM 3 skipped and rec( ) is called again with the argument ( x – 1 ), i.e. 2. This is a recursive call. Since x is still not equal to 1, rec( ) is called yet another time, with argument (2 - 1). This time as x is 1, control goes back to previous rec( ) with the value 1, and f is evaluated as 2. Similarly, each rec( ) evaluates its f from the returned value, and finally 6 is returned to main( ). The sequence would be grasped better by following the arrows shown in Figure 5.4. Let it be clear that while executing the program there do not exist so many copies of the function rec( ). These have been shown in the figure just to help you keep track of how the control flows during successive recursive calls. Recursion may seem strange and complicated at first glance, but it is often the most direct way to code an algorithm, and once you are familiar with recursion, the clearest way of doing so.