RECURSION

L27-L28

To learn and the following concepts

To design a recursive algorithmTo solve problems using recursion To understand the relationship and

difference between recursion and iteration

Objectives

• A recursive function is a function that invokes/calls itself directly or indirectly.

• Useful programming technique • Enables you to develop a natural, straightforward, simple solution

to a problem that would otherwise be difficult to solve.

Useful for many tasks, like sorting and mathematical functions like factorial, fibonacci, exponentiation, GCD, Tower of Hanoi, etc.

Recursion - Introduction

Recursive Thinking Recursion Process

A child couldn't sleep, so her mother told a story about a little frog,

who couldn't sleep, so the frog's mother told a story about a little bear, who couldn't sleep, so bear's mother told a story about a

little weasel ...who fell asleep. ...and the little bear fell asleep; ...and the little frog fell asleep; ...and the child fell asleep.

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Steps to Design a Recursive Algorithm Base case:

for a small value of n, it can be solved directly

Recursive case(s): Smaller versions of the same problem

Algorithmic steps: Identify the base case and provide a solution to it Reduce the problem to smaller versions of itself Move towards the base case using smaller versions

Let us consider the Iterative code …void main() { int i, n, sum=0; cout<<"Enter the limit"; cin>>n;cout<<"\nThe sum is"<<fnSum(n);}

int fnSum(int n){ int sum=0; for(i=1;i<=n;i++) //Iteration

sum=sum+i; return (sum);}

Let us consider the Recursive code …void main() { int i, n, sum=0; cout<<"Enter the limit"; cin>>n;cout<<"\nThe sum is"<<fnSum(n);}

int fnSum(int x) { if (x == 1) //base case return 1; else return fnSum(x-1) + x; //recursive case }

So factorial(5)= 5* factorial(4)

= 4* factorial(3) = 3*factorial(2) = 2* factorial(1) =

1*factorial(0) = 1

Factorial of a natural number– a classical recursive

example

Factorial- recursive procedure #include <iostream.h>

long factorial (long n) {

if (n ==0) //base case return (1);

return (n * factorial (n-1));

}

void main () {

long number; cout << "Please type a number: "; cin >> number; cout << number << "! = " << factorial (number);}

Recursion - How is it doing!

factorial(0) = 1 factorial(n) = n * factorial(n-1) [for n>0]

long rFact (long a) { if (a ==0) return (1);

return (a * rFact (a-1)); }

rFact(5)

rFact(4) x

Notice that the recursion isn’t finished at the bottom --

It must unwind all the way back to the top in

order to be done.

5

rFact(3) x4

rFact(2) x3

rFact(1) x2

rFact(0) x1 =1

=1

=2

=6

=24

120

= 1

= 2

= 6

= 24

= 120

int rfibo(int n) {

if (n <= 1) return n; else return (rfibo(n-1) + rfibo(n-2)); }

Fibonacci Numbers: Recursion

Output: n = 4

fib = 3

Fibonacci series is 0,1, 1, 2, 3, 5, 8 …

Fibonacci Numbers: Recursion

fibonacci(0) = 0 fibonacci(1) = 1 fibonacci(n) = fibonacci(n-1) + fibonacci(n-2) [for n>1]

So fibonacci(4) = fibonacci(3) + fibonacci(2)

(fibonacci(2) + fibonacci(1)) + (fibonacci(1) + fibonacci(0))

((fibonacci(1) + fibonacci(0)) + 1) + (1 + 0) ( 1 + 0 ) + 1) + (1 + 0) = 3

Recursive Calls initiated by Fib(4)

Fibonacci Series using Recursionint rfibo(int);

void main(void){ int n,i, a[20], fibo;

cout<<"enter any num to n\n"; cin>>n;

cout<<“Fibonacci series “; for (i=1; i<=n; i++) { fibo = rfib(i); cout<<"\n"<<fibo; }}

int rfibo(int n) { if (n <= 1) return n; else return (rfibo(n-1) + rfibo(n-2)); }

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Static Variable • The value of static variable persists until the end of the program.

• Static variables can be declared as static int x;

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void fnStat( );void main() { int i; for( i= 1; i<=3; i++) fnStat( );}

void fnStat( ){static int x = 0;

x = x + 1; cout<<\nx= “<<x;}

Output:x = 1x = 2x = 3

Output:x = 1x = 1x = 1

static int x = 0;

Static Variable

int rev(int num){ static int n = 0; if(num > 0) n = (n* 10) + (num%10) ; else return n; return rev(num/10);}

Extra problem: Reversing a Number: Recursion

Output: num = 234

rev = 432

int gcd(int x, int y) {

if (x == 0) return (y); if (y==0) return (x); return gcd(y, x % y); }

GCD: Recursion

Output:x= 24 , y = 9gcd = 3

gcd(24,9) Control In gcd fn on call

gcd(9,24%9) gcd(9, 6) gcd(6,9%6) gcd(6, 3) gcd(3,6%3) gcd(3, 0) return values return 3 return 3 return 3 return 3

Recursion - Should I or Shouldn’t I?

• Pros– Recursion is a

natural fit for some types of problems

• Cons– Recursive programs

typically use a large amount of computer memory and the greater the recursion, the more memory used

– Recursive programs can be confusing to develop and extremely complicated to debug

RECURSION Iteration

Uses more storage space requirement

Less storage space requirement

Overhead during runtime Less Overhead during runtime

Runs slower Runs faster

a better choice, a more elegant solution for recursive problems

Less elegant solution for recursive problems

Recursion Vs Iteration

Recursion – Do’s

• You must include a termination condition or Base Condition in recursive function; Otherwise your recursive function will run “forever” or infinite.

• Each successive call to the recursive function must be nearer to the base condition