Detail About Recursion and its Type

Article is originally written by Zeeshan Amjad

Introduction

Here I am going to give a detail about Recursion in C++. Definition: Recursion is the process where a function is called itself but stack frame will be out of limit because function call will be infinite times. So a termination condition is mandatory to a recursion.

Many complex problem can be solved by recursion in a simple code. But it's too much costly than iterative. because in every recursion call one stack frame will formed.You all already know that about it's cost. but if problem is very complex than no way to solve except recursion.

Background

First recursion came into mathematics and then came into Computer science. Idea of it's use that first broke your problem into subproblems and solve it by using recursion.

The code

In C++, Recursion can be divided into two types:
(a)Run- Time Recursion: Normal as in C
(b)Compile- Time Recursion: By using Template

Each of these can be also divided into following types:

1. Linear Recursion
2. Binary Recursion
3. Tail Recursion
4. Mutual Recursion
5. Nested Recursion


1. Linear Recursion: This recursion is the most commonly used. In this recursion a function call itself in a simple manner and by termination condition it terminates. This process called 'Winding' and when it returns to caller that is called 'Un-Winding'. Termination condition also known as Base condition.

Example: Factorial calculation by linear recursion

Run-Time Version

Code:

 int Fact(long n)
{
 	if(0>n)
               return -1;
	if(0 == n)
	   return 1;
	else
{
      return ( n* Fact(n-1));
}
}

Winding Process:

Function called Function return

Fact(6) 6*Fact(5)
Fact(5) 5*Fact(4)
Fact(4) 4*Fact(3)
Fact(3) 3* Fact(2)
Fact(2) 2* Fact(1)
Fact(1) 1* Fact(0)
Terminating Point
Fact(0) 1

Unwinding Process

Fact(1) 1*1
Fact(2) 2*1
Fact(3) 3*2*1
Fact(4) 4*3*2*1
Fact(5) 5*4*3*2*1
Fact(6) 6*5*4*3*2*1


Compile-Time Version

Code:

// template for Base Condition
template <>
struct Fact<0>
{
   enum 
  { 
      factVal = 1 
   };
};

template <int n>
struct Fact
{
   // Recursion call by linear method
   enum
  { 
value = n * Fact<n - 1>::factVal
   };
}; 

To test it how it's working at compile time, just call
cout << Fact<-1>::factVal ;
And compile it then compiler error will come, because no template for -1.

2. Binary Recursion: Binary Recursion is a process where function is called twice at a time inplace of once at a time. Mostly it's using in data structure like operations for tree as traversal, finding height, merging, etc.

Example: Fibonacci number

Run Time Version Code

Code:

int FibNum(int n)
{
   // Base conditions
      if (n < 1)
         return -1;
      if (1 == n || 2 == n)
        return 1;

   // Recursive call by Binary Method
     return FibNum(n - 1) + FibNum(n - 2);   // At a time two recursive function called so               
                                                                      //   binary
}

Compile Time Version Code

Code:

// Base Conditions
template<>
struct FibNum<2>
{
   enum { val = 1 };
};
template <>
struct FibNum<1>
{
   enum { val = 1 };
};

// Recursive call by Binary Method
template <int n>
struct FibNum
{   
   enum { val= FibNum<n - 1>::val + FibNum<n - 2>::val };
};

3. Tail Recursion: In this method, recursive function is called at the last. So it's more efficient than linear recursion method. Means you can say termination point will come(100%) only you have to put that condition.

Example: Fibonacci number

Run Time Version Code

Code:

int FibNum(int n, int x, int y)
{   
   if (1 == n)    // Base Condition
   {
      return y;
   }
   else        // Recursive call by Tail method
  {
      return FibNum(n-1, y, x+y);
   }
}

Compile Time Version Code

Code:

template <int n, int x, int y>
struct FibNum
{
   // Recursive call By tail method
   enum 
  { 
        val = FibNum<n-1, y, (x+y)>::val
   };
};

// Base Condition or Termination 
template<int x, int y>
struct FibNum<1, x, y>
{
   enum 
  {
      val = y 
   };
};

4. Mutual Recursion: Functions calling each other. Let's say FunA calling FunB and FunB calling FunA recursively. This is not actually not recursive but it's doing same as recursive. So you can say Programming languages which are not supporting recursive calls, mutual recursion can be applied there to fulfill the requirement of recursion. Base condition can be applied to any into one or more than one or all functions.

Example: To find Even Or Odd number

Run Time Version Code

Code:

bool IsOddNumber(int n)
{
   // Base or Termination Condition
   if (0 == n)
      return 0;
   else
      // Recursive call by Mutual Method
      return IsEvenNumber(n - 1);
}
bool IsEvenNumber(int n)
{
   // Base or Termination Condition
   if (0 == n)
      return 1;
   else
      // Recursive call by Mutual Method
      return IsOddNumber(n - 1);
}

Compile Time Version Code

Code:

// Base Or Termination Conditions
template <>
struct IsOddNumber<0>
{
   enum 
  {
     val = 0 
  };
};
template <>
struct IsEvenNumber<0>
{
   enum 
  { 
      val = 1 
  };
};

// Recursive calls by Mutual Method

template <int n>
struct IsOddNumber
{
   enum 
  { 
         val = n == 0 ? 0 : IsEvenNumber<n - 1>::val 
   };
};


template <int n>
struct IsEvenNumber
{
   enum 
  { 
       val = n == 0 ? 1 : IsOddNumber<n - 1>::val
  };
}; 

5.Nested Recursion: It's very different than all recursions. All recursion can be converted to iterative (loop) except nested recursion. You can understand this recursion by example of Ackermann function.

Example: Ackermann function

Run Time Version Code

Code:

int Ackermann(int x, int y)
{
      // Base or Termination Condition
    if (0 == x)
   {
      return y + 1;
 }   
// Error Handling condition
   if (x < 0  ||  y < 0)
  {
      return -1;
   }  
// Recursive call by Linear method 
   else if (x > 0 && 0 == y) 
  {
      return Ackermann(x-1, 1);
  }
   // Recursive call by Nested method
   else
  {
      return Ackermann(x-1, Ackermann(x, y-1));
  }
}

Compile Time Version Code

Code:

// Base Or Termination condition
template <int y>
struct Ackermann<0, y>
{
   enum { val = y + 1 };
};

   // Recursive Call by Linear Method
template <int x>
struct Ackermann<x, 0>
{
   enum 
   { 
            val = Ackermann<x-1, 1>::val
   };
};
// Recursive Call by Nested Method
template <int x, int y>
struct Ackermann
{
   Enum
   { 
         val = Ackermann<x-1, Ackermann<x, y-1> ::val>::val 
   };
};

References

"Teaching recursion using recursively generated geometric design"
by Aaron Gordon

 

Really good article, impressive I must say.

 

very good article

 

nice one

 

hi i am hire

 

Thank you.

 

thx.....so good

 

Interesting.

Now take a look at these with the posted date

http://www.codeproject.com/KB/cpp/Recursion_Prmr_CPP_01.aspx
http://www.codeguru.com/cpp/cpp/algorithms/math/article.php/c15111/

and then decide article of the month.

Have fun.

 

If you think it is then anyone can write more than donzens article daily.

This shows the credibility of these sites itself.

This decision is for your site credibility and for future authors who decided to write for you or avoid your site.

After few mins google i came across one more article

http://www.dreamincode.net/forums/showtopic45816.htm

In addition to the above article, I recommend you to please take a look at this one too

http://www.codeproject.com/KB/atl/atl_underthehood_.aspx

Regards

 

I had also seen this same topic in other sites by other author but content is not the same. Copyright rule is that content should not be copied.

What do you think ? There are many articles on the same topic but contents are different...First thing article topic is not a Research Topic Okay.....

If you are talking about this topic "Recursion" there are thousands articles but contents are different...Topic does not matter for articles.

I will take an example UNIX kernal is there...But LINUX group is saying we are totally different than Unix but they also modified some unix code and launch new operating system" Linux"...Same example as solaris is also flavour of Unix.

 

Thanks dableena for your post. Now let me clear few things that you might mix up. First of all take a look at all of my post when did I actually claim about copy right?

Any computer person can easily identify that this article is not exactly copied but the main idea, theme, and even all the examples except “Tail recursion” are same. You can’t say it is copied, but it is heavily based on mentioned articles without even mentioning it.

I just wanted to say one thing,

Give credit where it is due

 

But here we are not goin for PHD. 10000 articles can be on the same topics i have already mentioned.

Yes you are right!!! It may be not to win as winner of this month for this article. This total depend on system how r u implementing. But you cannt mention as " originally written zeesan in asadullah.ansari article." This is his wish that he mention or not .He knows better. You have to take permission of author then you can modify his article.
You know just go read BOOK as mention by him
"Teaching recursion using recursively generated geometric design"
by Aaron Gordon

same explanation by this book also. Can you blame to those writers also who mention this book as in his articles.

 

I am sorry debleena, but I still couldn’t understand what you wanted to say, please explain it to me. What my intention was I already mentioned in my previous mail.I am still confuse what do you want to say in this statement

What total are you talking about this? And what is depends on system? Do you want to say that recursion depends on system?

Can you please explain when did I mention this thing? What should be written or not written with this article, I have already left that decision to the moderator. And you may also understand that I am not a moderator to change others post.

For your kind information this is not a book, this is in fact an article published in “Journal of Computing Science in Colleges” in October 2006. Did you actually get a chance to see this article (or so called book) yourself? If you are a member of ACM then you can access it form ACM Digital Library. And for your kind information I have actually read that article. This 7 page articles has nothing to do with recursion type it only deals with “Recursively-Generated Geometric Designs” and discussed "Nested Triangles, Nested Polygons and Fractals" I can’t post the whole article here, but for your reference let me copy its abstract here

You can read this abstract yourself from here

http://portal.acm.org/citation.cfm?id=1181811.1181830

Even more interesting is that this reference is also copied from this article.

http://www.codeproject.com/KB/cpp/Recursion_Prmr_CPP_01.aspx

This reference was mentions properly in the above article by explaining what reader should expect in it, not adding the reference just for the sake of something in the reference section. Here is the excerpt from the above article that refers to this.

Here number 3 is the reference of “Teaching recursion using recursively generated geometric design, Aaron Gordon”

If you have time then I recommend you to take a look at its second part before someone write an article based on it without even giving the credit where it is due, from here

http://www.codeproject.com/KB/cpp/Recursion_Prmr_CPP_02.aspx

Can you please explain a little bit what you want to ask or say to me?