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  Theory of Recursion and their types

Recursion:

Recursion is very cool. Most algorithms require being implemented recursively. A popular combinatorial brute force algorithm, backtracking, generally implemented recursively.

Recursion is a function or procedure or method which calls itself again with a smaller range of arguments (that is, break a problem into simpler problems) or with distinct arguments (it is ineffective if you use recursion with similar arguments). It keeps calling itself till something that is so simple/simpler than before (that we termed base case), that can be solved simply or till an exit condition takes place.

A recursion should stop (really all programs should terminate). In order to do so, a valid base case or end-condition should be reached. Inappropriate coding will leads to stack overflow error.

You can find out whether a function is recursive or not by looking at the source code. If in a procedure or function, it calls itself again, it is a recursive type.

When a method or function or procedure is called:

•    caller is suspended,
•    "state" of caller saved,
•    new space allocated for variables of new method.

Recursion is an influential and elegant method which can be employed to resolve a problem by resolving a smaller problem of similar type. Most of the problems in Computer Science include recursion and some of them are in nature recursive.

If one problem can be resolved in both way (that is, recursive or iterative), then selecting iterative version is a good idea as it is faster and does not consume a lot of memory. Illustration: Factorial, Fibonacci and so on.

Though, there are as well problems which can only be resolved in recursive way or more efficient in recursive type or when it is iterative, solutions are hard to conceptualize. Illustrations: Tower of Hanoi, Searching (that is, DFS or BFS) and so on.

Types of recursion:

There are two types of recursion: Linear recursion and multiple branch (or Tree) recursion.

A) Linear recursion is the recursion whose order of growth is linear (that is, not branched). Illustration of Linear recursion is Factorial, defined by fac (n)=n * fac (n-1).

B) The tree recursion will branch to more than one node each step, growing up very rapidly. It gives us a flexible ways to resolve some logically difficult trouble. It can be employed to perform a Complete Search (that is, to make the computer to do trial and error). This recursion type has a quadratic or cubic or more order of growth and thus, it is not appropriate for resolving "big" problems. It is limited to small. Illustrations: resolving Tower of Hanoi, Searching (DFS and BFS), Fibonacci number and so forth.

Few compilers can make some kind of recursion to be iterative. One illustration is tail-recursion elimination. Tail-recursive is a kind of recursive that the recursive call is last command in that function or procedure.

Tips on Recursion:

A) Recursion is very alike to the mathematical induction.

B) You initially see how you can resolve the base case, for n=0 and for n=1.

C) Then you suppose that you know how to resolve the problem of size n-1, and you look for a way of receiving the solution for problem size n from the solution of size n-1.

Whenever constructing a recursive solution, maintain the following questions in mind:

A) How can you state the problem in term of smaller problem of similar type?

B) How does each and every recursive call reduce the size of problem?

C) What instance of the trouble can serve as the base case?

D) As the problem size reduces, will you reach this base case?

The sample of a very standard recursion, Factorial (in Java):

static int factorial(int n) {
   if (n==0)
      return 1;
   else
      return n*factorial(n-1);
}

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