Recursive Algorithm

A recursive algorithm is an algorithm which calls itself with "smaller (or simpler)" input values, and which obtains the result for the current input by applying simple operations to the returned value for the smaller (or simpler) input. More generally if a problem can be solved utilizing solutions to smaller versions of the same problem, and the smaller versions reduce to easily solvable cases, then one can use a recursive algorithm to solve that problem. For example, the elements of a recursively defined set, or the value of a recursively defined function can be obtained by a recursive algorithm. 

If a set or a function is defined recursively, then a recursive algorithm to compute its members or values mirrors the definition. Initial steps of the recursive algorithm correspond to the basis clause of the recursive definition and they identify the basis elements. They are then followed by steps corresponding to the inductive clause, which reduce the computation for an element of one generation to that of elements of the immediately preceding generation. 

In general, recursive computer programs require more memory and computation compared with iterative algorithms, but they are simpler and for many cases a natural way of thinking about the problem. 

Example 1: Algorithm for finding the k-th even natural number 
Note here that this can be solved very easily by simply outputting 2*(k - 1) for a given k . The purpose here, however, is to illustrate the basic idea of recursion rather than solving the problem. 

Algorithm 1:   Even(positive integer k) 
Input: k , a positive integer 
Output: k-th even natural number (the first even being 0) 

Algorithm: 
if k = 1, then return 0; 
else return Even(k-1) + 2 . 

Here the computation of Even(k) is reduced to that of Even for a smaller input value, that is Even(k-1). Even(k) eventually becomes Even(1) which is 0 by the first line. For example, to compute Even(3),Algorithm Even(k) is called with k = 2. In the computation of Even(2), Algorithm Even(k) is called with k = 1. Since Even(1) = 0, 0 is returned for the computation of Even(2), and Even(2) = Even(1) + 2= 2 is obtained. This value 2 for Even(2) is now returned to the computation of Even(3), and Even(3) = Even(2) + 2 = 4 is obtained. 
As can be seen by comparing this algorithm with the recursive definition of the set of nonnegative even numbers, the first line of the algorithm corresponds to the basis clause of the definition, and the second line corresponds to the inductive clause. 

By way of comparison, let us see how the same problem can be solved by an iterative algorithm. 

Algorithm 1-a:   Even(positive integer k) 
Input: k, a positive integer 
Output: k-th even natural number (the first even being 0) 

Algorithm: 
int   i, even; 
i := 1;
even := 0;
while( i < k ) { 
          even := even + 2; 
          i := i + 1; 
} 
return even . 


Example 2: Algorithm for computing the k-th power of 2 

Algorithm 2   Power_of_2(natural number k) 
Input: k , a natural number 
Output: k-th power of 2 

Algorithm: 
if k = 0, then return 1; 
else return 2*Power_of_2(k - 1) . 

By way of comparison, let us see how the same problem can be solved by an iterative algorithm. 

Algorithm 2-a   Power_of_2(natural number k) 
Input: k , a natural number 
Output: k-th power of 2 

Algorithm: 
int   i, power; 
i := 0;
power := 1;
while( i < k ) { 
          power := power * 2; 
          i := i + 1; 
} 
return power . 


The next example does not have any corresponding recursive definition. It shows a recursive way of solving a problem. 

Example 3: Recursive Algorithm for Sequential Search 

Algorithm 3   SeqSearch(L, i, j, x) 

Input: L is an array, i and j are positive integers, i  j, and x is the key to be searched for in L. 
Output: If x is in L between indexes i and j, then output its index, else output 0. 

Algorithm: 
if i  j , then 
{ 
   if L(i) = x, then return i ; 
   else return SeqSearch(L, i+1, j, x) 
} 
else return 0. 


Recursive algorithms can also be used to test objects for membership in a set. 

Example 4: Algorithm for testing whether or not a number x is a natural number 

Algorithm 4   Natural(a number x) 
Input: A number x 
Output: "Yes" if x is a natural number, else "No" 

Algorithm: 
if x < 0,   then return "No" 
else 
    if x = 0,   then return "Yes" 
    else return Natural( x - 1 ) 

Example 5: Algorithm for testing whether or not an expression w is a proposition(propositional form) 

Algorithm 5   Proposition( a string w ) 

Input: A string w 
Output: "Yes" if w is a proposition, else "No" 

Algorithm: 
if w is 1(true), 0(false), or a propositional variable, then return "Yes" 
else if w = ~w₁, then return Proposition(w₁) 
   else 
     if ( w = w₁  w₂ or w₁  w₂ or w₁  w₂ or w₁  w₂ ) and 
       Proposition(w₁) = Yes and   Proposition(w₂) = Yes 
     then return Yes 
     else return No 
end