Number of positions where a letter can be inserted such that a string becomes palindrome

Given a string str, we need to find the no. of positions where a letter(lowercase) can be inserted so that string becomes a palindrome. 

Examples: 

Input : str = "abca"
Output : possible palindromic strings: 
 1) acbca (at position 2)
 2) abcba (at position 4)
 Hence, the output is 2.

Input : str = "aaa"
Output : possible palindromic strings:
 1) aaaa
 2) aaaa
 3) aaaa
 4) aaaa
 Hence, the output is 4. 

Naive Approach: This approach is to insert all 26 alphabets at every position possible i.e., N+1 positions and check at every position if this insertion makes it a palindrome and increase the count.

Efficient Approach: First you have to observe that we have to make insertion only at the point when the character at that point violates the palindrome condition i.e., . Now, there will be Two cases based on the above fact: 

Case I: What if the given string is already a palindrome 

Then we can only insert at the position such that the insertion does not violate the palindrome. 

1. If the length is even then we can always insert any letter in the middle. 
2. If the length is odd then we can insert the letter which is in middle, to the left or right to it. 
3. In both the cases we can insert the letter which is in middle(let it be 'CH'), at positions equals to: 
   (no.of consecutive CH's to the left of middle letter)*2. 

Case II: If it is not a palindrome 

As mentioned above we should start inserting at position where , So we increase the count and check for the cases if insertion at any other position makes it a palindrome. 

1. If is a palindrome, then we can insert* at any position before until , K in range .(*letter = S[N-i-1]) 
2. If is a palindrome, then we can insert* at any position after until , K in range .(*letter = S[i]) 

In all the cases we keep increasing the count.

Implementation:

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