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:

C++

// CPP code to find the no.of positions where a
// letter can be inserted to make it a palindrome
#include <bits/stdc++.h>
using namespace std;

// Function to check if the string is palindrome
bool isPalindrome(string &s, int i, int j)
{
    int p = j;
    for (int k = i; k <= p; k++) {
        if (s[k] != s[p])
            return false;
        p--;
    }
    return true;
}

int countWays(string &s)
{
    // to know the length of string
    int n = s.length();
    int count = 0;

    // if the given string is a palindrome(Case-I)
    if (isPalindrome(s, 0, n - 1))
    {
        // Sub-case-III)
        for (int i = n / 2; i < n; i++)
        {
            if (s[i] == s[i + 1])
                count++;
            else
                break;
        }
        if (n % 2 == 0) // if the length is even
        {
            count++;
            count = 2 * count + 1; // sub-case-I
        } else
            count = 2 * count + 2; // sub-case-II
    } else {
        for (int i = 0; i < n / 2; i++) {

            // insertion point
            if (s[i] != s[n - 1 - i])
            {
                int j = n - 1 - i;

                // Case-I
                if (isPalindrome(s, i, n - 2 - i))
                {
                    for (int k = i - 1; k >= 0; k--) {
                        if (s[k] != s[j])
                            break;
                        count++;
                    }
                    count++;
                }

                // Case-II
                if (isPalindrome(s, i + 1, n - 1 - i))
                {
                    for (int k = n - i; k < n; k++) {
                        if (s[k] != s[i])
                            break;
                        count++;
                    }
                    count++;
                }
                break;
            }
        }
    }

    return count;
}

// Driver code
int main()
{
    string s = "abca";
    cout << countWays(s) << endl;
    return 0;
}

Java

// Java code to find the no.of positions where a
// letter can be inserted to make it a palindrome

import java.io.*;

class GFG {

    // Function to check if the string is palindrome
    static boolean isPalindrome(String s, int i, int j)
    {
        int p = j;
        for (int k = i; k <= p; k++) {
            if (s.charAt(k) != s.charAt(p))
                return false;
            p--;
        }

        return true;
    }

    static int countWays(String s)
    {

        // to know the length of string
        int n = s.length();
        int count = 0;

        // if the given string is a palindrome(Case-I)
        if (isPalindrome(s, 0, n - 1)) {

            // Sub-case-III)
            for (int i = n / 2; i < n; i++) {
                if (s.charAt(i) == s.charAt(i + 1))
                    count++;
                else
                    break;
            }

            if (n % 2 == 0) // if the length is even
            {
                count++;
                count = 2 * count + 1; // sub-case-I
            }
            else
                count = 2 * count + 2; // sub-case-II
        }
        else {
            for (int i = 0; i < n / 2; i++) {

                // insertion point
                if (s.charAt(i) != s.charAt(n - 1 - i)) {
                    int j = n - 1 - i;

                    // Case-I
                    if (isPalindrome(s, i, n - 2 - i)) {
                        for (int k = i - 1; k >= 0; k--) {
                            if (s.charAt(k) != s.charAt(j))
                                break;
                            count++;
                        }
                        count++;
                    }

                    // Case-II
                    if (isPalindrome(s, i + 1, n - 1 - i)) {
                        for (int k = n - i; k < n; k++) {
                            if (s.charAt(k) != s.charAt(i))
                                break;
                            count++;
                        }
                        count++;
                    }
                    break;
                }
            }
        }

        return count;
    }

    // Driver code
    public static void main(String[] args)
    {
        String s = "abca";
        System.out.println(countWays(s));
    }
}

// This code is contributed by vt_m.

Python 3

# Python 3 code to find the no.of positions
# where a letter can be inserted to make it
# a palindrome

# Function to check if the string
# is palindrome
def isPalindrome(s, i, j):

    p = j
    for k in range(i, p + 1):
        if (s[k] != s[p]):
            return False
        p -= 1

    return True

def countWays(s):

    # to know the length of string
    n = len(s)
    count = 0

    # if the given string is a palindrome(Case-I)
    if (isPalindrome(s, 0, n - 1)) :

        # Sub-case-III)
        for i in range(n // 2, n):

            if (s[i] == s[i + 1]):
                count += 1
            else:
                break

        if (n % 2 == 0): # if the length is even
            count += 1
            count = 2 * count + 1 # sub-case-I
        else:
            count = 2 * count + 2 # sub-case-II
    else :
        for i in range(n // 2) :

            # insertion point
            if (s[i] != s[n - 1 - i]) :
                j = n - 1 - i

                # Case-I
                if (isPalindrome(s, i, n - 2 - i)) :
                    for k in range(i - 1, -1, -1):
                        if (s[k] != s[j]):
                            break
                        count += 1

                    count += 1

                # Case-II
                if (isPalindrome(s, i + 1, n - 1 - i)) :
                    for k in range(n - i, n) :
                        if (s[k] != s[i]):
                            break
                        count += 1

                    count += 1

                break

    return count

# Driver code
if __name__ == "__main__":

    s = "abca"
    print(countWays(s))

# This code is contributed by ita_c

Output

2