Minimum number of characters to insert to make a palindrome

Given a string, how do you find the minimum number of characters to insert to make it a palindrome?
For example given a string “abbc”, the number of characters to insert is 2, and the resulting palindrome is “cabbac”
Take another example “abc”, we need to insert two characters at a minimum to make it a palindrome “abcba”
If the given string is already a palindrome, the result is 0.

Let us analyze the problem to find out a solution. Given a string, we have two possible cases
  • Case 1:The first and last characters are equal. In this case, the minimum number of characters to insert depends on the sub-string between the first and last characters. 
    • For example consider the string “abca”, the value depends on the sub-string “bc”
  • case 2:The first and last characters are not equal. In this case, we have to make the first and last characters equal to make it a palindrome. We can do this in two ways. Either we can insert a new character before the first character, or next to the last character. 
    • For example consider “gcc”, to make first and last characters equal, we can insert a “c” first or insert a “g” last. It either becomes “cgcc” or “gccg”. since “gccg” is minimal way of making a palindrome, We just need to insert 1 character.
Let us denote the minimum number of characters to insert as MCI. It can be defined as follows. str is a string of length n.

MCI(str) = MCI( str[1:n-2] ) if str[0] = str[n-1]
         = Min( MCI(str[0:n-2]), MCI(str[1:n-1]))+1 otherwise

A naive recursive implementation is straight forward. To implement dynamic programming solution, we create a a two dimensional table of size n*n.

table[i][j] indicates MCI of the sub-string str[i:j] the table is calculated using the following rules.

table[i][i] is always 0 because a single character is a palindrome.

table[i][j] = table[i+1][j-1] if str[i] = str[j]
            = Min(table[i+1][j], table[i,j-1])+1 otherwise

We construct this table in bottom up manner i.e solve the smaller problems first and use their result to solve bigger problems. Finally table[0][n-1] contains the required value.

Below is the dynamic programming implementation in C++.