Question

I have given word like abca. I want to know how many letters do I need to add to make it palindrome.
In this case its 1, because if I add b, I get abcba.

Answer 1

First, let’s consider an inefficient recursive solution:

Suppose the string is of the form aSb, where a and b are letters and S is a substring.

If a==b, then f(aSb) = f(S).

If a!=b, then you need to add a letter: either add an a at the end, or add a b in the front. We need to try both and see which is better. So in this case, f(aSb) = 1 + min(f(aS), f(Sb)).

This can be implemented with a recursive function which will take exponential time to run.

To improve performance, note that this function will only be called with substrings of the original string. There are only O(n^2) such substrings. So by memoizing the results of this function, we reduce the time taken to O(n^2), at the cost of O(n^2) space.