Given a string, find out if the string is K-Palindrome or not. A k-palindrome string transforms into a palindrome on removing at most k characters from it.
Input : String - abcdecba, k = 1 Output : Yes String can become palindrome by remo- -ving 1 character i.e. either d or e) Input : String - abcdeca, K = 2 Output : Yes Can become palindrome by removing 2 characters b and e. Input : String - acdcb, K = 1 Output : No String can not become palindrome by removing only one character.
If we carefully analyze the problem, the task is to transform the given string into its reverse by removing at most K characters from it. The problem is basically a variation of Edit Distance. We can modify the Edit Distance problem to consider given string and its reverse as input and only operation allowed is deletion. Since given string is compared with its reverse, we will do at most N deletions from first string and N deletions from second string to make them equal. Therefore, for a string to be k-palindrome, 2*N <= 2*K should hold true. Below are the detailed steps of algorithm - Process all characters one by one staring from either from left or right sides of both strings. Let us traverse from the right corner, there are two possibilities for every pair of character being traversed.
- If last characters of two strings are same, we ignore last characters and get count for remaining strings. So we recur for lengths m-1 and n-1 where m is length of str1 and n is length of str2.
- If last characters are not same, we consider remove operation on last character of first string and last character of second string, recursively compute minimum cost for the operations and take minimum of two values.
- Remove last char from str1: Recur for m-1 and n.
- Remove last char from str2: Recur for m and n-1.
Below is Naive recursive implementation of above approach.
The time complexity of above solution is exponential. In worst case, we may end up doing O(2n) operations. The worst case happens string contains all distinct characters.
This problem has both properties (see this and this) of a dynamic programming problem. Like other typical Dynamic Programming(DP) problems, re-computations of same subproblems can be avoided by constructing a temporary array that stores results of subproblems .
Below is Bottom-up implementation of above recursive approach :
Time complexity of above solution is O(m x n). We can improve time complexity by making use of the fact that only k deletions are allowed. Auxiliary space used is O(m x n).
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