To make a given string palindromic, you need to remove the minimum possible number of characters. The idea is to find the longest palindromic subsequence in the string and subtract its length from the length of the original string.
Here's an example algorithm to find the minimum size substring to be removed:
Initialize a table of size N x N, where N is the length of the string.
Let's assume the input string is called "s".
Iterate over the string "s" from the last character to the first character.
For each character "s[i]", iterate over the string "s" from the current character "i" to the last character.
If "s[i]" is equal to "s[j]" (where "j" is the current iteration index), set the table entry for the indices "i" and "j" to be the value of the table entry for the indices "i+1" and "j-1" plus 2.
Otherwise, set the table entry for the indices "i" and "j" to be the maximum of the table entry for the indices "i+1" and "j" or the table entry for the indices "i" and "j-1".
After completing the above steps, the value in the table entry for indices 0 and N-1 will give you the length of the longest palindromic subsequence in the string "s".
Subtract this length from the length of the original string to get the minimum size substring to be removed.
Here's a Python implementation of the above algorithm:
def min_substring_to_remove(s):
n = len(s)
table = [[0] * n for _ in range(n)]
for i in range(n-1, -1, -1):
table[i][i] = 1
for j in range(i+1, n):
if s[i] == s[j]:
table[i][j] = table[i+1][j-1] + 2
else:
table[i][j] = max(table[i+1][j], table[i][j-1])
longest_palindrome = table[0][n-1]
min_substring = n - longest_palindrome
return min_substring
# Example usage
string = "abca"
result = min_substring_to_remove(string)
print(result) # Output: 1
In the example usage, the input string is "abca". The longest palindromic subsequence is "aba" with a length of 3. Therefore, you need to remove one character (either 'b' or 'c') to make the string palindromic.
Certainly! Let's dive deeper into the algorithm and walk through an example to illustrate its workings.
Consider the input string "abca" again:
We initialize a table of size 4x4 (length of the string) with zeros:
[[0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]]
We start iterating over the string from the last character to the first character:
For i = 3, we have s[i] = 'a':
We initialize table[3][3] as 1 (a single character is a palindrome of length 1).
Then, we move to the next iteration.
For i = 2, we have s[i] = 'c':
We initialize table[2][2] as 1.
Since s[i] != s[j] (where j is the current iteration index), we set table[2][3] as the maximum of table[3][3] and table[2][2], which is 1.
Then, we move to the next iteration.
For i = 1, we have s[i] = 'b':
We initialize table[1][1] as 1.
Since s[i] != s[j], we set table[1][2] as the maximum of table[2][2] and table[1][1], which is 1.
Since s[i] != s[j], we set table[1][3] as the maximum of table[2][3] and table[1][2], which is 1.
Then, we move to the next iteration.
For i = 0, we have s[i] = 'a':
We initialize table[0][0] as 1.
Since s[i] == s[j], we set table[0][3] as table[1][2] + 2, which is 3 (length of "aba").
Then, we set table[0][2] as the maximum of table[1][2] and table[0][3], which is 3.
Since s[i] == s[j], we set table[0][1] as table[1][1] + 2, which is 3 (length of "aa").
Then, we set table[0][3] as the maximum of table[0][2] and table[1][3], which is 3.
Finally, we move to the next iteration.
The table after completing the iterations:
[[1, 1, 1, 3], [0, 1, 1, 1], [0, 0, 1, 1], [0, 0, 0, 1]]
The value in table[0][3] is 3, which represents the length of the longest palindromic subsequence ("aba").
To make the string palindromic, we need to remove the minimum possible