# Longest Palindrome

Clarify the problem:

- The problem requires finding the length of the longest palindromic substring within a given string.
- We need to implement a function that takes a string as input and returns an integer representing the length of the longest palindromic substring.

Analyze the problem:

- Input: A string.
- Output: An integer representing the length of the longest palindromic substring.
- Constraints:
- The input string may be empty.
- The input string may contain uppercase and lowercase letters, digits, and special characters.
- The problem asks for a substring, which means the characters of the palindrome must be contiguous.

Design an algorithm:

- We can solve this problem using the dynamic programming approach.
- We need to create a 2D array,
`dp`

, of size (n, n), where n is the length of the input string. - The value
`dp[i][j]`

will represent whether the substring from index i to j is a palindrome. - We can fill in the values of
`dp`

by iterating over the string from the last character to the first character. - To determine if a substring from index i to j is a palindrome, we check if the characters at indices i and j are equal and if the substring between i+1 and j-1 is a palindrome.
- We initialize the diagonal elements of
`dp`

as True since single characters are palindromes. - We iterate over the string, starting from the last character, and for each character, iterate over the remaining characters after it.
- If the current characters are equal and the substring between them is a palindrome, we set
`dp[i][j]`

as True. - We keep track of the longest palindromic substring encountered so far and update it whenever we find a longer palindrome.
- Finally, we return the length of the longest palindromic substring.

Explain your approach:

- We will implement a function called
`longestPalindrome`

that takes a string as input. - We initialize
`n`

as the length of the string. - We create a 2D array,
`dp`

, of size (n, n), and initialize all elements as False. - We set the diagonal elements of
`dp`

as True since single characters are palindromes. - We initialize
`max_length`

as 1, representing the length of the longest palindromic substring encountered so far. - We iterate over the string from the last character to the first character using the variable
`i`

. - Inside the outer loop, we iterate over the characters after
`i`

using the variable`j`

. - If the current characters at indices
`i`

and`j`

are equal and the substring between them is a palindrome (dp[i+1][j-1] is True), we set dp[i][j] as True. - If dp[i][j] is True, we update
`max_length`

to be the maximum of`max_length`

and`j - i + 1`

, representing the length of the current palindromic substring. - Finally, we return
`max_length`

as the length of the longest palindromic substring.

- We will implement a function called
Write clean and readable code:

python`def longestPalindrome(s): n = len(s) dp = [[False] * n for _ in range(n)] max_length = 1 # Initialize diagonal elements as True for i in range(n): dp[i][i] = True # Iterate over the string for i in range(n - 1, -1, -1): for j in range(i + 1, n): if s[i] == s[j] and (j - i == 1 or dp[i + 1][j - 1]): dp[i][j] = True max_length = max(max_length, j - i + 1) return max_length`

Test your code:

python`# Test case 1: Example case s = "babad" # The longest palindromic substring is "bab" or "aba". # Both substrings have a length of 3. print(longestPalindrome(s)) # Output: 3 # Test case 2: Empty string s = "" # The input string is empty, so the longest palindromic substring is an empty string. print(longestPalindrome(s)) # Output: 0 # Test case 3: Single character s = "a" # The input string has only one character, which is a palindrome itself. print(longestPalindrome(s)) # Output: 1 # Test case 4: All characters are the same s = "bbb" # The longest palindromic substring is "bbb". # The length of the substring is 3. print(longestPalindrome(s)) # Output: 3 # Test case 5: String with multiple palindromic substrings s = "cbbd" # The longest palindromic substring is "bb". # The length of the substring is 2. print(longestPalindrome(s)) # Output: 2`

Optimize if necessary:

- The solution using dynamic programming has a time complexity of O(n^2), where n is the length of the input string. We need to fill in the dp table of size (n, n) and each cell takes constant time to calculate.
- The space complexity is also O(n^2) since we use a dp table of size (n, n).

Handle error cases:

- We need to consider the case where the input string is empty. In this case, we can simply return 0 as the length of the longest palindromic substring.

Discuss complexity analysis:

- Let n be the length of the input string.
- The time complexity of the solution is O(n^2) since we iterate over the string twice.
- The space complexity is O(n^2) since we use a dp table of size (n, n).