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.

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).