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Finding the longest palindromic substring is a classic problem in computer science and can be solved efficiently using dynamic programming. A palindromic substring is a substring that reads the same forwards and backward.
To find the longest palindromic substring, you can use the following approach:
Initialize a 2-dimensional boolean array to keep track of whether substrings are palindromes. Let
dp[i][j]
beTrue
if the substring from indexi
toj
is a palindrome, otherwiseFalse
.For each character
s[i]
,dp[i][i]
is alwaysTrue
since a single character is always a palindrome.For each character
s[i]
ands[i+1]
, ifs[i] == s[i+1]
, thendp[i][i+1]
isTrue
as they form a palindrome.For substrings of length greater than 2 (i.e.,
len > 2
), you can use the recursive formula:cssCopy code
dp[i][j] = dp[i+1][j-1] and s[i] == s[j]
Keep track of the longest palindromic substring while calculating the
dp
array.Here's a Python function that implements the algorithm:
pythonCopy code
def longest_palindromic_substring(s): n = len(s) dp = [[False] * n for _ in range(n)] start, max_length = 0, 1 # All substrings of length 1 are palindromes for i in range(n): dp[i][i] = True # Check for substrings of length 2 for i in range(n - 1): if s[i] == s[i + 1]: dp[i][i + 1] = True start = i max_length = 2 # Check for substrings of length greater than 2 for length in range(3, n + 1): for i in range(n - length + 1): j = i + length - 1 if s[i] == s[j] and dp[i + 1][j - 1]: dp[i][j] = True start = i max_length = length return s[start:start + max_length] # Example usage: s = "babad" result = longest_palindromic_substring(s) print(result)
Output:
sqlCopy code
"bab" or "aba" (both are valid solutions)
The function will return the longest palindromic substring of the input string
s
. In the example provided, both "bab" and "aba" are valid solutions, and the function can return either of them.
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