Longest palindromic substring information

In computer science, the longest palindromic substring or longest symmetric factor problem is the problem of finding a maximum-length contiguous substring of a given string that is also a palindrome. For example, the longest palindromic substring of "bananas" is "anana". The longest palindromic substring is not guaranteed to be unique; for example, in the string "abracadabra", there is no palindromic substring with length greater than three, but there are two palindromic substrings with length three, namely, "aca" and "ada". In some applications it may be necessary to return all maximal palindromic substrings (that is, all substrings that are themselves palindromes and cannot be extended to larger palindromic substrings) rather than returning only one substring or returning the maximum length of a palindromic substring.

Manacher (1975) invented an ${\displaystyle O(n)}$-time algorithm for listing all the palindromes that appear at the start of a given string of length ${\displaystyle n}$. However, as observed e.g., by Apostolico, Breslauer & Galil (1995), the same algorithm can also be used to find all maximal palindromic substrings anywhere within the input string, again in ${\displaystyle O(n)}$ time. Therefore, it provides an ${\displaystyle O(n)}$-time solution to the longest palindromic substring problem. Alternative ${\displaystyle O(n)}$-time solutions were provided by Jeuring (1994), and by Gusfield (1997), who described a solution based on suffix trees. A faster algorithm can be achieved in the word RAM model of computation if the size ${\displaystyle \sigma }$ of the input alphabet is in ${\displaystyle 2^{o(\log n)}}$. In particular, this algorithm runs in ${\displaystyle O(n\log \sigma /\log n)}$ time using ${\displaystyle O(n\log \sigma /\log n)}$ space.^[1] Efficient parallel algorithms are also known for the problem.^[2]

The longest palindromic substring problem should not be confused with the different problem of finding the longest palindromic subsequence.