# 15-2 Longest palindrome subsequence

A palindrome is a nonempty string over some alphabet that reads the same forward and backward. Examples of palindromes are all strings of length $1$, $\text{civic}$, $\text{racecar}$, and $\text{aibohphobia}$ (fear of palindromes).

Give an efficient algorithm to find the longest palindrome that is a subsequence of a given input string. For example, given the input $\text{character}$, your algorithm should return $\text{carac}$. What is the running time of your algorithm?

Let $A[1..n]$ denote the array which contains the given word. First note that for a palindrome to be a subsequence we must be able to divide the input word at some position $i$, and then solve the longest common subsequence problem on $A[1..i]$ and $A[i + 1..n]$, possibly adding in an extra letter to account for palindromes with a central letter. Since there are $n$ places at which we could split the input word and the $\text{LCS}$ problem takes time $O(n^2)$, we can solve the palindrome problem in time $O(n^3)$.