Version I - simple DP

dp[i][j] is the longest palindromic subsequences of substring i .. j

public class Solution {
    public int longestPalindromeSubseq(String s) {
        if (s.length() <= 1) return s.length();

        int len = s.length();
        int[][] dp = new int[len][len];

        for (int i = 0; i < len; ++i) {
            dp[i][i] = 1;
        }

        for (int i = len - 1; i >= 0; --i) {
            for (int j = i + 1; j < len; ++j) {
                if (s.charAt(i) == s.charAt(j)) {
                    dp[i][j] = dp[i + 1][j - 1] + 2;
                } else {
                    dp[i][j] = Math.max(dp[i + 1][j], dp[i][j - 1]);
                }
            }
        }

        return dp[0][len - 1];
    }
}