Manacher's Algorithm

string longestPalinderome(string s) {
    int N = s.size();
    int id = 0, mx = 0;
    vector<int> dp(2 * N + 1);
    for (int i = 0; i < 2 * N + 1; ++i) {
        int j = 2 * id - i;
        if (mx > i) {
            dp[i] = min(mx - i, dp[j]);
        }
        int left = i - dp[i], right = i + dp[i];
        for (; left >= 0 && right <= 2 * N; --left, ++right) {
            if ((left & 1) == 0 || s[left / 2] == s[right / 2]) {
                ++dp[i];
            }
            else 
                break;
        }
        if (dp[i] + i > mx) {
            id = i;
            mx = dp[i] + i;
        }
    }
    int res = 0;
    for (int i = 0; i < 2 * N + 1; ++i) {
        if (dp[i] > dp[res])
            res = i;
    }
    return s.substr(res / 2 - (dp[res] - 1) / 2, dp[res] - 1);
}