Problem:
Score : 100 points

Problem Statement

Taro and Jiro will play the following game against each other.

Initially, they are given a sequence a = (a_1, a_2, \ldots, a_N).
Until a becomes empty, the two players perform the following operation alternately, starting from Taro:

Remove the element at the beginning or the end of a. The player earns x points, where x is the removed element.

Let X and Y be Taro's and Jiro's total score at the end of the game, respectively.
Taro tries to maximize X - Y, while Jiro tries to minimize X - Y.

Assuming that the two players play optimally, find the resulting value of X - Y.

Constraints

All values in input are integers.

1 \leq N \leq 3000

1 \leq a_i \leq 10^9

Input

Input is given from Standard Input in the following format:

N
a_1 a_2 \ldots a_N

Output

Print the resulting value of X - Y, assuming that the two players play optimally.

Sample Input 1

4
10 80 90 30

Sample Output 1

10

The game proceeds as follows when the two players play optimally (the element being removed is written bold):

Taro: (10, 80, 90, 30) → (10, 80, 90)

Jiro: (10, 80, 90) → (10, 80)

Taro: (10, 80) → (10)

Jiro: (10) → ()

Here, X = 30 + 80 = 110 and Y = 90 + 10 = 100.

Sample Input 2

3
10 100 10

Sample Output 2

-80

The game proceeds, for example, as follows when the two players play optimally:

Taro: (10, 100, 10) → (100, 10)

Jiro: (100, 10) → (10)

Taro: (10) → ()

Here, X = 10 + 10 = 20 and Y = 100.

Sample Input 3

1
10

Sample Output 3

10

Sample Input 4

10
1000000000 1 1000000000 1 1000000000 1 1000000000 1 1000000000 1

Sample Output 4

4999999995

The answer may not fit into a 32-bit integer type.

Sample Input 5

6
4 2 9 7 1 5

Sample Output 5

2

The game proceeds, for example, as follows when the two players play optimally:

Taro: (4, 2, 9, 7, 1, 5) → (4, 2, 9, 7, 1)

Jiro: (4, 2, 9, 7, 1) → (2, 9, 7, 1)

Taro: (2, 9, 7, 1) → (2, 9, 7)

Jiro: (2, 9, 7) → (2, 9)

Taro: (2, 9) → (2)

Jiro: (2) → ()

Here, X = 5 + 1 + 9 = 15 and Y = 4 + 7 + 2 = 13.

Solution in Java8 (OpenJDK 1.8.0) (judge verdict: Time Limit Exceeded)

import java.io.*;
import java.util.StringTokenizer;

public class Main {

    public static void main(String[] args) throws IOException {
        init();

        int n = nextInt();
        int[] values = new int[n];
        for (int i = 0; i < values.length; i++) values[i] = nextInt();

        long[][][] optimalScores = new long[n][n][2];

        for (int l = 1; l <= n; l++) {
            for (int i = n - 1; i >= 0; i--) {
                int j = i + l - 1;
                if (j >= n) continue;
                if (l % 2 == n % 2) {
                    long takeI = values[i] + (i + 1 < n ? optimalScores[i + 1][j][0] : 0);
                    long takeJ = values[j] + (j - 1 >= 0 ? optimalScores[i][j - 1][0] : 0);
                    if (takeI > takeJ) {
                        optimalScores[i][j][0] = takeI;
                        optimalScores[i][j][1] = (i + 1 < n ? optimalScores[i + 1][j][1] : 0);
                    } else {
                        optimalScores[i][j][0] = takeJ;
                        optimalScores[i][j][1] = (j - 1 >= 0 ? optimalScores[i][j - 1][1] : 0);
                    }
                } else {
                    long takeI = values[i] + (i + 1 < n ? optimalScores[i + 1][j][1] : 0);
                    long takeJ = values[j] + (j - 1 >= 0 ? optimalScores[i][j - 1][1] : 0);
                    if (takeI > takeJ) {
                        optimalScores[i][j][1] = takeI;
                        optimalScores[i][j][0] = (i + 1 < n ? optimalScores[i + 1][j][0] : 0);
                    } else {
                        optimalScores[i][j][1] = takeJ;
                        optimalScores[i][j][0] = (j - 1 >= 0 ? optimalScores[i][j - 1][0] : 0);
                    }
                }
            }
        }

        System.out.println(optimalScores[0][n - 1][0] - optimalScores[0][n - 1][1]);
    }

    //Input Reader
    private static BufferedReader reader;
    private static StringTokenizer tokenizer;

    private static void init() {
        reader = new BufferedReader(new InputStreamReader(System.in));
        tokenizer = new StringTokenizer("");
    }

    private static String next() throws IOException {
        String read;
        while (!tokenizer.hasMoreTokens()) {
            read = reader.readLine();
            if (read == null || read.equals(""))
                return "-1";
            tokenizer = new StringTokenizer(read);
        }

        return tokenizer.nextToken();
    }

    private static int nextInt() throws IOException {
        return Integer.parseInt(next());
    }
}