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 C++14 (GCC 5.4.1) (judge verdict: Accepted)


#include <bits/stdc++.h>
using namespace std;

typedef long long ll;
 
using tpl = tuple<int, int, int>;
typedef pair<int,int> pii;

void fastStream(){cin.tie(0);std::ios_base::sync_with_stdio(0);}

int N;
int A[3010];
ll dp[3010][3010];

ll rec(int from, int to){
    if(dp[from][to] >= 0) return dp[from][to];
    ll res = 0;
    if(from > to){
        res = 0;
    }
    else{
        bool is_taro = (N - (to - from + 1)) % 2 == 0;
        if(is_taro){
            res = max(res, rec(from + 1, to) + A[from]);
            res = max(res, rec(from, to - 1) + A[to]);
        }
        else{
            res = rec(from + 1, to);
            res = min(res, rec(from, to - 1));
        }
    }
    return dp[from][to] = res;
}

int main(){

    cin >> N;
    
    ll tot = 0;
    for(int i = 0; i < N; i++){
        cin >> A[i];
        tot += A[i];
    }
    memset(dp, -1, sizeof(dp));
    ll s = rec(0, N - 1);
    cout << s - (tot - s) << endl;

    return 0;
}