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: Runtime Error)

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

//dp[i][j] = max a person can get from strones from i to  j

int main()
{
	long long int n;
	cin >> n;
	vector<long long int> x(n);
	
	for(long long int i=0;i<n;i++)
		cin>>x[i];

	vector<long long int> ab(n+1);
	vector<vector<long long int> > dp(n+1,ab);

	for(long long int i=1;i<n;i++)
		{
			dp[i][i]=x[i-1];
			
			if(x[i-1]>x[i])
				dp[i][i+1]=x[i-1];
			else
				dp[i][i+1]=x[i];
		}

	dp[0][0]=0;
	dp[n][n]=x[n-1];

	

	for(long long int len=2;len<=n;len++)
	{
		for(long long int i=1;i<=n-len;i++)
		{
			
			long long int j=i+len;	
			
			long long int a=min(dp[i+2][j],dp[i+1][j-1]) + x[i-1];
			long long int b=min(dp[i][j-2],dp[i+1][j-1]) + x[j-1];
			
			dp[i][j]=max(a,b);
		}
	}
	long long int a=dp[1][n];
	long long int b=min(dp[2][n],dp[1][n-1]);
	cout<<a-b;
	return 0;

}