Problem:
Score : 700 points

Problem Statement

You are given N integers A_1, A_2, ..., A_N.

Consider the sums of all non-empty subsequences of A. There are 2^N - 1 such sums, an odd number.

Let the list of these sums in non-decreasing order be S_1, S_2, ..., S_{2^N - 1}.

Find the median of this list, S_{2^{N-1}}.

Constraints

1 \leq N \leq 2000

1 \leq A_i \leq 2000

All input values are integers.

Input

Input is given from Standard Input in the following format:

N
A_1 A_2 ... A_N

Output

Print the median of the sorted list of the sums of all non-empty subsequences of A.

Sample Input 1

3
1 2 1

Sample Output 1

2

In this case, S = (1, 1, 2, 2, 3, 3, 4). Its median is S_4 = 2.

Sample Input 2

1
58

Sample Output 2

58

In this case, S = (58).

Solution in C++14 (GCC 5.4.1) (judge verdict: Time Limit Exceeded)

#include <iostream>
#include <algorithm>

#define maxNr 2010

int main()
{
	int n, a[maxNr];
	bool dp[maxNr][maxNr] = { 0 };
	std::cin >> n;
	
	int sum = 0;
	for (int i = 0; i < n; ++i)
	{
		std::cin >> a[i];
		sum += a[i];
	}
	
	std::sort(a, a + n);
	
	for (int j = 1; j < ((sum + 1) / 2); ++j)
	{
		int val = j % 2001;
		if (a[0] == j)
			dp[0][val] = 1;
		else
			dp[0][val] = 0;
		
		for (int i = 1; i < n; ++i)
		{
			dp[i][val] |= (j == a[i]);
			dp[i][val] |= dp[i - 1][val];
			
			if (j > a[i])
				dp[i][val] |= dp[i - 1][(j - a[i]) % 2001];
		}
	}
	
	for (int j = (sum + 1) / 2; ; ++j)
	{
		int val = j % 2001;
		if (a[0] == j)
		{
			std::cout << j;
			return 0;
		}
		
		for (int i = 1; i < n; ++i)
		{
			dp[i][val] |= (j == a[i]);
			dp[i][val] |= dp[i - 1][val];
			
			if (j > a[i])
				dp[i][val] |= dp[i - 1][(j - a[i]) % 2001];
			
			if (dp[i][val])
			{
				std::cout << j;
				return 0;
			}
		}
	}
	
	/*
	int max = 1 << n;
	int otherArr[10010] = { 0 };
	int cnt = 0;
	for (int i = 1; i < max; ++i)
	{
		int sum = 0;
		for (int j = 0; j < n; ++j)
		{
			if (i & (1 << j))
				sum += a[j];
		}
		
		otherArr[i - 1] = sum;
	}
	
	std::sort(otherArr, otherArr + max - 1);
	for (int i = 0; i < max - 1; ++i)
	{
		std::cout << otherArr[i] << " ";
	}
	
	std::cout << "\nAND THE ANSWER IS: ";
	std::cout << otherArr[(max - 1) / 2];*/
}