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

import java.util.ArrayList;
import java.util.Arrays;
import java.util.List;
import java.util.Scanner;
import java.util.stream.IntStream;

class Main{
    public static void main(String[] args){
        Scanner scan = new Scanner(System.in);
        int N = scan.nextInt();
        int[] A = IntStream.range(0,N).map(i->scan.nextInt()).toArray();

        int sumA = 0;
        for(int i=0;i<N;++i)sumA+=A[i];
        boolean dp[] = new boolean[4000001];
        int minvalue = Integer.MAX_VALUE;

        dp[0]=true;
        for(int i=0;i<N;++i){
            boolean newdp[] = new boolean[4000001];
            for(int j=0;j<=4000000;++j){
                if(!dp[j])continue;
                newdp[Math.abs(j-A[i])]=true;
                if(j+A[i]<=4000000)newdp[j+A[i]]=true;
            }
            dp = newdp;
        }
        for(int i=0;i<4000000;++i){
            if(dp[i]){
                minvalue = Math.min(minvalue, i);
                break;
            }
        }


        System.out.println((sumA+minvalue)/2);


    }
}