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 Rust (1.15.1) (judge verdict: Accepted)

use std::io::Read;

fn run() {
    let mut s = String::new();
    std::io::stdin().read_to_string(&mut s).unwrap();
    let mut it = s.trim().split_whitespace();
    it.next();
    let w = std::mem::size_of::<usize>() * 8;
    let mut dp = vec![0usize; (2000 * 2000 + 1 + w - 1) / w];
    let mut sum = 0;
    dp[0] = 1;
    for s in it {
        let a: usize = s.parse().unwrap();
        let (q, r) = (a / w, a % w);
        if r == 0 {
            for i in (0..(sum / w + 1)).rev() {
                dp[i + q] |= dp[i];
            }
        } else {
            for i in (0..(sum / w + 1)).rev() {
                dp[i + q + 1] |= dp[i] >> (w - r);
                dp[i + q] |= dp[i] << r;
            }
        }
        sum += a;
    }
    let mut ans = (sum + 1) / 2;
    loop {
        let (q, r) = (ans / w, ans % w);
        if (dp[q] >> r) & 1 == 1 {
            println!("{}", ans);
            return;
        }
        ans += 1;
    }
}

fn main() {
    run();
}