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)

macro_rules ! input { ( source = $ s : expr , $ ( $ r : tt ) * ) => { let mut iter = $ s . split_whitespace ( ) ; input_inner ! { iter , $ ( $ r ) * } } ; ( iter = $ iter : ident , $ ( $ r : tt ) * ) => { let s = { use std :: io :: Read ; let mut s = String :: new ( ) ; std :: io :: stdin ( ) . read_to_string ( & mut s ) . unwrap ( ) ; s } ; let mut $ iter = s . split_whitespace ( ) ; input_inner ! { $ iter , $ ( $ r ) * } } ; ( $ ( $ r : tt ) * ) => { let s = { use std :: io :: Read ; let mut s = String :: new ( ) ; std :: io :: stdin ( ) . read_to_string ( & mut s ) . unwrap ( ) ; s } ; let mut iter = s . split_whitespace ( ) ; input_inner ! { iter , $ ( $ r ) * } } ; }
macro_rules ! input_inner { ( $ iter : expr ) => { } ; ( $ iter : expr , ) => { } ; ( $ iter : expr , mut $ var : ident : $ t : tt $ ( $ r : tt ) * ) => { let mut $ var = read_value ! ( $ iter , $ t ) ; input_inner ! { $ iter $ ( $ r ) * } } ; ( $ iter : expr , mut $ var : ident $ ( $ r : tt ) * ) => { input_inner ! { $ iter , mut $ var : usize $ ( $ r ) * } } ; ( $ iter : expr , $ var : ident : $ t : tt $ ( $ r : tt ) * ) => { let $ var = read_value ! ( $ iter , $ t ) ; input_inner ! { $ iter $ ( $ r ) * } } ; ( $ iter : expr , $ var : ident $ ( $ r : tt ) * ) => { input_inner ! { $ iter , $ var : usize $ ( $ r ) * } } ; }
macro_rules ! read_value { ( $ iter : expr , ( $ ( $ t : tt ) ,* ) ) => { ( $ ( read_value ! ( $ iter , $ t ) ) ,* ) } ; ( $ iter : expr , [ $ t : tt ; $ len : expr ] ) => { ( 0 ..$ len ) . map ( | _ | read_value ! ( $ iter , $ t ) ) . collect ::< Vec < _ >> ( ) } ; ( $ iter : expr , { chars : $ base : expr } ) => { read_value ! ( $ iter , String ) . chars ( ) . map ( | c | ( c as u8 - $ base as u8 ) as usize ) . collect ::< Vec < usize >> ( ) } ; ( $ iter : expr , { char : $ base : expr } ) => { read_value ! ( $ iter , { chars : $ base } ) [ 0 ] } ; ( $ iter : expr , chars ) => { read_value ! ( $ iter , String ) . chars ( ) . collect ::< Vec < char >> ( ) } ; ( $ iter : expr , char ) => { read_value ! ( $ iter , chars ) [ 0 ] } ; ( $ iter : expr , usize1 ) => { read_value ! ( $ iter , usize ) - 1 } ; ( $ iter : expr , $ t : ty ) => { $ iter . next ( ) . unwrap ( ) . parse ::<$ t > ( ) . unwrap ( ) } ; }
#[derive(Clone, Debug, Default, PartialEq, Eq, PartialOrd, Ord, Hash)]
pub struct BitSet {
    size: usize,
    bits: Vec<u64>,
}
impl BitSet {
    pub fn new(size: usize) -> Self {
        BitSet {
            size: size,
            bits: vec![0; (size + 63) / 64],
        }
    }
    #[inline]
    pub fn get(&self, i: usize) -> bool {
        self.bits[i >> 6] & 1 << (i & 63) != 0
    }
    #[inline]
    pub fn set(&mut self, i: usize, b: bool) {
        if b {
            self.bits[i >> 6] |= 1 << (i & 63);
        } else {
            self.bits[i >> 6] &= !(1 << (i & 63));
        }
    }
    #[inline]
    pub fn count_ones(&self) -> u64 {
        self.bits.iter().map(|x| x.count_ones() as u64).sum()
    }
    #[inline]
    pub fn count_zeros(&self) -> u64 {
        self.size as u64 - self.count_ones()
    }
    #[inline]
    pub fn trim(&mut self) {
        if self.size & 63 != 0 {
            if let Some(x) = self.bits.last_mut() {
                *x &= 0xffffffffffffffff >> 64 - (self.size & 63);
            }
        }
    }
    #[inline]
    pub fn shl_or_assign(&mut self, rhs: usize) {
        let n = self.bits.len();
        let k = rhs >> 6;
        let d = rhs & 63;
        if k < n {
            if d == 0 {
                for i in (0..n - k).rev() {
                    self.bits[i + k] |= self.bits[i];
                }
            } else {
                for i in (1..n - k).rev() {
                    self.bits[i + k] |= self.bits[i] << d | self.bits[i - 1] >> 64 - d;
                }
                self.bits[k] |= self.bits[0] << d;
            }
            self.trim();
        }
    }
    #[inline]
    pub fn shr_or_assign(&mut self, rhs: usize) {
        let n = self.bits.len();
        let k = rhs >> 6;
        let d = rhs & 63;
        if k < n {
            if d == 0 {
                for i in k..n {
                    *unsafe { self.bits.get_unchecked_mut(i - k) } |=
                        *unsafe { self.bits.get_unchecked(i) };
                }
            } else {
                for i in k..n - 1 {
                    *unsafe { self.bits.get_unchecked_mut(i - k) } |=
                        *unsafe { self.bits.get_unchecked(i) } >> d
                            | *unsafe { self.bits.get_unchecked(i + 1) } << 64 - d;
                }
                *unsafe { self.bits.get_unchecked_mut(n - k - 1) } |=
                    *unsafe { self.bits.get_unchecked(n - 1) } >> d;
            }
        }
    }
}
impl std::ops::ShlAssign<usize> for BitSet {
    fn shl_assign(&mut self, rhs: usize) {
        let n = self.bits.len();
        let k = rhs >> 6;
        let d = rhs & 63;
        if k >= n {
            for x in self.bits.iter_mut() {
                *x = 0;
            }
        } else {
            if d == 0 {
                for i in (0..n - k).rev() {
                    self.bits[i + k] = self.bits[i];
                }
            } else {
                for i in (1..n - k).rev() {
                    self.bits[i + k] = self.bits[i] << d | self.bits[i - 1] >> 64 - d;
                }
                self.bits[k] = self.bits[0] << d;
            }
            for x in self.bits[..k].iter_mut() {
                *x = 0;
            }
            self.trim();
        }
    }
}
impl std::ops::Shl<usize> for BitSet {
    type Output = Self;
    fn shl(mut self, rhs: usize) -> Self::Output {
        self <<= rhs;
        self
    }
}
impl std::ops::ShrAssign<usize> for BitSet {
    fn shr_assign(&mut self, rhs: usize) {
        let n = self.bits.len();
        let k = rhs >> 6;
        let d = rhs & 63;
        if k >= n {
            for x in self.bits.iter_mut() {
                *x = 0;
            }
        } else {
            if d == 0 {
                for i in k..n {
                    self.bits[i - k] = self.bits[i];
                }
            } else {
                for i in k..n - 1 {
                    self.bits[i - k] = self.bits[i] >> d | self.bits[i + 1] << 64 - d;
                }
                self.bits[n - k - 1] = self.bits[n - 1] >> d;
            }
            for x in self.bits[n - k..].iter_mut() {
                *x = 0;
            }
        }
    }
}
impl std::ops::Shr<usize> for BitSet {
    type Output = Self;
    fn shr(mut self, rhs: usize) -> Self::Output {
        self >>= rhs;
        self
    }
}
impl<'a> std::ops::BitOrAssign<&'a BitSet> for BitSet {
    fn bitor_assign(&mut self, rhs: &'a Self) {
        for (l, r) in self.bits.iter_mut().zip(rhs.bits.iter()) {
            *l |= *r;
        }
        self.trim();
    }
}
impl<'a> std::ops::BitOr<&'a BitSet> for BitSet {
    type Output = Self;
    fn bitor(mut self, rhs: &'a Self) -> Self::Output {
        self |= rhs;
        self
    }
}
fn main() {
    input! { n, a: [usize; n] };
    let s = a.iter().sum::<usize>();
    let mut bits = BitSet::new(s + 1);
    bits.set(s, true);
    for x in a.into_iter() {
        bits.shr_or_assign(x);
    }
    println!("{}", ((s + 1) / 2..).find(|&i| bits.get(i)).unwrap());
}