Problem:
Score : 700 points

Problem Statement

There is an integer sequence of length 2^N: A_0, A_1, ..., A_{2^N-1}. (Note that the sequence is 0-indexed.)

For every integer K satisfying 1 \leq K \leq 2^N-1, solve the following problem:

Let i and j be integers. Find the maximum value of A_i + A_j where 0 \leq i < j \leq 2^N-1 and (i or j) \leq K.
Here, or denotes the bitwise OR.

Constraints

1 \leq N \leq 18

1 \leq A_i \leq 10^9

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N
A_0 A_1 ... A_{2^N-1}

Output

Print 2^N-1 lines.
In the i-th line, print the answer of the problem above for K=i.

Sample Input 1

2
1 2 3 1

Sample Output 1

3
4
5

For K=1, the only possible pair of i and j is (i,j)=(0,1), so the answer is A_0+A_1=1+2=3.

For K=2, the possible pairs of i and j are (i,j)=(0,1),(0,2).
When (i,j)=(0,2), A_i+A_j=1+3=4. This is the maximum value, so the answer is 4.

For K=3, the possible pairs of i and j are (i,j)=(0,1),(0,2),(0,3),(1,2),(1,3),(2,3) .
When (i,j)=(1,2), A_i+A_j=2+3=5. This is the maximum value, so the answer is 5.

Sample Input 2

3
10 71 84 33 6 47 23 25

Sample Output 2

81
94
155
155
155
155
155

Sample Input 3

4
75 26 45 72 81 47 97 97 2 2 25 82 84 17 56 32

Sample Output 3

101
120
147
156
156
178
194
194
194
194
194
194
194
194
194

Solution in Rust (1.15.1) (judge verdict: Accepted)

use io::*;
use std::cmp::max;

fn maxp(a: (i64, i64), b: (i64, i64)) -> (i64, i64) {
    if a.0 > b.0 {
        (a.0, max(a.1, b.0))
    } else {
        (b.0, max(a.0, b.1))
    }
}

fn solve(pos: usize, p: usize, tbl: &mut Vec<Vec<(i64, i64)>>, a: &Vec<i64>) -> (i64, i64) {
    if pos >= tbl.len() {
        return (a[p],-1);
    }

    if tbl[pos][p].0 >= 0 {
        return tbl[pos][p];
    }

    let ret = solve(pos+1,p, tbl, a);
    let ret = if p&(1<<pos) != 0 {
        maxp(ret, solve(pos+1, p&!(1<<pos),tbl,a))
    } else {ret};

    tbl[pos][p] = ret;
    ret
}

fn main() {
    let con = read_string();
    let mut sc = Scanner::new(&con);

    let n: usize = sc.next();
    let a: Vec<i64> = sc.next_vec(1 << n);

    let mut tbl: Vec<Vec<(i64, i64)>> = vec![vec![(-1, -1); 1<<n]; n];

    let mut ans = 0;

    for k in 1..1 << n {
        // println!("* k: {}", cnv(k));
        let t = solve(0, k, &mut tbl, &a);
        ans = max(ans, t.0 + t.1);
        println!("{}", ans);
    }
}

//-----

pub mod io {
    use std;
    use std::str::FromStr;

    pub struct Scanner<'a> {
        iter: std::str::SplitWhitespace<'a>,
    }

    impl<'a> Scanner<'a> {
        pub fn new(s: &'a str) -> Scanner<'a> {
            Scanner {
                iter: s.split_whitespace(),
            }
        }

        pub fn next<T: FromStr>(&mut self) -> T {
            let s = self.iter.next().unwrap();
            if let Ok(v) = s.parse::<T>() {
                v
            } else {
                panic!("Parse error")
            }
        }

        pub fn next_vec_len<T: FromStr>(&mut self) -> Vec<T> {
            let n: usize = self.next();
            self.next_vec(n)
        }

        pub fn next_vec<T: FromStr>(&mut self, n: usize) -> Vec<T> {
            (0..n).map(|_| self.next()).collect()
        }
    }

    pub fn read_string() -> String {
        use std::io::Read;

        let mut s = String::new();
        std::io::stdin().read_to_string(&mut s).unwrap();
        s
    }

    pub fn read_line() -> String {
        let mut s = String::new();
        std::io::stdin().read_line(&mut s).unwrap();
        s.trim_right().to_owned()
    }
}