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 std::io::*;
use std::str::FromStr;

fn read<T: FromStr>() -> T {
    let stdin = stdin();
    let stdin = stdin.lock();
    let token: String = stdin
        .bytes()
        .map(|c| c.expect("failed to read char") as char)
        .skip_while(|c| c.is_whitespace())
        .take_while(|c| !c.is_whitespace())
        .collect();
    token.parse().ok().expect("failed to parse token")
}

type T = (i32, usize);
type P = (T, T);

fn update1(p: P, t: T) -> P {
    if (p.0).1 == t.1 || (p.1).1 == t.1 {
        return p;
    }
    if t > p.0 {
        return (t, p.0);
    }
    if t > p.1 {
        return (p.0, t);
    }
    return p;
}

fn update2(p: P, q: P) -> P {
    return update1(update1(p, q.0), q.1);
}

fn main() {
    let n: usize = read();
    let m = 1 << n;
    let a: Vec<i32> = (0..m).map({|_| read()}).collect();

    let mut dp: Vec<P> = vec![((0, m), (0, m)); m];

    for i in 0..m {
        dp[i] = update1(dp[i], (a[i], i));
    }

    for i in 0..m {
        for b in 0..n {
            let j = i | (1 << b);

            if j == b {
                continue;
            }
            dp[j] = update2(dp[j], dp[i]);
        }
    }

    let mut ans = 0;
    for k in 1..m {
        ans = std::cmp::max(ans, (dp[k].0).0 + (dp[k].1).0);
        // println!("{}: {:?}", k, dp[k]);
        println!("{}", ans);
    }

    //println!("{}", ans);
}