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)

#[allow(unused_imports)]
use std::cmp::{min,max};
#[allow(unused_imports)]
use std::collections::BTreeMap;
#[allow(unused_imports)]
use std::ops::*;
#[allow(unused_imports)]
use std::collections::BinaryHeap;

fn main() {
    let n = read_i64() as usize;
    let xs = read_vec_i64();
    let mut dp = vec![(None,None); 1<<n];
    for (i,&x) in xs.iter().enumerate() {
        dp[i].0 = Some(i);
    }
    let f = |xyi : (Option<usize>, Option<usize>), zi: Option<usize>| {
        match zi {
            Some(zi) => {
                match xyi.0 {
                    Some(xi) => {
                        match xyi.1 {
                            Some(yi) => {
                                if xs[zi] > xs[xi] {
                                    (Some(zi), Some(xi))
                                }
                                else if xs[zi] > xs[yi] && xi != zi {
                                    (Some(xi), Some(zi))
                                }
                                else {
                                    xyi
                                }
                            }
                            None => {
                                if xi != zi {
                                    if xs[xi] > xs[zi] {
                                        (Some(xi), Some(zi))
                                    }
                                    else {
                                        (Some(zi), Some(xi))
                                    }
                                }
                                else {
                                    (Some(xi), None)
                                }
                            }
                        }
                    }
                    None => {
                        (Some(zi), None)
                    }
                }
            }
            None => {
                xyi
            }
        }
    };
    for i in 0..(1<<n) {
        for k in 0..n {
            if i & (1 << k) != 0 {
                continue;
            }
            dp[i|(1<<k)] = f(dp[i|(1<<k)], dp[i].0);
            dp[i|(1<<k)] = f(dp[i|(1<<k)], dp[i].1);
        }
    }
    let mut res = 0;
    for i in 1..(1<<n) {
        res = max(res, xs[dp[i].0.unwrap()] + xs[dp[i].1.unwrap()]);
        println!("{}", res);
    }
}

#[allow(dead_code)]
fn read_line() -> String {
    let mut ret = String::new();
    std::io::stdin().read_line(&mut ret).ok();
    ret.pop();
    return ret;
}

#[allow(dead_code)]
fn read_i64() -> i64 {
    let ss = read_line();
    return ss.parse::<i64>().unwrap();
}

#[allow(dead_code)]
fn read_vec_i64() -> Vec<i64> {
    let mut res = vec![];
    let ss = read_line();
    for ts in ss.split_whitespace() {
        let x = ts.parse::<i64>().unwrap();
        res.push(x);
    }
    return res;
}

use std::fmt::Display;
#[allow(dead_code)]
fn write_vec<T: Display>(xs: &Vec<T>) {
    if xs.len() == 0 {
        println!("");
        return;
    }
    print!("{}", xs[0]);
    for i in 1..xs.len() {
        print!(" {}", xs[i]);
    }
    println!("");
}