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 C# (Mono 4.6.2.0) (judge verdict: Accepted)

using System;
using System.IO;
using System.Linq;
using System.Numerics;
using System.Collections.Generic;
using System.Text;
using System.Text.RegularExpressions;
using System.Threading.Tasks;
using System.Diagnostics;
using static System.Math;
using MethodImplAttribute = System.Runtime.CompilerServices.MethodImplAttribute;
using MethodImplOptions = System.Runtime.CompilerServices.MethodImplOptions;

public static class P
{
    static int[] a;
    public static void Main()
    {
        var n = int.Parse(Console.ReadLine());
        a = Console.ReadLine().Split().Select(int.Parse).ToArray();

        Top2[] top = Enumerable.Repeat(0, a.Length).Select(_ => new Top2()).ToArray();

        for (int i = 0; i < a.Length; i++)
        {
            var top2 = new Top2();
            for (int ind = 1; ind <= i; ind <<= 1)
                if ((i & ind) != 0) top[i].Update(top[i - ind]);
            top[i].Update(i);
        }
        int[] res = new int[a.Length];
        for (int i = 1; i < res.Length; i++)
            res[i] = Max(res[i - 1], top[i].First + top[i].Second);

        Console.WriteLine(string.Join("\n", res.Skip(1)));
    }

    class Top2
    {
        public int FirstInd = -1;
        public int SecondInd = -1;
        public int First { get { return a[FirstInd]; } }
        public int Second { get { return a[SecondInd]; } }
        public void Update(Top2 another)
        {
            Update(another.FirstInd);
            Update(another.SecondInd);
        }
        public void Update(int ind)
        {
            if (ind == -1 || ind == FirstInd || ind == SecondInd) return;
            if (FirstInd == -1 || First < a[ind])
            {
                SecondInd = FirstInd;
                FirstInd = ind;
            }
            else if (SecondInd == -1 || Second < a[ind])
            {
                SecondInd = ind;
            }
        }
    }
}