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 Scala (2.11.7) (judge verdict: Time Limit Exceeded)

import java.io._
import java.util.StringTokenizer
import scala.util.control.Breaks.{breakable,break}

object Main extends App {
  val in = new InputReader(System.in)
  val pw = new PrintWriter(System.out)

  val N = in.next().toInt
  val A = new Array[Int](1 << N)
  for (i <- 0 to (1 << N) - 1)
    A(i) = in.next().toInt


  //
  val dp = Array.fill[(Int, Int)](1 << N)((-1, -1))
  dp(0) = (0, 0)
  dp(1) = (0, 1)
  var ans = A(0) + A(1)
  println(ans)


  import scala.collection.mutable.PriorityQueue

  for (i <- 2 to (1 << N) - 1) {
    val pq = new PriorityQueue[(Int, Int)]
    pq.enqueue(A(i) -> i)

    for (k <- 0 to N) {
      val hoge = i & 1 << k
      if(hoge != 0) {
        val now = i ^ hoge
        val (a, b) = dp(now)
        pq.enqueue(A(a) -> a)
        pq.enqueue(A(b) -> b)
      }
    }

    var lst: List[Int] = Nil
    while(lst.length < 2) {
      val (_, ind) = pq.dequeue
      if(!lst.contains(ind))
        lst = ind :: lst
    }

    dp(i) = (lst(0), lst(1))
    ans = Math.max(ans, lst.map(ind => A(ind)).sum)
    println(ans)
  }
}


class InputReader(stream: InputStream) {
  private val reader = new BufferedReader(new InputStreamReader(stream))
  private var tokenizer: StringTokenizer = new StringTokenizer(reader.readLine())
  def next(): String = {
    while (!tokenizer.hasMoreTokens()) {
      tokenizer = new StringTokenizer(reader.readLine())
    }
    tokenizer.nextToken()
  }
}