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 Kotlin (1.0.0) (judge verdict: Accepted)

// Reads multiple ints in a line, separate by a space.
// It works even if the actual number of ints is smaller than n.
// WARNING: This function does not work for -2^31.
// NOTE: This function does not handle error cases.
fun fastReadInts(n: Int): IntArray {
  val xs = IntArray(n)
  val s = readLine()!!
  var currentValue = 0
  var isNegative = false
  var currentIndex = 0
  for (c in s) {
    if (c == ' ') {
      xs[currentIndex] = currentValue * if (isNegative) -1 else 1
      currentValue = 0
      isNegative = false
      currentIndex++
    } else if (c == '-') {
      isNegative = true
    } else {
      currentValue *= 10
      currentValue += c - '0'
    }
  }
  xs[currentIndex] = currentValue * if (isNegative) -1 else 1
  return xs
}

fun filterList(es: List<IndexedValue<Int>>): List<IndexedValue<Int>> {
  val sortedEs = es.toMutableList().sortedByDescending { it.value }
  val output = mutableListOf<IndexedValue<Int>>()
  output.add(sortedEs[0])
  output.add(sortedEs.first { it.index != sortedEs[0].index })
  return output
}

fun main(args: Array<String>) {
  val n = readLine()!!.toInt()
  val xs = fastReadInts(1 shl n)
  val ys = Array(1 shl n , { listOf<IndexedValue<Int>>() })
  ys[0] = listOf(IndexedValue(0, xs[0]))
  ys[1] = listOf(IndexedValue(0, xs[0]), IndexedValue(1, xs[1]))
  for (k in 2 until (1 shl n)) {
    val list = mutableListOf(IndexedValue(k, xs[k]))
    for (i in 0 until 20) {
      if (k or (1 shl i) == k) {
        list.addAll(ys[k - (1 shl i)])
      }
    }
    // println("k: $k list: $list")
    val filtered = filterList(list)
    ys[k] = filtered
  }

  val zs = ys.map { it.sumBy { it.value } }
  val ws = IntArray(1 shl n)
  for (k in 1 until (1 shl n)) {
    ws[k] = Math.max(ws[k - 1], zs[k])
  }

  val output = StringBuilder()
  for (k in 1 until (1 shl n)) {
    output.append(ws[k])
    output.append('\n')
  }

  print(output)
}