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 Go (1.6) (judge verdict: Time Limit Exceeded)

package main
import (
  "bufio"
  "os"
  "strconv"
  "strings"
  "fmt"
  "sort"
)

var reader = bufio.NewReaderSize(os.Stdin, 1000000)
var writer = bufio.NewWriter(os.Stdout)
func NextLine() string {
  buffer := make([]byte, 0)
  for true {
    line, isPrefix, err := reader.ReadLine()
    if err != nil { panic(err) }
    buffer = append(buffer, line...)
    if !isPrefix { break }
  }
  return string(buffer)
}
func NextInt() int {
  n, _ := strconv.Atoi(NextLine())
  return n
}
func NextIntVec() []int {
  L := strings.Split(NextLine(), " ")
  N := make([]int, len(L))
  for i := range N {
    N[i], _ = strconv.Atoi(L[i])
  }
  return N
}
func Write(s interface{}) {
  fmt.Fprintln(writer, s)
}
func WriteIntVec(A []int) {
  S := make([]string, len(A))
  for i, a := range A {
    S[i] = strconv.Itoa(a)
  }
  Write(strings.Join(S, " "))
}
func Output() {
  _ = writer.Flush()
}

type Items [][]int
func (I Items) Len() int { return len(I) }
func (I Items) Less(i, j int) bool {
  if I[i][0] == I[j][0] { return I[i][1] < I[j][1] }
  return I[i][0] < I[j][0]
}
func (I Items) Swap(i, j int) { I[i], I[j] = I[j], I[i] }

func Unique(A *[][]int) {
  N := len((*A))
  for i := N - 1; 0 < i; i-- {
    if (*A)[i][0] == (*A)[i - 1][0] && (*A)[i][1] == (*A)[i - 1][1] {
      (*A) = append((*A)[:i - 1], (*A)[i:]...)
    }
  }
}

func Max(x, y int) int {
  if x < y { return y }
  return x
}

func main() {
  N := NextInt()
  A := NextIntVec()
  B := make([][][]int, len(A))
  for i, a := range A {
    B[i] = make([][]int, 0)
    B[i] = append(B[i], []int{a, i})
  }
  for i, b := range B {
    for j := 0; j < N; j++ {
      k := i | (1 << uint(j))
      B[k] = append(B[k], b...)
      sort.Sort(sort.Reverse(Items(B[k])))
      Unique(&B[k])
      B[k] = B[k][:2]
    }
  }
  ans := 0
  for _, b := range B[1:] {
    ans = Max(b[0][0] + b[1][0], ans)
    Write(ans)
  }
  Output()
}