#include <bits/stdc++.h>
using namespace std;
const int m = 22;
int pow_two[m];
const int maxk = 19;
const int maxn = 7e5;
int n, k;
int a[maxn];
vector<bool> bits_arr[maxn];
void compute(int d, int l, int r, vector<int>& min_xor, vector<int>& max_xor,
             vector<int>& min_dist) {
  assert((r - l + 1) > 0);
  int ll = l, lr = -1, rl = -1, rr = r;
  for (int i = l; i <= r; i++) {
    if ((bits_arr[i][d] == 1) && (rl == -1))
      rl = i;
    else if (bits_arr[i][d] == 0)
      lr = i;
  }
  bool left_empty = (lr == -1), right_empty = (rl == -1);
  if (d == 0) {
    if (left_empty || right_empty) {
      min_xor[0] = max_xor[0] = ((left_empty) ? 1 : 0);
      max_xor[1] = min_xor[1] = ((left_empty) ? 0 : 1);
      min_dist[0] = min_dist[1] = ((r - l) > 0) ? 0 : INT_MAX;
    } else {
      min_xor[0] = min_xor[1] = 0;
      max_xor[0] = max_xor[1] = 1;
      min_dist[0] = min_dist[1] = (((!left_empty) && ((lr - ll) > 0)) ||
                                   ((!right_empty) && ((rr - rl) > 0)))
                                      ? 0
                                      : 1;
    }
  } else {
    int sz = pow_two[d];
    vector<int> left_min_xor(sz), left_max_xor(sz), left_min_dist(sz);
    vector<int> right_min_xor(sz), right_max_xor(sz), right_min_dist(sz);
    if (!left_empty)
      compute(d - 1, ll, lr, left_min_xor, left_max_xor, left_min_dist);
    if (!right_empty)
      compute(d - 1, rl, rr, right_min_xor, right_max_xor, right_min_dist);
    for (int x = 0; x < pow_two[d + 1]; x++) {
      bool msb_one = (x >= pow_two[d]);
      int y = (msb_one) ? (x - pow_two[d]) : x;
      min_dist[x] = INT_MAX;
      if (!left_empty) min_dist[x] = min(min_dist[x], left_min_dist[y]);
      if (!right_empty) min_dist[x] = min(min_dist[x], right_min_dist[y]);
      if ((!left_empty) && (!right_empty)) {
        int min_cross_diff =
            (msb_one) ? (pow_two[d] + left_min_xor[y] - right_max_xor[y])
                      : (pow_two[d] + right_min_xor[y] - left_max_xor[y]);
        min_dist[x] = min(min_dist[x], min_cross_diff);
      }
      min_xor[x] = INT_MAX;
      max_xor[x] = INT_MIN;
      if (!left_empty) {
        min_xor[x] = min(min_xor[x], (msb_one) ? (pow_two[d] + left_min_xor[y])
                                               : (left_min_xor[y]));
        max_xor[x] = max(max_xor[x], (msb_one) ? (pow_two[d] + left_max_xor[y])
                                               : (left_max_xor[y]));
      }
      if (!right_empty) {
        min_xor[x] =
            min(min_xor[x], (msb_one) ? (right_min_xor[y])
                                      : (pow_two[d] + right_min_xor[y]));
        max_xor[x] =
            max(max_xor[x], (msb_one) ? (right_max_xor[y])
                                      : (pow_two[d] + right_max_xor[y]));
      }
    }
  }
}
void sort_and_get_bits_arr() {
  sort(a, a + n);
  for (int i = 0; i < n; i++) {
    vector<bool> b;
    int x = a[i];
    for (int j = 0; j < k; j++) {
      b.push_back((bool)(x % 2));
      x /= 2;
    }
    bits_arr[i] = b;
  }
}
int main() {
  for (int i = 0; i < m; i++)
    pow_two[i] = ((i == 0) ? 1 : (pow_two[i - 1] * 2));
  cin >> n >> k;
  for (int i = 0; i < n; i++) cin >> a[i];
  sort_and_get_bits_arr();
  int sz = pow_two[k];
  vector<int> min_xor(sz), max_xor(sz), min_dist(sz);
  compute(k - 1, 0, n - 1, min_xor, max_xor, min_dist);
  for (int x = 0; x < pow_two[k]; x++) cout << min_dist[x] << ' ';
  cout << endl;
}

### Prompt

Your challenge is to write a Cpp solution to the following problem:

You are given an array a consisting of n distinct elements and an integer k. Each element in the array is a non-negative integer not exceeding 2^k-1.

Let's define the XOR distance for a number x as the value of 

$$$f(x) = min_{i = 1}^{n} min_{j = i + 1}^{n} |(a_i ⊕ x) - (a_j ⊕ x)|,$$$

where ⊕ denotes [the bitwise XOR operation](https://en.wikipedia.org/wiki/Bitwise_operation#XOR).

For every integer x from 0 to 2^k-1, you have to calculate f(x).

Input

The first line contains two integers n and k (1 ≤ k ≤ 19; 2 ≤ n ≤ 2^k).

The second line contains n integers a_1, a_2, ..., a_n (0 ≤ a_i ≤ 2^k-1). All these integers are distinct.

Output

Print 2^k integers. The i-th of them should be equal to f(i-1).

Examples

Input


3 3
6 0 3


Output


3 1 1 2 2 1 1 3 


Input


3 4
13 4 2


Output


2 2 6 6 3 1 2 2 2 2 1 3 6 6 2 2 

Note

Consider the first example:

  * for x = 0, if we apply bitwise XOR to the elements of the array with x, we get the array [6, 0, 3], and the minimum absolute difference of two elements is 3; 
  * for x = 1, if we apply bitwise XOR to the elements of the array with x, we get the array [7, 1, 2], and the minimum absolute difference of two elements is 1; 
  * for x = 2, if we apply bitwise XOR to the elements of the array with x, we get the array [4, 2, 1], and the minimum absolute difference of two elements is 1; 
  * for x = 3, if we apply bitwise XOR to the elements of the array with x, we get the array [5, 3, 0], and the minimum absolute difference of two elements is 2; 
  * for x = 4, if we apply bitwise XOR to the elements of the array with x, we get the array [2, 4, 7], and the minimum absolute difference of two elements is 2; 
  * for x = 5, if we apply bitwise XOR to the elements of the array with x, we get the array [3, 5, 6], and the minimum absolute difference of two elements is 1; 
  * for x = 6, if we apply bitwise XOR to the elements of the array with x, we get the array [0, 6, 5], and the minimum absolute difference of two elements is 1; 
  * for x = 7, if we apply bitwise XOR to the elements of the array with x, we get the array [1, 7, 4], and the minimum absolute difference of two elements is 3. 

### Response

```cpp
#include <bits/stdc++.h>
using namespace std;
const int m = 22;
int pow_two[m];
const int maxk = 19;
const int maxn = 7e5;
int n, k;
int a[maxn];
vector<bool> bits_arr[maxn];
void compute(int d, int l, int r, vector<int>& min_xor, vector<int>& max_xor,
             vector<int>& min_dist) {
  assert((r - l + 1) > 0);
  int ll = l, lr = -1, rl = -1, rr = r;
  for (int i = l; i <= r; i++) {
    if ((bits_arr[i][d] == 1) && (rl == -1))
      rl = i;
    else if (bits_arr[i][d] == 0)
      lr = i;
  }
  bool left_empty = (lr == -1), right_empty = (rl == -1);
  if (d == 0) {
    if (left_empty || right_empty) {
      min_xor[0] = max_xor[0] = ((left_empty) ? 1 : 0);
      max_xor[1] = min_xor[1] = ((left_empty) ? 0 : 1);
      min_dist[0] = min_dist[1] = ((r - l) > 0) ? 0 : INT_MAX;
    } else {
      min_xor[0] = min_xor[1] = 0;
      max_xor[0] = max_xor[1] = 1;
      min_dist[0] = min_dist[1] = (((!left_empty) && ((lr - ll) > 0)) ||
                                   ((!right_empty) && ((rr - rl) > 0)))
                                      ? 0
                                      : 1;
    }
  } else {
    int sz = pow_two[d];
    vector<int> left_min_xor(sz), left_max_xor(sz), left_min_dist(sz);
    vector<int> right_min_xor(sz), right_max_xor(sz), right_min_dist(sz);
    if (!left_empty)
      compute(d - 1, ll, lr, left_min_xor, left_max_xor, left_min_dist);
    if (!right_empty)
      compute(d - 1, rl, rr, right_min_xor, right_max_xor, right_min_dist);
    for (int x = 0; x < pow_two[d + 1]; x++) {
      bool msb_one = (x >= pow_two[d]);
      int y = (msb_one) ? (x - pow_two[d]) : x;
      min_dist[x] = INT_MAX;
      if (!left_empty) min_dist[x] = min(min_dist[x], left_min_dist[y]);
      if (!right_empty) min_dist[x] = min(min_dist[x], right_min_dist[y]);
      if ((!left_empty) && (!right_empty)) {
        int min_cross_diff =
            (msb_one) ? (pow_two[d] + left_min_xor[y] - right_max_xor[y])
                      : (pow_two[d] + right_min_xor[y] - left_max_xor[y]);
        min_dist[x] = min(min_dist[x], min_cross_diff);
      }
      min_xor[x] = INT_MAX;
      max_xor[x] = INT_MIN;
      if (!left_empty) {
        min_xor[x] = min(min_xor[x], (msb_one) ? (pow_two[d] + left_min_xor[y])
                                               : (left_min_xor[y]));
        max_xor[x] = max(max_xor[x], (msb_one) ? (pow_two[d] + left_max_xor[y])
                                               : (left_max_xor[y]));
      }
      if (!right_empty) {
        min_xor[x] =
            min(min_xor[x], (msb_one) ? (right_min_xor[y])
                                      : (pow_two[d] + right_min_xor[y]));
        max_xor[x] =
            max(max_xor[x], (msb_one) ? (right_max_xor[y])
                                      : (pow_two[d] + right_max_xor[y]));
      }
    }
  }
}
void sort_and_get_bits_arr() {
  sort(a, a + n);
  for (int i = 0; i < n; i++) {
    vector<bool> b;
    int x = a[i];
    for (int j = 0; j < k; j++) {
      b.push_back((bool)(x % 2));
      x /= 2;
    }
    bits_arr[i] = b;
  }
}
int main() {
  for (int i = 0; i < m; i++)
    pow_two[i] = ((i == 0) ? 1 : (pow_two[i - 1] * 2));
  cin >> n >> k;
  for (int i = 0; i < n; i++) cin >> a[i];
  sort_and_get_bits_arr();
  int sz = pow_two[k];
  vector<int> min_xor(sz), max_xor(sz), min_dist(sz);
  compute(k - 1, 0, n - 1, min_xor, max_xor, min_dist);
  for (int x = 0; x < pow_two[k]; x++) cout << min_dist[x] << ' ';
  cout << endl;
}

```