GCD Groups 2 - VietMX's Blog

You are given an array of $n$ integers. You need to split all integers into two groups so that the GCD of all integers in the first group is equal to one and the GCD of all integers in the second group is equal to one.

The GCD of a group of integers is the largest non-negative integer that divides all the integers in the group.

Both groups have to be non-empty.

Input

The first line contains a single integer $n$ ( $2 \leq n \leq 10^5$ ).

The second line contains $n$ integers $a_1$ , $a_2$ , $\ldots$ , $a_n$ ( $1 \leq a_i \leq 10^9$ ) — the elements of the array.

Output

In the first line print “YES” (without quotes), if it is possible to split the integers into two groups as required, and “NO” (without quotes) otherwise.

If it is possible to split the integers, in the second line print $n$ integers, where the $i$ -th integer is equal to $1$ if the integer $a_i$ should be in the first group, and $2$ otherwise.

If there are multiple solutions, print any.

Examples

input

4
2 3 6 7

output

YES
2 2 1 1

input

5
6 15 35 77 22

output

YES
2 1 2 1 1

input

5
6 10 15 1000 75

output

NO

Solution:

#include <bits/stdc++.h>
 
using namespace std;
 
namespace factorizer {
 
vector<int> least = {0, 1};
vector<int> primes;
int precalculated = 1;
 
void RunLinearSieve(int n) {
n = max(n, 1);
least.assign(n + 1, 0);
primes.clear();
for (int i = 2; i <= n; i++) {
    if (least[i] == 0) {
      least[i] = i;
      primes.push_back(i);
    }
    for (int x : primes) {
      if (x > least[i] || i * x > n) {
break;
}
least[i * x] = x;
}
}
precalculated = n;
}
 
void RunSieve(int n) {
RunLinearSieve(n);
}
 
template <typename T>
vector<pair<T, int>> Factorize(T x) {
vector<pair<T, int>> ret;
for (T i : primes) {
T t = x / i;
if (i > t) {
break;
}
if (x == t * i) {
int cnt = 0;
while (x % i == 0) {
x /= i;
cnt++;
}
ret.emplace_back(i, cnt);
}
}
if (x > 1) {
ret.emplace_back(x, 1);
}
return ret;
}
 
}  // namespace factorizer
 
vector<int> GetFirsts(const vector<pair<int, int>>& a) {
vector<int> b;
for (auto& p : a) {
b.push_back(p.first);
}
return b;
}
 
mt19937 rng((unsigned int) chrono::steady_clock::now().time_since_epoch().count());
 
const int N = 100010;
const int MX = 555;
 
int delta[N][2];
int nxt[N][23];
int dist[MX * MX];
int pr[MX * MX];
vector<int> here[N];
 
int main() {
ios::sync_with_stdio(false);
cin.tie(0);
int n;
cin >> n;
vector<pair<int, int>> aa(n);
for (int i = 0; i < n; i++) {
    cin >> aa[i].first;
aa[i].second = i;
}
shuffle(aa.begin(), aa.end(), rng);
vector<int> a(n);
vector<int> real_id(n);
for (int i = 0; i < n; i++) {
    a[i] = aa[i].first;
    real_id[i] = aa[i].second;
  }
  factorizer::RunSieve(40000);
  for (int start = 1; start < n; start++) {
    vector<vector<int>> f = {GetFirsts(factorizer::Factorize(a[0])), GetFirsts(factorizer::Factorize(a[start]))};
vector<int> sz = {(int) f[0].size(), (int) f[1].size()};
for (int i = start + 1; i < n; i++) {
      for (int r = 0; r < 2; r++) {
        delta[i][r] = 0;
        for (int j = 0; j < sz[r]; j++) {
          if (a[i] % f[r][j] == 0) {
            delta[i][r] |= (1 << j);
          }
        }
      }
    }
    for (int j = 0; j < sz[0] + sz[1]; j++) {
      nxt[n][j] = n;
    }
    for (int i = n - 1; i >= start + 1; i--) {
for (int j = 0; j < sz[0]; j++) {
        if ((delta[i][0] & (1 << j)) == 0) {
          nxt[i][j] = i;
        } else {
          nxt[i][j] = nxt[i + 1][j];
        }
      }
      for (int j = 0; j < sz[1]; j++) {
        if ((delta[i][1] & (1 << j)) == 0) {
          nxt[i][sz[0] + j] = i;
        } else {
          nxt[i][sz[0] + j] = nxt[i + 1][sz[0] + j];
        }
      }
    }
    for (int i = 0; i <= n; i++) {
      here[i].clear();
    }
    for (int t = 0; t < (1 << (sz[0] + sz[1])); t++) {
      dist[t] = n + 1;
    }
    int init = (1 << (sz[0] + sz[1])) - 1;
    dist[init] = start + 1;
    here[start + 1].push_back(init);
    for (int at = start + 1; at < n; at++) {
      for (int t : here[at]) {
        if (dist[t] != at) {
          continue;
        }
        for (int bit = 0; bit < sz[0]; bit++) {
          if ((t & (1 << bit)) == 0) {
            continue;
          }
          int to = nxt[at][bit];
          if (to == n) {
            continue;
          }
          int nt = t & (delta[to][0] | (((1 << (sz[0] + sz[1])) - 1) ^ ((1 << sz[0]) - 1)));
          if (to + 1 < dist[nt]) {
            dist[nt] = to + 1;
            pr[nt] = t;
            here[to + 1].push_back(nt);
          }
        }
        for (int bit = sz[0]; bit < sz[0] + sz[1]; bit++) {
          if ((t & (1 << bit)) == 0) {
            continue;
          }
          int to = nxt[at][bit];
          if (to == n) {
            continue;
          }
          int nt = t & ((delta[to][1] << sz[0]) | ((1 << sz[0]) - 1));
          if (to + 1 < dist[nt]) {
            dist[nt] = to + 1;
            pr[nt] = ~t;
            here[to + 1].push_back(nt);
          }
        }
      }
    }
    if (dist[0] <= n) {
      cout << "YES" << '\n';
      vector<int> res(n);
for (int i = 0; i < start; i++) {
        res[real_id[i]] = 1;
      }
      res[real_id[start]] = 2;
      int t = 0;
      while (t != init) {
        int id = real_id[dist[t] - 1];
        if (pr[t] >= 0) {
res[id] = 1;
t = pr[t];
} else {
res[id] = 2;
t = ~pr[t];
}
}
for (int i = 0; i < n; i++) {
        if (res[i] == 0) {
          res[i] = rng() % 2 + 1;
        }
      }
      for (int i = 0; i < n; i++) {
        if (i > 0) {
cout << " ";
        }
        cout << res[i];
      }
      cout << '\n';
      return 0;
    }
    if (__gcd(a[0], a[start]) == a[0]) {
      break;
    }
    a[0] = __gcd(a[0], a[start]);
  }
  cout << "NO" << '\n';
  return 0;
}