[Light OJ] 1356 - Prime Independence

Author            : Dipu Kumar Mohanto 
                    CSE, Batch - 6
                    BRUR.
Problem Statement : 1356 - Prime Independence
Source            : Light Online Judge
Category          : Graph Theory, Number Theory
Algorithm         : Maximum Bipartite Matching, Maximum Independent Set, Fopcropt Carp Algorithm
                    Prime Factorization
Verdict           : Accepted
 
#include "bits/stdc++.h"
 
using namespace std;
 
#define FI              freopen("in.txt", "r", stdin)
#define FO              freopen("out.txt", "w", stdout)
#define FAST            ios_base::sync_with_stdio(false), cin.tie(NULL), cout.tie(NULL)
 
#define FOR(i, n)       for (int i = 1; i <= n; i++)
#define For(i, n)       for (int i = 0; i < n; i++)
#define ROF(i, n)       for (int i = n; i >= 1; i--)
#define Rof(i, n)       for (int i = n-1; i >= 0; i--)
#define FORI(i, n)      for (auto i : n)
 
#define ll              long long
#define ull             unsigned long long
#define vi              vector <int>
#define vl              vector <ll>
#define pii             pair <int, int>
#define pll             pair <ll, ll>
#define mk              make_pair
#define ff              first
#define ss              second
#define eb              emplace_back
#define em              emplace
#define pb              push_back
#define ppb             pop_back
#define All(a)          a.begin(), a.end()
#define memo(a, b)      memset(a, b, sizeof a)
#define Sort(a)         sort(All(a))
#define ED(a)           Sort(a), a.erase(unique(All(a)), a.end())
#define rev(a)          reverse(All(a))
#define sz(a)           (int)a.size()
#define max3(a, b, c)   max(a, max(b, c))
#define min3(a, b, c)   min(a, min(b, c))
#define maxAll(a)       *max_element(All(a))
#define minAll(a)       *min_element(All(a))
#define allUpper(a)     transform(All(a), a.begin(), :: toupper)
#define allLower(a)     transform(All(a), a.begin(), :: tolower)
#define endl            '\n'
#define nl              puts("")
#define ub              upper_bound
#define lb              lower_bound
#define Exp             exp(1.0)
#define PIE             2*acos(0.0)
#define Sin(a)          sin(((a)*PIE)/180.0)
#define EPS             1e-9
 
// Important Functions
// mo compare function : friend bool operator < (mo p, mo q) { int pb = p.l / block; int qb = q.l / block; if (pb != qb) return p.l < q.l; return (p.r < q.r) ^ (p.l / block % 2); }
// factorial MOD inverse : ll inv[maxn]; inline ll factorialInverse() { inv[1] = 1; for (int i = 2; i < maxn; i++) inv[i] = (MOD – (MOD / i)) * inv[MOD % i) % MOD; }
 
/**
  * uses of PBDS:
  * 1) *X.find_by_order(k) : returns k'th element
  * 2) X.order_of_key(k)   : count total element less than k
  * 3) All other operations same as set
**/
#include "ext/pb_ds/assoc_container.hpp"
#include "ext/pb_ds/tree_policy.hpp"
using namespace __gnu_pbds;
 
template <typename T> using orderset = tree <T, null_type, less <T>, rb_tree_tag, tree_order_statistics_node_update>;
 
/**
  * uses of rope data structure:
  * Initialize                 : rope <data_type> Rope;
  * Insert single element      : Rope.push_back()
  * Insert a block of elements : Rope.insert(position after insert, newRope)
  * Erase a block of elements  : Rope.erase(starting position of deletion, size of deletion)
  * Other Operations           : Same as std::vector
  * Iterator                   : Rope.mutable_begin(), Rope.mutable_end(), Rope.mutable_rbegin(), Rope.mutable_rend()
  * Caution                    : Never use -- Rope.begin() iterator
**/
#include "ext/rope"
using namespace __gnu_cxx;
 
// rope <int> Rope;
 
// int dr[] = {1, -1, 0, 0}; // 4 Direction
// int dc[] = {0, 0, 1, -1};
// int dr[] = {0, 0, 1, -1, 1, 1, -1, -1}; // 8 Direction
// int dc[] = {1, -1, 0, 0, 1, -1, 1, -1};
// int dr[] = {-1, 1, -2, -2, -1, 1, 2, 2}; // knight Moves
// int dc[] = {-2, -2, -1, 1, 2, 2, 1, -1};
 
#define trace1(x)                           cerr << #x << ": " << x << endl;
#define trace2(x, y)                        cerr << #x << ": " << x << " | " << #y << ": " << y << endl;
#define trace3(x, y, z)                     cerr << #x << ": " << x << " | " << #y << ": " << y << " | " << #z << ": " << z << endl;
#define trace4(a, b, c, d)                  cerr << #a << ": " << a << " | " << #b << ": " << b << " | " << #c << ": " << c << " | " << #d << ": " << d << endl;
#define trace5(a, b, c, d, e)               cerr << #a << ": " << a << " | " << #b << ": " << b << " | " << #c << ": " << c << " | " << #d << ": " << d << " | " << #e << ": " << e << endl;
#define trace6(a, b, c, d, e, f)            cerr << #a << ": " << a << " | " << #b << ": " << b << " | " << #c << ": " << c << " | " << #d << ": " << d << " | " << #e << ": " << e << " | " << #f << ": " << f << endl;
 
inline int setbit(int mask, int pos)      { return mask |= (1 << pos); }
inline int resetbit(int mask, int pos)    { return mask &= ~(1 << pos); }
inline int togglebit(int mask, int pos)   { return mask ^= (1 << pos); }
inline bool checkbit(int mask, int pos)   { return (bool)(mask & (1 << pos)); }
 
#define popcount(mask)                       __builtin_popcount(mask) // count set bit
#define popcountLL(mask)                     __builtin_popcountll(mask) // for long long
 
inline int read()                         { int a; scanf("%d", &a); return a; }
inline ll readLL()                        { ll a; scanf("%lld", &a); return a; }
inline double readDD()                    { double a; scanf("%lf", &a); return a; }
 
template <typename T> string toString(T num) { stringstream ss; ss << num; return ss.str(); }
int toInt(string s)                          { int num; istringstream iss(s); iss >> num; return num;  }
ll toLLong(string s)                         { ll num; istringstream iss(s); iss >> num; return num; }
 
#define inf             1e17
#define mod             1000000007
 
static const int maxn = 4e4 + 5;
static const int logn = 18;
static const int mx = 500000 + 6;
 
vi primefactor[mx], primeList;
bool isprime[mx];
 
inline void primeFactorization()
{
      for (int i = 2; i < mx; i++)
      {
            if (isprime[i]) continue;
            primeList.eb(i);
            for (int j = 1; j*i < mx; j++)
            {
                  isprime[i*j] = 1;
                  primefactor[i*j].eb(i);
            }
      }
}
 
inline void seive()
{
      memo(isprime, 1);
      isprime[0] = isprime[1] = 0;
      for (int i = 4; i < mx; i += 2) isprime[i] = 0;
      for (int i = 3; i*i <= mx; i += 2)
      {
            if (isprime[i])
            {
                  for (int j = i*i; j < mx; j += i+i)
                  {
                        isprime[j] = 0;
                  }
            }
      }
      primeList.eb(2);
      for (int i = 3; i < mx; i += 2) if (isprime[i]) primeList.eb(i);
}
 
vi graph[maxn];
int n, arr[maxn], position[mx];
 
inline void clean()
{
      For(i, maxn) graph[i].clear();
      memo(position, -1);
}
 
/**            Hopcropt Carp Algorithm       **/
 
#define inf           1 << 28
#define NILL          0
 
static const int MAXN = 4e4 + 5;
 
int elements_in_left_side; // no.of elements in left side
int elements_in_right_side; // no.of elements in right side
 
// int pairU[MAXN], pairV[MAXN], dist[MAXN];
int match[MAXN], dist[MAXN]; // if we cannot divide the graph in two parts
                             // use only match[]
 
inline void initialize_hopcroptcarp()
{
      for (int i = 0; i < MAXN; i++)
      {
//            pairU[i] = NILL;
//            pairV[i] = NILL;
            match[i] = NILL;
      }
}
 
inline bool bfs()
{
      queue <int> PQ;
      for (int u = 1; u <= elements_in_left_side; u++)
      {
            if (match[u] == NILL)
            {
                  dist[u] = NILL;
                  PQ.push(u);
            }
            else
            {
                  dist[u] = inf;
            }
      }
      dist[NILL] = inf;
      while (!PQ.empty())
      {
            int u = PQ.front(); PQ.pop();
            if (u != NILL && dist[u] < dist[NILL])
            {
                  for (int v : graph[u])
                  {
                        if (dist[ match[v] ] == inf)
                        {
                              dist[ match[v] ] = dist[u] + 1;
                              PQ.push(match[v]);
                        }
                  }
            }
      }
      if (dist[NILL] != inf) return true;
      else return false;
}
 
inline bool dfs(int u)
{
      if (u != NILL)
      {
            for (int v : graph[u])
            {
                  if (dist[ match[v] ] == dist[u] + 1)
                  {
                        if (dfs(match[v]))
                        {
                              match[u] = v;
                              match[v] = u;
                              return true;
                        }
                  }
            }
            dist[u] = inf;
            return false;
      }
      return true;
}
 
inline int hopcropt_carp()
{
      int max_match = 0;
      while (bfs())
      {
            for (int u = 1; u <= elements_in_left_side; u++)
            {
                  if (match[u] == NILL && dfs(u)) max_match++;
            }
      }
      return max_match;
}
 
int main()
{
//      FI;
      seive();
//      primeFactorization(); // this cause MLE
      int tc = read();
      FOR(tcase, tc)
      {
            clean();
            n = read();
            int maxnum = -1;
            FOR(i, n)
            {
                  arr[i] = read();
                  position[ arr[i] ] = i;
                  maxnum = max(maxnum, arr[i]);
            }
            FOR(i, n)
            {
                  int num = arr[i];
//                  FORI(pf, primefactor[num])
//                  {
//                        if (position[ num/pf ] != -1)
//                        {
//                              int u = position[ num/pf ];
//                              int v = i;
//                              graph[u].eb(v);
//                              graph[v].eb(u);
//                        }
//                  }
                  for (int p : primeList)
                  {
                        int num2 = p * num;
                        if (num2 > maxnum) break;
                        if (position[num2] != -1)
                        {
                              int u = i;
                              int v = position[num2];
                              graph[u].eb(v);
                              graph[v].eb(u);
                        }
                  }
            }
            elements_in_left_side = n;
            elements_in_right_side = n;
            initialize_hopcroptcarp();
            int maxmatch = hopcropt_carp();
            int ans = n - maxmatch;
            printf("Case %d: %d\n", tcase, ans);
      }
}