Algorithm


C. k-Multiple Free Set
time limit per test
2 seconds
memory limit per test
256 megabytes
input
standard input
output
standard output

k-multiple free set is a set of integers where there is no pair of integers where one is equal to another integer multiplied by k. That is, there are no two integers x and y (x < y) from the set, such that y = x·k.

You're given a set of n distinct positive integers. Your task is to find the size of it's largest k-multiple free subset.

Input

The first line of the input contains two integers n and k (1 ≤ n ≤ 105, 1 ≤ k ≤ 109). The next line contains a list of n distinct positive integers a1, a2, ..., an (1 ≤ ai ≤ 109).

All the numbers in the lines are separated by single spaces.

Output

On the only line of the output print the size of the largest k-multiple free subset of {a1, a2, ..., an}.

Examples
input
Copy
6 2
2 3 6 5 4 10
output
Copy
3
Note

In the sample input one of the possible maximum 2-multiple free subsets is {4, 5, 6}.



 

Code Examples

#1 Code Example with C++ Programming

Code - C++ Programming

#include <bits/stdc++.h>

using namespace std;

int const N = 1e5 + 1;
int n, k, a[N], nxt[N], res;

int main() {
  scanf("%d %d", &n, &k);
  for(int i = 0; i < n; ++i)
  	scanf("%d", a + i);
  sort(a, a + n);

  memset(nxt, -1, sizeof nxt);
  for(int i = 0, idx; i < n - 1; ++i) {
  	idx = lower_bound(a, a + n, 1ll * a[i] * k) - a;
  	if(idx == i || idx == n || 1ll * a[i] * k != a[idx])
  		continue;
  	nxt[i] = idx;
  }

  for(int i = 0, cur, prv, cnt; i < n; ++i) {
  	if(nxt[i] == 1e9)
  		continue;
  	if(nxt[i] == -1) {
  		++res;
  		continue;
  	}
  	cur = prv = i, cnt = 0;
  	while(nxt[cur] != -1) {
  		if(cnt % 2 == 0)
  			++res;
  		prv = cur, cur = nxt[cur], nxt[prv] = 1e9, ++cnt;
  	}
  	if(cnt + 1 & 1)
  		++res;
  	nxt[cur] = 1e9;
  }

  printf("%d\n", res);

  return 0;
}
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Input

x
+
cmd
6 2
2 3 6 5 4 10

Output

x
+
cmd
3
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Demonstration


Codeforces Solution- k-Multiple Free Set-Solution in C, C++, Java, Python

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