Problem:
Score : 700 points

Problem Statement

Snuke's mother gave Snuke an undirected graph consisting of N vertices numbered 0 to N-1 and M edges.
This graph was connected and contained no parallel edges or self-loops.

One day, Snuke broke this graph.
Fortunately, he remembered Q clues about the graph.
The i-th clue (0 \leq i \leq Q-1) is represented as integers A_i,B_i,C_i and means the following:

If C_i=0: there was exactly one simple path (a path that never visits the same vertex twice) from Vertex A_i to B_i.

If C_i=1: there were two or more simple paths from Vertex A_i to B_i.

Snuke is not sure if his memory is correct, and worried whether there is a graph that matches these Q clues.
Determine if there exists a graph that matches Snuke's memory.

Constraints

2 \leq N \leq 10^5

N-1 \leq M \leq N \times (N-1)/2

1 \leq Q \leq 10^5

0 \leq A_i,B_i \leq N-1

A_i \neq B_i

0 \leq C_i \leq 1

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N M Q
A_0 B_0 C_0
A_1 B_1 C_1
\vdots
A_{Q-1} B_{Q-1} C_{Q-1}

Output

If there exists a graph that matches Snuke's memory, print Yes; otherwise, print No.

Sample Input 1

5 5 3
0 1 0
1 2 1
2 3 0

Sample Output 1

Yes

For example, consider a graph with edges (0,1),(1,2),(1,4),(2,3),(2,4). This graph matches the clues.

Sample Input 2

4 4 3
0 1 0
1 2 1
2 3 0

Sample Output 2

No

Sample Input 3

10 9 9
7 6 0
4 5 1
9 7 0
2 9 0
2 3 0
4 1 0
8 0 0
9 1 0
3 0 0

Sample Output 3

No

Solution in C++14 (GCC 5.4.1) (judge verdict: Wrong Answer)

#include <iostream>
#include <vector>

struct DJSet {
  int n;
  std::vector<int> upper;

  DJSet(int n) : upper(n) {
    for (int i=0;i<n;++i) upper[i]=-1;
  }

  int root(int x) {
    return upper[x]<0?x:(upper[x]=root(upper[x]));
  }

  bool equiv(int x, int y) {
    return root(x)==root(y);
  }

  void setUnion(int x, int y) {
    x=root(x);
    y=root(y);
    if (x==y) return;
    if (upper[x]>upper[y]) {
      x^=y;
      y^=x;
      x^=y;
    }
    upper[x]+=upper[y];
    upper[y]=x;
  }
};

int main() {
  int N, M, Q;
  std::cin >> N >> M >> Q;
  DJSet ds(N);
  std::vector<int> A(Q);
  std::vector<int> B(Q);
  std::vector<int> C(Q);
  for (int i=0;i<Q;++i) {
    std::cin >> A[i] >> B[i] >> C[i];
    if (C[i]==0) {
      ds.setUnion(A[i], B[i]);
    }
  }
  bool flag=false;
  for (int i=0;i<Q;++i) {
    if (C[i]==1) {
      flag=true;
      if (ds.equiv(A[i], B[i])) {
	std::cout << "No" << std::endl;
	return 0;
      }
    }
  }
  int num=0;
  for (int i=0;i<N;++i) {
    if (ds.upper[i]<0) ++num;
  }
  if (N-1+num*(num-1)/2<M) {
    std::cout << "No" << std::endl;
  }
  if (M<N-1+(flag?1:0)) {
    std::cout << "No" << std::endl;
    return 0;
  }
  if (num==2) {
    std::cout << "No" << std::endl;
    return 0;
  }
  std::cout << "Yes" << std::endl;
  return 0;
}