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 Java8 (OpenJDK 1.8.0) (judge verdict: Accepted)

import java.io.OutputStream;
import java.io.IOException;
import java.io.InputStream;
import java.io.OutputStream;
import java.io.IOException;
import java.util.ArrayList;
import java.io.UncheckedIOException;
import java.util.List;
import java.io.Closeable;
import java.io.Writer;
import java.io.OutputStreamWriter;
import java.io.InputStream;

/**
 * Built using CHelper plug-in
 * Actual solution is at the top
 */
public class Main {
    public static void main(String[] args) throws Exception {
        Thread thread = new Thread(null, new TaskAdapter(), "", 1 << 27);
        thread.start();
        thread.join();
    }

    static class TaskAdapter implements Runnable {
        @Override
        public void run() {
            InputStream inputStream = System.in;
            OutputStream outputStream = System.out;
            FastInput in = new FastInput(inputStream);
            FastOutput out = new FastOutput(outputStream);
            DUniquePath solver = new DUniquePath();
            solver.solve(1, in, out);
            out.close();
        }
    }

    static class DUniquePath {
        FastOutput out;

        public void solve(int testNumber, FastInput in, FastOutput out) {
            this.out = out;
            int n = in.readInt();
            long m = in.readLong();
            int q = in.readInt();

            DSU dsu = new DSU(n);
            List<int[]> type0 = new ArrayList<>(q);
            List<int[]> type1 = new ArrayList<>(q);
            for (int i = 0; i < q; i++) {
                int[] ab = new int[]{in.readInt(), in.readInt()};
                if (in.readInt() == 0) {
                    type0.add(ab);
                } else {
                    type1.add(ab);
                }
            }

            for (int[] ab : type0) {
                dsu.merge(ab[0], ab[1]);
            }

            for (int[] ab : type1) {
                if (dsu.find(ab[0]) == dsu.find(ab[1])) {
                    no();
                    return;
                }
            }

            int setCnt = 0;
            for (int i = 0; i < n; i++) {
                if (dsu.find(i) == i) {
                    setCnt++;
                }
            }

            int edge = n - setCnt;
            if (setCnt <= 2 && !type1.isEmpty()) {
                no();
                return;
            }

            long max = 0;
            long min = 0;

            if (!type1.isEmpty()) {
                boolean[] circle = new boolean[n];
                for (int[] ab : type1) {
                    circle[dsu.find(ab[0])] = true;
                    circle[dsu.find(ab[1])] = true;
                }

                int circleCnt = 0;
                for (boolean b : circle) {
                    if (b) {
                        circleCnt++;
                    }
                }
                circleCnt = Math.max(circleCnt, 3);
                min = circleCnt;
            }

            max = choose2(setCnt);

            if (edge + min <= m && m <= edge + max) {
                yes();
                return;
            }
            no();
        }

        public long choose2(long n) {
            return n * (n - 1) / 2;
        }

        public void no() {
            out.println("No");
        }

        public void yes() {
            out.println("Yes");
        }

    }

    static class DSU {
        int[] p;
        int[] rank;

        public DSU(int n) {
            p = new int[n];
            rank = new int[n];
            reset();
        }

        public void reset() {
            for (int i = 0; i < p.length; i++) {
                p[i] = i;
                rank[i] = 0;
            }
        }

        public int find(int a) {
            return p[a] == p[p[a]] ? p[a] : (p[a] = find(p[a]));
        }

        public void merge(int a, int b) {
            a = find(a);
            b = find(b);
            if (a == b) {
                return;
            }
            if (rank[a] == rank[b]) {
                rank[a]++;
            }
            if (rank[a] > rank[b]) {
                p[b] = a;
            } else {
                p[a] = b;
            }
        }

    }

    static class FastOutput implements AutoCloseable, Closeable, Appendable {
        private StringBuilder cache = new StringBuilder(10 << 20);
        private final Writer os;

        public FastOutput append(CharSequence csq) {
            cache.append(csq);
            return this;
        }

        public FastOutput append(CharSequence csq, int start, int end) {
            cache.append(csq, start, end);
            return this;
        }

        public FastOutput(Writer os) {
            this.os = os;
        }

        public FastOutput(OutputStream os) {
            this(new OutputStreamWriter(os));
        }

        public FastOutput append(char c) {
            cache.append(c);
            return this;
        }

        public FastOutput println(String c) {
            cache.append(c);
            println();
            return this;
        }

        public FastOutput println() {
            cache.append(System.lineSeparator());
            return this;
        }

        public FastOutput flush() {
            try {
                os.append(cache);
                os.flush();
                cache.setLength(0);
            } catch (IOException e) {
                throw new UncheckedIOException(e);
            }
            return this;
        }

        public void close() {
            flush();
            try {
                os.close();
            } catch (IOException e) {
                throw new UncheckedIOException(e);
            }
        }

        public String toString() {
            return cache.toString();
        }

    }

    static class FastInput {
        private final InputStream is;
        private byte[] buf = new byte[1 << 20];
        private int bufLen;
        private int bufOffset;
        private int next;

        public FastInput(InputStream is) {
            this.is = is;
        }

        private int read() {
            while (bufLen == bufOffset) {
                bufOffset = 0;
                try {
                    bufLen = is.read(buf);
                } catch (IOException e) {
                    bufLen = -1;
                }
                if (bufLen == -1) {
                    return -1;
                }
            }
            return buf[bufOffset++];
        }

        public void skipBlank() {
            while (next >= 0 && next <= 32) {
                next = read();
            }
        }

        public int readInt() {
            int sign = 1;

            skipBlank();
            if (next == '+' || next == '-') {
                sign = next == '+' ? 1 : -1;
                next = read();
            }

            int val = 0;
            if (sign == 1) {
                while (next >= '0' && next <= '9') {
                    val = val * 10 + next - '0';
                    next = read();
                }
            } else {
                while (next >= '0' && next <= '9') {
                    val = val * 10 - next + '0';
                    next = read();
                }
            }

            return val;
        }

        public long readLong() {
            int sign = 1;

            skipBlank();
            if (next == '+' || next == '-') {
                sign = next == '+' ? 1 : -1;
                next = read();
            }

            long val = 0;
            if (sign == 1) {
                while (next >= '0' && next <= '9') {
                    val = val * 10 + next - '0';
                    next = read();
                }
            } else {
                while (next >= '0' && next <= '9') {
                    val = val * 10 - next + '0';
                    next = read();
                }
            }

            return val;
        }

    }
}