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 Rust (1.15.1) (judge verdict: Accepted)

// https://atcoder.jp/contests/agc038/tasks/agc038_d
//
#![allow(unused_imports)]
use std::io::*;
use std::fmt::*;
use std::str::*;
use std::cmp::*;
use std::collections::*;

macro_rules! input {
    (source = $s:expr, $($r:tt)*) => {
        let mut iter = $s.split_whitespace();
        input_inner!{iter, $($r)*}
    };
    ($($r:tt)*) => {
        let s = {
            use std::io::Read;
            let mut s = String::new();
            std::io::stdin().read_to_string(&mut s).unwrap();
            s
        };
        let mut iter = s.split_whitespace();
        input_inner!{iter, $($r)*}
    };
}

macro_rules! input_inner {
    ($iter:expr) => {};
    ($iter:expr, ) => {};

    ($iter:expr, $var:ident : $t:tt $($r:tt)*) => {
        let $var = read_value!($iter, $t);
        input_inner!{$iter $($r)*}
    };
}

macro_rules! read_value {
    ($iter:expr, ( $($t:tt),* )) => {
        ( $(read_value!($iter, $t)),* )
    };

    ($iter:expr, [ $t:tt ; $len:expr ]) => {
        (0..$len).map(|_| read_value!($iter, $t)).collect::<Vec<_>>()
    };

    ($iter:expr, [ next / $t:tt ]) => {
        {
            let len = read_value!($iter, usize);
            (0..len).map(|_| read_value!($iter, $t)).collect::<Vec<_>>()
        }
    };

    ($iter:expr, chars) => {
        read_value!($iter, String).chars().collect::<Vec<char>>()
    };

    ($iter:expr, usize1) => {
        read_value!($iter, usize) - 1
    };

    ($iter:expr, $t:ty) => {
        $iter.next().unwrap().parse::<$t>().expect("Parse error")
    };
}

#[allow(unused_macros)]
macro_rules! dvec {
    ($t:expr ; $len:expr) => {
        vec![$t; $len]
    };

    ($t:expr ; $len:expr, $($rest:expr),*) => {
        vec![dvec!($t; $($rest),*); $len]
    };
}

#[allow(unused_macros)]
macro_rules! ifv {
    ($t:expr, $a:expr, $b: expr) => {
        if $t { $a } else { $b }
    }
}

#[allow(unused_macros)]
macro_rules! fill {
    ($t:expr, $v:expr) => {
        for i in 0..$t.len() {
            $t[i] = $v;
        }
    };
}

#[allow(unused_macros)]
macro_rules! join {
    ($t:expr, $glue:expr) => {
        $t.into_iter().map(|w| w.to_string()).collect::<Vec<_>>().join($glue)
    };
}

#[allow(unused_macros)]
macro_rules! debug {
    ($($a:expr),*) => {
        println!(concat!($(stringify!($a), " = {:?}, "),*), $($a),*);
    }
}

struct UnionFind {
      n: usize,
      parent: Vec<usize>,
      count: Vec<usize>
}

impl UnionFind {
     fn new(n: usize) -> Self {
           let mut p = vec![0; n];
           for i in 0..n {
                 p[i] = i;
           }
           UnionFind { n: n, parent: p, count: vec![1; n] }
     }

     fn find(&mut self, a: usize) -> usize {
           if self.parent[a] == a {
                 return a;
           }
           let par = self.parent[a];
           self.parent[a] = self.find(par);
           self.parent[a]
     }

     fn unite(&mut self, a: usize, b: usize) {
           let a = self.find(a);
           let b = self.find(b);
           if a == b {
                 return;
           }
           let ca = self.count[a];
           let cb = self.count[b];
           let total = ca + cb;
           self.count[b] = total;
           self.count[a] = total;
           if ca < cb {
                 self.parent[a] = b;
           } else {
                 self.parent[b] = a;
           }
     }

     fn same(&mut self, a: usize, b: usize) -> bool {
           self.find(a) == self.find(b)
     }

     fn count(&mut self, a: usize) -> usize {
           let a = self.find(a);
           self.count[a]
     }
}

type Info = (usize, usize, i32);

fn solve(n: usize, m: usize, mut zeros: Vec<Info>, mut ones: Vec<Info>) -> bool {
    zeros.dedup();
    ones.dedup();

    let mut uf0 = UnionFind::new(n);
    let mut fedges = 0;

    let mut cmp = n;
    for z in zeros {
        if uf0.same(z.0, z.1) {
            continue;
        }
        uf0.unite(z.0, z.1);
        fedges += 1;
        cmp -= 1;
    }
    if ones.len() == 0 {
        let wedge = cmp * (cmp - 1) / 2;
        return m <= wedge + fedges;
    }
    for o in ones {
        if uf0.same(o.0, o.1) {
            return false;
        }
    }

    if cmp == 2 {
        return false;
    }

    let wedge = cmp * (cmp - 1) / 2;
    return m <= wedge + fedges && m >= n;
}

fn main() {
    input! {
        n: usize, m: usize, q: usize,
        info: [(usize, usize, i32); q]
    };
    let zeros = info.clone().into_iter().filter(|&(a, b, c)| c == 0).collect::<Vec<_>>();
    let ones  = info.clone().into_iter().filter(|&(a, b, c)| c == 1).collect::<Vec<_>>();

    println!("{}", ifv!(solve(n, m, zeros, ones), "Yes", "No"));
}