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 Fortran (gfortran v4.8.4) (judge verdict: Accepted)

module mod_tree_set
  implicit none
  type t_node
    private
    type(t_node), pointer :: left => null(), right => null()
    integer :: level = 1
    integer :: e = 0
  end type
  type t_tree_set
    private
    type(t_node), pointer :: root => null()
    integer :: deflt = -1
  contains
    procedure :: size => size_of
    procedure :: clear => clear
    procedure :: add => add
    procedure :: remove => remove
    procedure :: contain => contain
    procedure :: first => firste
    procedure :: poll_first => poll_first
    procedure :: last => laste
    procedure :: poll_last => poll_last
    procedure :: floor => floore
    procedure :: lower => lowere
    procedure :: ceiling => ceilinge
    procedure :: higher => highere
    procedure :: element_list => element_list
  end type
contains
  function new_node(e) result(n)
    integer, intent(in) :: e
    type(t_node), pointer :: n
    n => null()
    allocate(n)
    n%e = e
  end
  integer function level(n)
    type(t_node), pointer, intent(in) :: n
    level = 0
    if (.not.associated(n)) return
    level = n%level
  end
  logical function is_leaf(n)
    type(t_node), pointer, intent(in) :: n
    is_leaf = associated(n) .and. .not.associated(n%left) .and. &
    & .not.associated(n%right)
  end
  recursive function tree_size(n) result(s)
    type(t_node), pointer, intent(in) :: n
    integer :: s
    s = 0
    if (.not.associated(n)) return
    s = 1+tree_size(n%left)+tree_size(n%right)
  end
  function skew(n) result(l)
    type(t_node), pointer, intent(in) :: n
    type(t_node), pointer :: l
    l => n
    if (.not.(associated(n) .and. associated(n%left) .and. &
    &   n%left%level == n%level)) return
    l => n%left
    n%left => l%right
    l%right => n
  end
  function split(n) result(r)
    type(t_node), pointer, intent(in) :: n
    type(t_node), pointer :: r
    r => n
    if (.not.(associated(n) .and. associated(n%right) .and. &
    &   associated(n%right%right) .and. n%right%right%level == n%level)) return
    r => n%right
    n%right => r%left
    r%left => n
    r%level = r%level+1
  end
  function predecessor(n) result(p)
    type(t_node), pointer, intent(in) :: n
    type(t_node), pointer :: p
    p => null()
    if (.not.associated(n%left)) return
    p => n%left
    do while (associated(p%right))
      p => p%right
    end do
  end
  function successor(n) result(s)
    type(t_node), pointer, intent(in) :: n
    type(t_node), pointer :: s
    s => null()
    if (.not.associated(n%right)) return
    s => n%right
    do while (associated(s%left))
      s => s%left
    end do
  end
  recursive function insert(n,e) result(t)
    type(t_node), pointer, intent(in) :: n
    integer, intent(in) :: e
    type(t_node), pointer :: t
    t => new_node(e)
    if (.not.associated(n)) return
    t => n
    if (e < t%e) then
      t%left => insert(t%left,e)
    else if (e > t%e) then
      t%right => insert(t%right,e)
    end if
    t => skew(t)
    t => split(t)
  end
  recursive function delete(n,e) result(t)
    type(t_node), pointer, intent(in) :: n
    integer, intent(in) :: e
    type(t_node), pointer :: t, l
    t => n
    if (.not.associated(n)) return
    if (e < t%e) then
      t%left => delete(t%left,e)
    else if (e > t%e) then
      t%right => delete(t%right,e)
    else
      if (is_leaf(t)) then
        t => null()
        return
      end if
      if (.not.associated(t%left)) then
        l => successor(t)
        t%right => delete(t%right,l%e)
        t%e = l%e
      else
        l => predecessor(t)
        t%left => delete(t%left,l%e)
        t%e = l%e
      end if
    end if
    t => decrease_level(t)
    t => skew(t)
    t%right => skew(t%right)
    if (associated(t%right)) t%right%right => skew(t%right%right)
    t => split(t)
    t%right => split(t%right)
  end
  function decrease_level(n) result(t)
    type(t_node), pointer, intent(in) :: n
    type(t_node), pointer :: t
    integer :: should_be
    t => n
    should_be = min(level(t%left),level(t%right))+1
    if (t%level > should_be) then
      t%level = should_be
      if (level(t%right) > should_be) t%right%level = should_be
    end if
  end
  recursive subroutine release_tree(t)
    type(t_node), pointer, intent(inout) :: t
    if (.not.associated(t)) return
    call release_tree(t%left)
    call release_tree(t%right)
    deallocate(t)
  end
  recursive subroutine get_element(t,es,num)
    type(t_node), pointer, intent(in) :: t
    integer, intent(inout) :: es(:)
    integer, intent(inout) :: num
    if (.not.associated(t)) return
    call get_element(t%left,es,num)
    num = num+1
    es(num) = t%e
    call get_element(t%right,es,num)
  end
  integer function size_of(this)
    class(t_tree_set), intent(in) :: this
    size_of = tree_size(this%root)
  end
  subroutine clear(this)
    class(t_tree_set), intent(inout) :: this
    call release_tree(this%root)
    this%root => null()
  end
  subroutine set_default(this,deflt)
    class(t_tree_set), intent(inout) :: this
    integer, intent(in) :: deflt
    this%deflt = deflt
  end
  subroutine add(this,e)
    class(t_tree_set), intent(inout) :: this
    integer, intent(in) :: e
    this%root => insert(this%root,e)
  end
  subroutine remove(this,e)
    class(t_tree_set), intent(inout) :: this
    integer, intent(in) :: e
    this%root => delete(this%root,e)
  end
  function contain(this,e) result(ret)
    class(t_tree_set), intent(inout) :: this
    integer, intent(in) :: e
    type(t_node), pointer :: n
    logical :: ret
    ret = .false.
    n => this%root
    do while (associated(n))
      if (e < n%e) then
        n => n%left
      else if (e > n%e) then
        n => n%right
      else
        ret = .true.
        return
      end if
    end do
  end
  function firste(this) result(ret)
    class(t_tree_set), intent(inout) :: this
    type(t_node), pointer :: n
    integer :: ret
    ret = this%deflt
    n => this%root
    if (.not.associated(n)) return
    do while (associated(n%left))
      n => n%left
    end do
    ret = n%e
  end
  function poll_first(this) result(ret)
    class(t_tree_set), intent(inout) :: this
    type(t_node), pointer :: n
    integer :: ret
    ret = firste(this)
    if (ret /= this%deflt) this%root => delete(this%root,ret)
  end
  function laste(this) result(ret)
    class(t_tree_set), intent(inout) :: this
    type(t_node), pointer :: n
    integer :: ret
    ret = this%deflt
    n => this%root
    if (.not.associated(n)) return
    do while (associated(n%right))
      n => n%right
    end do
    ret = n%e
  end
  function poll_last(this) result(ret)
    class(t_tree_set), intent(inout) :: this
    type(t_node), pointer :: n
    integer :: ret
    ret = laste(this)
    if (ret /= this%deflt) this%root => delete(this%root,ret)
  end
  function floore(this,e) result(ret)
    class(t_tree_set), intent(inout) :: this
    integer, intent(in) :: e
    type(t_node), pointer :: n
    integer :: ret
    ret = this%deflt
    n => this%root
    do while (associated(n))
      if (e < n%e) then
        n => n%left
      else if (e > n%e) then
        if (ret == this%deflt) ret = n%e
        if (e-ret > e-n%e) ret = n%e
        n => n%right
      else
        ret = n%e
        return
      end if
    end do
  end
  function lowere(this,e) result(ret)
    class(t_tree_set), intent(inout) :: this
    integer, intent(in) :: e
    integer :: ret
    ret = floore(this,e-1)
  end
  function ceilinge(this,e) result(ret)
    class(t_tree_set), intent(inout) :: this
    integer, intent(in) :: e
    type(t_node), pointer :: n
    integer :: ret
    ret = this%deflt
    n => this%root
    do while (associated(n))
      if (e < n%e) then
        if (ret == this%deflt) ret = n%e
        if (e-ret < e-n%e) ret = n%e
        n => n%left
      else if (e > n%e) then
        n => n%right
      else
        ret = n%e
        return
      end if
    end do
  end
  function highere(this,e) result(ret)
    class(t_tree_set), intent(inout) :: this
    integer, intent(in) :: e
    integer :: ret
    ret = ceilinge(this,e+1)
  end
  subroutine element_list(this,es,num)
    class(t_tree_set), intent(inout) :: this
    integer, intent(out) :: es(:)
    integer, intent(out) :: num
    num = 0
    call get_element(this%root,es,num)
  end
end module mod_tree_set
module mod_union_find
  implicit none
  type union_find
    integer :: n = 0
    integer, pointer :: par(:), rnk(:), siz(:) => null()
  contains
    procedure :: init => inituf
    procedure :: release => releaseuf
    procedure :: find => find
    procedure :: same => same
    procedure :: unite => unite
    procedure :: size => size_of
  end type
  interface union_find
    module procedure :: newuf
  end interface union_find
contains
  function newuf(n) result(ret)
    integer, intent(in) :: n
    type(union_find) :: ret
    integer :: i
    ret%n = n
    allocate(ret%par(n),ret%rnk(n),ret%siz(n))
    ret%rnk = 0
    ret%siz = 1
    ret%par = [(i,i=1,n)]
  end
  subroutine inituf(this,n)
    class(union_find), intent(inout) :: this
    integer, intent(in) :: n
    integer :: i
    if (this%n /= n) then
      if (this%n /= 0) deallocate(this%par,this%rnk,this%siz)
      allocate(this%par(n),this%rnk(n),this%siz(n))
    end if
    this%n = n
    this%rnk = 0
    this%siz = 1
    this%par = [(i,i=1,n)]
  end
  subroutine releaseuf(this)
    class(union_find), intent(inout) :: this
    if (this%n /= 0) deallocate(this%par,this%rnk,this%siz)
    this%n = 0
  end
  recursive function find(this,i) result(ret)
    class(union_find), intent(inout) :: this
    integer, intent(in) :: i
    integer :: ret
    if (this%par(i) /= i) this%par(i) = find(this,this%par(i))
    ret = this%par(i)
  end
  logical function same(this,i,j)
    class(union_find), intent(inout) :: this
    integer, intent(in) :: i, j
    same = find(this,i) == find(this,j)
  end
  subroutine unite(this,i,j)
    class(union_find), intent(inout) :: this
    integer, intent(in) :: i, j
    integer :: x, y
    x = find(this,i)
    y = find(this,j)
    if (x == y) return
    if (this%rnk(x) < this%rnk(y)) then
      this%par(x) = y
      this%siz(y) = this%siz(y)+this%siz(x)
    else
      this%par(y) = x
      this%siz(x) = this%siz(x)+this%siz(y)
      if (this%rnk(x) == this%rnk(y)) this%rnk(x) = this%rnk(x)+1
    end if
  end
  integer function size_of(this,i)
    class(union_find), intent(inout) :: this
    integer, intent(in) :: i
    size_of = this%siz(find(this,i))
  end
end module mod_union_find
program unique_path
  use mod_tree_set
  use mod_union_find
  implicit none
  type(t_tree_set) :: set
  type(union_find) :: uf
  integer :: n, q, a(100000) = 0, b(100000) = 0, c(100000) = 0, k, i
  integer(8) :: m
  read(*,*) n, m, q
  uf = union_find(n)
  do i = 1, q
    read(*,*) a(i), b(i), c(i)
    a(i) = a(i)+1
    b(i) = b(i)+1
    if (c(i) == 0) call uf%unite(a(i),b(i))
  end do
  if (m == n-1) then
    if (any(c(1:q) == 1)) then
      write(*,'(a)') "No"
    else
      write(*,'(a)') "Yes"
    end if
    stop
  end if
  do i = 1, q
    if (c(i) == 1 .and. uf%same(a(i),b(i))) then
      write(*,'(a)') "No"
      stop
    end if
  end do
  do i = 1, n
    call set%add(uf%find(i))
  end do
  k = set%size()
  if (m <= int(n,8)+int(k,8)*(int(k,8)-3_8)/2_8) then
    write(*,'(a)') "Yes"
  else
    write(*,'(a)') "No"
  end if
end program unique_path