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 Common Lisp (SBCL 1.1.14) (judge verdict: Runtime Error)

;; -*- coding: utf-8 -*-
(eval-when (:compile-toplevel :load-toplevel :execute)
  (sb-int:defconstant-eqx OPT
    #+swank '(optimize (speed 3) (safety 2))
    #-swank '(optimize (speed 3) (safety 0) (debug 0))
    #'equal)
  #+swank (ql:quickload '(:cl-debug-print :fiveam) :silent t)
  #-swank (set-dispatch-macro-character
           #\# #\> (lambda (s c p) (declare (ignore c p)) (read s nil nil t))))
#+swank (cl-syntax:use-syntax cl-debug-print:debug-print-syntax)
#-swank (disable-debugger) ; for CS Academy

;; BEGIN_INSERTED_CONTENTS
;;;
;;; Disjoint set by Union-Find algorithm
;;;

(defstruct (disjoint-set
            (:constructor make-disjoint-set
                (size &aux (data (make-array size :element-type 'fixnum :initial-element -1))))
            (:conc-name ds-))
  (data nil :type (simple-array fixnum (*))))

(declaim (ftype (function * (values (mod #.array-total-size-limit) &optional)) ds-root))
(defun ds-root (x disjoint-set)
  "Returns the root of X."
  (declare (optimize (speed 3))
           ((mod #.array-total-size-limit) x))
  (let ((data (ds-data disjoint-set)))
    (if (< (aref data x) 0)
        x
        (setf (aref data x)
              (ds-root (aref data x) disjoint-set)))))

(declaim (inline ds-unite!))
(defun ds-unite! (x1 x2 disjoint-set)
  "Destructively unites X1 and X2 and returns true iff X1 and X2 become
connected for the first time."
  (let ((root1 (ds-root x1 disjoint-set))
        (root2 (ds-root x2 disjoint-set)))
    (unless (= root1 root2)
      (let ((data (ds-data disjoint-set)))
        ;; ensure the size of root1 >= the size of root2
        (when (> (aref data root1) (aref data root2))
          (rotatef root1 root2))
        (incf (aref data root1) (aref data root2))
        (setf (aref data root2) root1)))))

(declaim (inline ds-connected-p))
(defun ds-connected-p (x1 x2 disjoint-set)
  "Returns true iff X1 and X2 have the same root."
  (= (ds-root x1 disjoint-set) (ds-root x2 disjoint-set)))

(declaim (inline ds-size))
(defun ds-size (x disjoint-set)
  "Returns the size of the connected component to which X belongs."
  (- (aref (ds-data disjoint-set)
           (ds-root x disjoint-set))))

(declaim (ftype (function * (values fixnum &optional)) read-fixnum))
(defun read-fixnum (&optional (in *standard-input*))
  (macrolet ((%read-byte ()
               `(the (unsigned-byte 8)
                     #+swank (char-code (read-char in nil #\Nul))
                     #-swank (sb-impl::ansi-stream-read-byte in nil #.(char-code #\Nul) nil))))
    (let* ((minus nil)
           (result (loop (let ((byte (%read-byte)))
                           (cond ((<= 48 byte 57)
                                  (return (- byte 48)))
                                 ((zerop byte) ; #\Nul
                                  (error "Read EOF or #\Nul."))
                                 ((= byte #.(char-code #\-))
                                  (setf minus t)))))))
      (declare ((integer 0 #.most-positive-fixnum) result))
      (loop
        (let* ((byte (%read-byte)))
          (if (<= 48 byte 57)
              (setq result (+ (- byte 48)
                              (* 10 (the (integer 0 #.(floor most-positive-fixnum 10)) result))))
              (return (if minus (- result) result))))))))

(defmacro dbg (&rest forms)
  #+swank
  (if (= (length forms) 1)
      `(format *error-output* "~A => ~A~%" ',(car forms) ,(car forms))
      `(format *error-output* "~A => ~A~%" ',forms `(,,@forms)))
  #-swank (declare (ignore forms)))

(defmacro define-int-types (&rest bits)
  `(progn
     ,@(mapcar (lambda (b) `(deftype ,(intern (format nil "UINT~A" b)) () '(unsigned-byte ,b))) bits)
     ,@(mapcar (lambda (b) `(deftype ,(intern (format nil "INT~A" b)) () '(signed-byte ,b))) bits)))
(define-int-types 2 4 7 8 15 16 31 32 62 63 64)

(declaim (inline println))
(defun println (obj &optional (stream *standard-output*))
  (let ((*read-default-float-format* 'double-float))
    (prog1 (princ obj stream) (terpri stream))))

(defconstant +mod+ 1000000007)

;;;
;;; Body
;;;

(declaim (inline =>))
(defun => (x y)
  (or (not x) y))

(defun main ()
  (let* ((n (read))
         (m (read))
         (q (read))
         (as (make-array q :element-type 'uint32))
         (bs (make-array q :element-type 'uint32))
         (cs (make-array q :element-type 'uint32))
         (dset (make-disjoint-set n))
         (table (make-hash-table :test #'eq :size n)))
    (declare (uint31 n m q))
    (dotimes (i q)
      (let ((a (read-fixnum ))
            (b (read-fixnum))
            (c (read-fixnum)))
        (setf (aref as i) a
              (aref bs i) b
              (aref cs i) c)
        (when (zerop c)
          (ds-unite! b c dset))))
    (dotimes (i n)
      (setf (gethash (ds-root i dset) table) t))
    (let ((k (hash-table-count table)))
      (declare (uint31 k))
      (write-line
       (cond ((= m (- n 1))
              (if (zerop (count 1 cs))
                  "Yes"
                  "No"))
             ((loop for i below q
                    for a = (aref as i)
                    for b = (aref bs i)
                    for c = (aref cs i)
                    thereis (and (= c 1)
                                 (ds-connected-p a b dset)))
              "No")
             (;; N-k+k <= M <= N-k + 1/2k(k-1)
              (<= m (+ n (floor (* k (- k 3)) 2)))
              "Yes")
             (t "No"))))))

#-swank (main)