Problem:
Score : 700 points

Problem Statement

We have a graph with N vertices and M edges, and there are two people on the graph: Takahashi and Aoki.

The i-th edge connects Vertex U_i and Vertex V_i.
The time it takes to traverse this edge is D_i minutes, regardless of direction and who traverses the edge (Takahashi or Aoki).

Takahashi departs Vertex S and Aoki departs Vertex T at the same time. Takahashi travels to Vertex T and Aoki travels to Vertex S, both in the shortest time possible.
Find the number of the pairs of ways for Takahashi and Aoki to choose their shortest paths such that they never meet (at a vertex or on an edge) during the travel, modulo 10^9 + 7.

Constraints

1 \leq N \leq 100 000

1 \leq M \leq 200 000

1 \leq S, T \leq N

S \neq T

1 \leq U_i, V_i \leq N (1 \leq i \leq M)

1 \leq D_i \leq 10^9 (1 \leq i \leq M)

If i \neq j, then (U_i, V_i) \neq (U_j, V_j) and (U_i, V_i) \neq (V_j, U_j).

U_i \neq V_i (1 \leq i \leq M)

D_i are integers.

The given graph is connected.

Input

Input is given from Standard Input in the following format:

N M
S T
U_1 V_1 D_1
U_2 V_2 D_2
:
U_M V_M D_M

Output

Print the answer.

Sample Input 1

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

Sample Output 1

2

There are two ways to choose shortest paths that satisfies the condition:

Takahashi chooses the path 1 \rightarrow 2 \rightarrow 3, and Aoki chooses the path 3 \rightarrow 4 \rightarrow 1.

Takahashi chooses the path 1 \rightarrow 4 \rightarrow 3, and Aoki chooses the path 3 \rightarrow 2 \rightarrow 1.

Sample Input 2

3 3
1 3
1 2 1
2 3 1
3 1 2

Sample Output 2

2

Sample Input 3

3 3
1 3
1 2 1
2 3 1
3 1 2

Sample Output 3

2

Sample Input 4

8 13
4 2
7 3 9
6 2 3
1 6 4
7 6 9
3 8 9
1 2 2
2 8 12
8 6 9
2 5 5
4 2 18
5 3 7
5 1 515371567
4 8 6

Sample Output 4

6

Solution in Common Lisp (SBCL 2.0.3) (judge verdict: Accepted)

#-swank
(unless (member :child-sbcl *features*)
  (quit
   :unix-status
   (process-exit-code
    (run-program *runtime-pathname*
                 `("--control-stack-size" "128MB"
                   "--noinform" "--disable-ldb" "--lose-on-corruption" "--end-runtime-options"
                   "--eval" "(push :child-sbcl *features*)"
                   "--script" ,(namestring *load-pathname*))
                 :output t :error t :input t))))
(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)) `(values ,(read s nil nil t)))))
#+swank (cl-syntax:use-syntax cl-debug-print:debug-print-syntax)

(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)

;; BEGIN_INSERTED_CONTENTS
(declaim (ftype (function * (values fixnum &optional)) read-fixnum))
(defun read-fixnum (&optional (in *standard-input*))
  "NOTE: cannot read -2^62"
  (declare #.opt)
  (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 #\-))
                                  (setq 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))))))))

;;;
;;; Arithmetic operations with static modulus
;;;

;; FIXME: Currently MOD* and MOD+ doesn't apply MOD when the number of
;; parameters is one.
(defmacro define-mod-operations (divisor)
  `(progn
     (defun mod* (&rest args)
       (reduce (lambda (x y) (mod (* x y) ,divisor)) args))

     (defun mod+ (&rest args)
       (reduce (lambda (x y) (mod (+ x y) ,divisor)) args))

     #+sbcl
     (eval-when (:compile-toplevel :load-toplevel :execute)
       (locally (declare (muffle-conditions warning))
         (sb-c:define-source-transform mod* (&rest args)
           (if (null args)
               1
               (reduce (lambda (x y) `(mod (* ,x ,y) ,',divisor)) args)))
         (sb-c:define-source-transform mod+ (&rest args)
           (if (null args)
               0
               (reduce (lambda (x y) `(mod (+ ,x ,y) ,',divisor)) args)))))

     (define-modify-macro incfmod (delta)
       (lambda (x y) (mod (+ x y) ,divisor)))

     (define-modify-macro decfmod (delta)
       (lambda (x y) (mod (- x y) ,divisor)))

     (define-modify-macro mulfmod (multiplier)
       (lambda (x y) (mod (* x y) ,divisor)))))

;;;
;;; Binary heap
;;;

(define-condition heap-empty-error (error)
  ((heap :initarg :heap :reader heap-empty-error-heap))
  (:report
   (lambda (condition stream)
     (format stream "Attempted to pop empty heap ~W" (heap-empty-error-heap condition)))))

(defmacro define-binary-heap (name &key (order '#'>) (element-type 'fixnum))
  "Defines a binary heap specialized for the given order and the element
type. This macro defines a structure of the given NAME and relevant functions:
MAKE-<NAME>, <NAME>-PUSH, <NAME>-POP, <NAME>-REINITIALIZE, <NAME>-EMPTY-P,
<NAME>-COUNT, and <NAME>-PEEK."
  (check-type name symbol)
  (let* ((string-name (string name))
         (fname-push (intern (format nil "~A-PUSH" string-name)))
         (fname-pop (intern (format nil "~A-POP" string-name)))
         (fname-reinitialize (intern (format nil "~A-REINITIALIZE" string-name)))
         (fname-empty-p (intern (format nil "~A-EMPTY-P" string-name)))
         (fname-count (intern (format nil "~A-COUNT" string-name)))
         (fname-peek (intern (format nil "~A-PEEK" string-name)))
         (fname-make (intern (format nil "MAKE-~A" string-name)))
         (acc-position (intern (format nil "~A-POSITION" string-name)))
         (acc-data (intern (format nil "~A-DATA" string-name))))
    `(progn
       (locally
           ;; prevent style warnings
           (declare #+sbcl (muffle-conditions style-warning))
         (defstruct (,name
                     (:constructor ,fname-make
                         (size
                          &aux
                          (data (make-array (1+ size)
                                            :element-type ',(if (eql element-type '*) t  element-type))))))
           (data nil :type (simple-array ,element-type (*)))
           (position 1 :type (integer 1 #.array-total-size-limit))))

       (declaim #+sbcl (sb-ext:maybe-inline ,fname-push))
       (defun ,fname-push (obj heap)
         "Adds OBJ to HEAP."
         (declare (optimize (speed 3))
                  (type ,name heap))
         (symbol-macrolet ((position (,acc-position heap)))
           (when (>= position (length (,acc-data heap)))
             (setf (,acc-data heap)
                   (adjust-array (,acc-data heap)
                                 (min (- array-total-size-limit 1)
                                      (* position 2)))))
           (let ((data (,acc-data heap)))
             (declare ((simple-array ,element-type (*)) data))
             (labels ((heapify (pos)
                        (declare (optimize (speed 3) (safety 0)))
                        (unless (= pos 1)
                          (let ((parent-pos (ash pos -1)))
                            (when (funcall ,order (aref data pos) (aref data parent-pos))
                              (rotatef (aref data pos) (aref data parent-pos))
                              (heapify parent-pos))))))
               (setf (aref data position) obj)
               (heapify position)
               (incf position)
               heap))))

       (declaim #+sbcl (sb-ext:maybe-inline ,fname-pop))
       (defun ,fname-pop (heap)
         "Removes and returns the element at the top of HEAP."
         (declare (optimize (speed 3))
                  (type ,name heap))
         (symbol-macrolet ((position (,acc-position heap)))
           (let ((data (,acc-data heap)))
             (declare ((simple-array ,element-type (*)) data))
             (labels ((heapify (pos)
                        (declare (optimize (speed 3) (safety 0))
                                 ((integer 1 #.array-total-size-limit) pos))
                        (let* ((child-pos1 (+ pos pos))
                               (child-pos2 (1+ child-pos1)))
                          (when (<= child-pos1 position)
                            (if (<= child-pos2 position)
                                (if (funcall ,order (aref data child-pos1) (aref data child-pos2))
                                    (unless (funcall ,order (aref data pos) (aref data child-pos1))
                                      (rotatef (aref data pos) (aref data child-pos1))
                                      (heapify child-pos1))
                                    (unless (funcall ,order (aref data pos) (aref data child-pos2))
                                      (rotatef (aref data pos) (aref data child-pos2))
                                      (heapify child-pos2)))
                                (unless (funcall ,order (aref data pos) (aref data child-pos1))
                                  (rotatef (aref data pos) (aref data child-pos1))))))))
               (when (= position 1)
                 (error 'heap-empty-error :heap heap))
               (prog1 (aref data 1)
                 (decf position)
                 (setf (aref data 1) (aref data position))
                 (heapify 1))))))

       (declaim (inline ,fname-reinitialize))
       (defun ,fname-reinitialize (heap)
         "Makes HEAP empty."
         (setf (,acc-position heap) 1)
         heap)

       (declaim (inline ,fname-empty-p))
       (defun ,fname-empty-p (heap)
         "Returns true iff HEAP is empty."
         (= 1 (,acc-position heap)))

       (declaim (inline ,fname-count))
       (defun ,fname-count (heap)
         "Returns the current number of the elements in HEAP."
         (- (,acc-position heap) 1))

       (declaim (inline ,fname-peek))
       (defun ,fname-peek (heap)
         "Returns the topmost element of HEAP."
         (if (= 1 (,acc-position heap))
             (error 'heap-empty-error :heap heap)
             (aref (,acc-data heap) 1))))))

(in-package :cl-user)

(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)))

(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
;;;

;; dist . v
(define-binary-heap heap
  :order (lambda (x y)
           (< (the fixnum (car x)) (the fixnum (car y))))
  :element-type list)

(define-mod-operations +mod+)

(defconstant +inf+ most-positive-fixnum)
(defun make-dists (n m src graph)
  (declare #.opt
           ((simple-array list (*)) graph))
  (let* ((dists (make-array n :element-type 'uint62 :initial-element +inf+))
         (que (make-heap m)))
    (setf (aref dists src) 0)
    (heap-push (cons 0 src) que)
    (loop until (heap-empty-p que)
          for (dist . v) of-type (uint62 . uint62) = (heap-pop que)
          when (= dist (aref dists v))
          do (loop for (next . d) of-type (uint62 . uint62) in (aref graph v)
                   for next-dist = (+ d dist)
                   when (< next-dist (aref dists next))
                   do (setf (aref dists next) next-dist)
                      (heap-push (cons next-dist next) que)))
    (let ((dp (make-array n :element-type 'uint31 :initial-element 0))
          (visited (make-array n :element-type 'bit :initial-element 0))
          (ords (make-array n :element-type 'uint31 :initial-element 0))
          (index (- n 1)))
      (declare (int32 index))
      (setf (aref dp src) 1)
      (sb-int:named-let dfs ((v src))
        (when (zerop (aref visited v))
          (setf (aref visited v) 1)
          (loop for (next . d) of-type (uint62 . uint62) in (aref graph v)
                when (= (+ (aref dists v) d) (aref dists next))
                do (dfs next))
          (setf (aref ords index) v
                index (- index 1))))
      (sb-int:dovector (v ords)
        (loop for (next . d) of-type (uint62 . uint62) in (aref graph v)
              when (= (+ (aref dists v) d) (aref dists next))
              do (incfmod (aref dp next) (aref dp v))))
      (values dists dp))))

(defun main ()
  (declare #.opt)
  (let* ((n (read))
         (m (read))
         (ss (- (read) 1))
         (tt (- (read) 1))
         (graph (make-array n :element-type 'list :initial-element nil)))
    (declare (uint31 n m ss tt))
    (dotimes (i m)
      (let ((u (- (read-fixnum) 1))
            (v (- (read-fixnum) 1))
            (d (read-fixnum)))
        (push (cons u d) (aref graph v))
        (push (cons v d) (aref graph u))))
    (multiple-value-bind (dists+ dp+) (make-dists n m ss graph)
      (multiple-value-bind (dists- dp-) (make-dists n m tt graph)
        (declare ((simple-array uint62 (*)) dists+ dists-)
                 ((simple-array uint31 (*)) dp+ dp-))
        (let ((l (aref dists+ tt))
              (res (mod* (aref dp+ tt) (aref dp- ss))))
          (declare (uint31 res))
          (dotimes (v n)
            (when (= (+ (aref dists+ v) (aref dists- v)) l)
              (when (= (* 2 (aref dists+ v)) l)
                (decfmod res (mod* (aref dp+ v) (aref dp- v)
                                   (aref dp+ v) (aref dp- v))))
              (when (< (* 2 (aref dists+ v)) l)
                (loop for (next . d) in (aref graph v)
                      when (and (< (* 2 (aref dists- next)) l)
                                (= l (+ (aref dists+ v) (aref dists- next) d)))
                      do (decfmod res (mod* (aref dp+ v) (aref dp- next)
                                            (aref dp+ v) (aref dp- next)))))))
          (println res))))))

#-swank (main)

;;;
;;; Test and benchmark
;;;

#+swank
(defun io-equal (in-string out-string &key (function #'main) (test #'equal))
  "Passes IN-STRING to *STANDARD-INPUT*, executes FUNCTION, and returns true if
the string output to *STANDARD-OUTPUT* is equal to OUT-STRING."
  (labels ((ensure-last-lf (s)
             (if (eql (uiop:last-char s) #\Linefeed)
                 s
                 (uiop:strcat s uiop:+lf+))))
    (funcall test
             (ensure-last-lf out-string)
             (with-output-to-string (out)
               (let ((*standard-output* out))
                 (with-input-from-string (*standard-input* (ensure-last-lf in-string))
                   (funcall function)))))))

#+swank
(defun get-clipbrd ()
  (with-output-to-string (out)
    #+os-windows (run-program "powershell.exe" '("-Command" "Get-Clipboard") :output out :search t)
    #+os-unix (run-program "xsel" '("-b" "-o") :output out :search t)))

#+swank (defparameter *this-pathname* (uiop:current-lisp-file-pathname))
#+swank (defparameter *dat-pathname* (uiop:merge-pathnames* "test.dat" *this-pathname*))

#+swank
(defun run (&optional thing (out *standard-output*))
  "THING := null | string | symbol | pathname

null: run #'MAIN using the text on clipboard as input.
string: run #'MAIN using the string as input.
symbol: alias of FIVEAM:RUN!.
pathname: run #'MAIN using the text file as input."
  (let ((*standard-output* out))
    (etypecase thing
      (null
       (with-input-from-string (*standard-input* (delete #\Return (get-clipbrd)))
         (main)))
      (string
       (with-input-from-string (*standard-input* (delete #\Return thing))
         (main)))
      (symbol (5am:run! thing))
      (pathname
       (with-open-file (*standard-input* thing)
         (main))))))

#+swank
(defun gen-dat ()
  (uiop:with-output-file (out *dat-pathname* :if-exists :supersede)
    (format out "")))

#+swank
(defun bench (&optional (out (make-broadcast-stream)))
  (time (run *dat-pathname* out)))

;; To run: (5am:run! :sample)
#+swank
(it.bese.fiveam:test :sample
  (it.bese.fiveam:is
   (common-lisp-user::io-equal "4 4
1 3
1 2 1
2 3 1
3 4 1
4 1 1
"
    "2
"))
  (it.bese.fiveam:is
   (common-lisp-user::io-equal "3 3
1 3
1 2 1
2 3 1
3 1 2
"
    "2
"))
  (it.bese.fiveam:is
   (common-lisp-user::io-equal "8 13
4 2
7 3 9
6 2 3
1 6 4
7 6 9
3 8 9
1 2 2
2 8 12
8 6 9
2 5 5
4 2 18
5 3 7
5 1 515371567
4 8 6
"
    "6
")))