Problem:
Score : 600 points

Problem Statement

There is a tree with N vertices numbered 1 to N.
The i-th edge in this tree connects Vertex a_i and Vertex b_i, and the color and length of that edge are c_i and d_i, respectively.
Here the color of each edge is represented by an integer between 1 and N-1 (inclusive). The same integer corresponds to the same color, and different integers correspond to different colors.

Answer the following Q queries:

Query j (1 \leq j \leq Q): assuming that the length of every edge whose color is x_j is changed to y_j, find the distance between Vertex u_j and Vertex v_j. (The changes of the lengths of edges do not affect the subsequent queries.)

Constraints

2 \leq N \leq 10^5

1 \leq Q \leq 10^5

1 \leq a_i, b_i \leq N

1 \leq c_i \leq N-1

1 \leq d_i \leq 10^4

1 \leq x_j \leq N-1

1 \leq y_j \leq 10^4

1 \leq u_j < v_j \leq N

The given graph is a tree.

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N Q
a_1 b_1 c_1 d_1
:
a_{N-1} b_{N-1} c_{N-1} d_{N-1}
x_1 y_1 u_1 v_1
:
x_Q y_Q u_Q v_Q

Output

Print Q lines. The j-th line (1 \leq j \leq Q) should contain the answer to Query j.

Sample Input 1

5 3
1 2 1 10
1 3 2 20
2 4 4 30
5 2 1 40
1 100 1 4
1 100 1 5
3 1000 3 4

Sample Output 1

130
200
60

The graph in this input is as follows:

Here the edges of Color 1 are shown as solid red lines, the edge of Color 2 is shown as a bold green line, and the edge of Color 4 is shown as a blue dashed line.

Query 1: Assuming that the length of every edge whose color is 1 is changed to 100, the distance between Vertex 1 and Vertex 4 is 100 + 30 = 130.

Query 2: Assuming that the length of every edge whose color is 1 is changed to 100, the distance between Vertex 1 and Vertex 5 is 100 + 100 = 200.

Query 3: Assuming that the length of every edge whose color is 3 is changed to 1000 (there is no such edge), the distance between Vertex 3 and Vertex 4 is 20 + 10 + 30 = 60. Note that the edges of Color 1 now have their original lengths.

Solution in Common Lisp (SBCL 1.1.14) (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
           ;; enclose the form with VALUES to avoid being captured by LOOP macro
           #\# #\> (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)
#-swank (disable-debugger) ; for CS Academy

;; BEGIN_INSERTED_CONTENTS
(defmacro with-buffered-stdout (&body body)
  "Buffers all outputs to *STANDARD-OUTPUT* in BODY and flushes them to
*STANDARD-OUTPUT* after BODY has been done (without error). Note that only
BASE-CHAR is allowed."
  (let ((out (gensym)))
    `(let ((,out (make-string-output-stream :element-type 'base-char)))
       (let ((*standard-output* ,out))
         ,@body)
       (write-string (get-output-stream-string ,out)))))

(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))))))))

;;;
;;; Lowest common ancestor of tree (or forest) by binary lifting
;;; build: O(nlog(n))
;;; query: O(log(n))
;;;

;; PAY ATTENTION TO THE STACK SIZE! THE CONSTRUCTOR DOES DFS.

(deftype lca-vertex-number () '(signed-byte 32))

(defstruct (lca-table
            (:constructor %make-lca-table
                (size
                 &aux
                 ;; requires 1 + log_2{size-1}
                 (max-level (+ 1 (integer-length (- size 2))))
                 (depths (make-array size
                                     :element-type 'lca-vertex-number
                                     :initial-element -1))
                 (parents (make-array (list size max-level)
                                      :element-type 'lca-vertex-number))))
            (:conc-name lca-))
  (max-level nil :type (integer 0 #.most-positive-fixnum))
  (depths nil :type (simple-array lca-vertex-number (*)))
  (parents nil :type (simple-array lca-vertex-number (* *))))

(defun make-lca-table (graph &key root (key #'identity))
  "GRAPH := vector of adjacency lists
ROOT := null | non-negative fixnum

If ROOT is null, this function traverses each connected component of GRAPH from
an arbitrarily picked vertex. Otherwise this function traverses GRAPH only from
ROOT; GRAPH must be tree in the latter case."
  (declare (optimize (speed 3))
           (vector graph)
           (function key)
           ((or null (integer 0 #.most-positive-fixnum)) root))
  (let* ((size (length graph))
         (lca-table (%make-lca-table size))
         (depths (lca-depths lca-table))
         (parents (lca-parents lca-table))
         (max-level (lca-max-level lca-table)))
    (labels ((dfs (v prev-v depth)
               (declare (lca-vertex-number v prev-v))
               (setf (aref depths v) depth)
               (setf (aref parents v 0) prev-v)
               (dolist (node (aref graph v))
                 (let ((dest (funcall key node)))
                   (declare (lca-vertex-number dest))
                   (unless (= dest prev-v)
                     (dfs dest v (+ 1 depth)))))))
      (if root
          (dfs root -1 0)
          (dotimes (v size)
            (when (= (aref depths v) -1)
              (dfs v -1 0))))
      (dotimes (k (- max-level 1))
        (dotimes (v size)
          (if (= -1 (aref parents v k))
              (setf (aref parents v (+ k 1)) -1)
              (setf (aref parents v (+ k 1))
                    (aref parents (aref parents v k) k)))))
      lca-table)))

(define-condition two-vertices-disconnected-error (error)
  ((lca-table :initarg :lca-table :accessor two-vertices-disconnected-error-lca-table)
   (vertex1 :initarg :vertex1 :accessor two-vertices-disconnected-error-vertex1)
   (vertex2 :initarg :vertex2 :accessor two-vertices-disconnected-error-vertex2))
  (:report
   (lambda (c s)
     (format s "~W and ~W are disconnected on lca-table ~W"
             (two-vertices-disconnected-error-vertex1 c)
             (two-vertices-disconnected-error-vertex2 c)
             (two-vertices-disconnected-error-lca-table c)))))

(declaim (ftype (function * (values (integer 0 #.most-positive-fixnum) &optional))
                lca-get-lca))
(defun lca-get-lca (lca-table vertex1 vertex2)
  "Returns the lowest common ancestor of the vertices VERTEX1 and VERTEX2."
  (declare (optimize (speed 3))
           ((and lca-vertex-number (integer 0)) vertex1 vertex2))
  (let* ((u vertex1)
         (v vertex2)
         (depths (lca-depths lca-table))
         (parents (lca-parents lca-table))
         (max-level (lca-max-level lca-table)))
    (declare (lca-vertex-number u v))
    ;; Ensures depth[u] <= depth[v]
    (when (> (aref depths u) (aref depths v))
      (rotatef u v))
    (dotimes (k max-level)
      (when (logbitp k (- (aref depths v) (aref depths u)))
        (setf v (aref parents v k))))
    (if (= u v)
        u
        (loop for k from (- max-level 1) downto 0
              unless (= (aref parents u k) (aref parents v k))
              do (setq u (aref parents u k)
                       v (aref parents v k))
              finally (if (= (aref parents u 0) -1)
                          (error 'two-vertices-disconnected-error
                                 :lca-table lca-table
                                 :vertex1 vertex1
                                 :vertex2 vertex2)
                          (return (aref parents u 0)))))))

(declaim (inline lca-distance))
(defun lca-distance (lca-table u v)
  "Returns the distance between two vertices U and V."
  (declare (optimize (speed 3)))
  (let ((depths (lca-depths lca-table))
        (lca (lca-get-lca lca-table u v)))
    (+ (- (aref depths u) (aref depths lca))
       (- (aref depths v) (aref depths lca)))))

(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
;;;

;; TODO:
;; - more sane handling of unbounded index
;; - handy function for initialization
;; - iteration; map
;; - abstraction
;; - printer

(defconstant +persistent-vector-log+ 16)

(declaim (inline %make-persistent-vector))
(defstruct (persistent-vector (:constructor %make-persistent-vector ())
                              (:conc-name %pv-))
  (value 0 :type fixnum)
  (children nil :type (or null (simple-vector #.+persistent-vector-log+))))

(defun pv-assoc (pvector index value)
  (declare ((or null persistent-vector) pvector)
           ((integer 0 #.most-positive-fixnum) index))
  (labels ((recur (pvector index)
             (declare ((or (integer 0 0) persistent-vector) pvector)
                      ((integer 0 #.most-positive-fixnum) index))
             (let ((res (%make-persistent-vector)))
               (if (eql 0 pvector)
                   (setf (%pv-children res)
                         (make-array +persistent-vector-log+ :initial-element 0))
                   (setf (%pv-children res) (copy-seq (%pv-children pvector))
                         (%pv-value res) (%pv-value pvector)))
               (if (zerop index)
                   (setf (%pv-value res) value)
                   (setf (aref (%pv-children res) (mod index +persistent-vector-log+))
                         (recur (aref (%pv-children res) (mod index +persistent-vector-log+))
                                (floor index +persistent-vector-log+))))
               res)))
    (recur (or pvector 0) index)))

(defun pv-ref (pvector index)
  (declare ((or null persistent-vector) pvector))
  (labels ((recur (pvector index)
             (declare ((or (integer 0 0) persistent-vector) pvector)
                      ((integer 0 #.most-positive-fixnum) index))
             (cond ((eql 0 pvector) 0)
                   ((zerop index) (%pv-value pvector))
                   (t (recur (aref (%pv-children pvector) (mod index +persistent-vector-log+))
                             (floor index +persistent-vector-log+))))))
    (recur (or pvector 0) index)))

(defun main ()
  (let* ((n (read))
         (q (read))
         (graph (make-array n :element-type 'list :initial-element nil))
         (wgraph (make-array n :element-type 'list :initial-element nil)))
    (dotimes (i (- n 1))
      (let ((a (- (read-fixnum) 1))
            (b (- (read-fixnum) 1))
            (c (- (read-fixnum) 1))
            (d (read-fixnum)))
        (push a (aref graph b))
        (push b (aref graph a))
        (push (list* a c d) (aref wgraph b))
        (push (list* b c d) (aref wgraph a))))
    (let ((lca-table (make-lca-table graph))
          (dists (make-array n :element-type 'uint32))
          (ccounts (make-array n :initial-element nil))
          (cdists (make-array n :initial-element nil)))
      (sb-int:named-let dfs ((v 0) (parent -1))
        (dolist (node (aref wgraph v))
          (destructuring-bind (child color . cost) node
            (unless (= child parent)
              (setf (aref dists child) (+ (aref dists v) cost))
              (setf (aref ccounts child)
                    (pv-assoc (aref ccounts v)
                              color
                              (+ 1 (pv-ref (aref ccounts v) color))))
              (setf (aref cdists child)
                    (pv-assoc (aref cdists v)
                              color
                              (+ cost (pv-ref (aref cdists v) color))))
              (dfs child v)))))
      (with-buffered-stdout
        (dotimes (_ q)
          (let* ((x (- (read-fixnum) 1))
                 (y (read-fixnum))
                 (u (- (read-fixnum) 1))
                 (v (- (read-fixnum) 1))
                 (lca (lca-get-lca lca-table u v))
                 (base (+ (aref dists u) (aref dists v) (* -2 (aref dists lca))))
                 (ccount (+ (pv-ref (aref ccounts u) x)
                            (pv-ref (aref ccounts v) x)
                            (* -2 (pv-ref (aref ccounts lca) x))))
                 (cdist (+ (pv-ref (aref cdists u) x)
                           (pv-ref (aref cdists v) x)
                           (* -2 (pv-ref (aref cdists lca) x)))))
            (dbg x y u v lca)
            (dbg base ccount cdist)
            (println (+ (- base cdist) (* ccount y)))))))))

#-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)
    (run-program "powershell.exe" '("-Command" "Get-Clipboard") :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 "5 3
1 2 1 10
1 3 2 20
2 4 4 30
5 2 1 40
1 100 1 4
1 100 1 5
3 1000 3 4
"
    "130
200
60
")))