Problem:
Score : 700 points

Problem Statement

Nuske has a grid with N rows and M columns of squares. The rows are numbered 1 through N from top to bottom, and the columns are numbered 1 through M from left to right.
Each square in the grid is painted in either blue or white. If S_{i,j} is 1, the square at the i-th row and j-th column is blue; if S_{i,j} is 0, the square is white.
For every pair of two blue square a and b, there is at most one path that starts from a, repeatedly proceeds to an adjacent (side by side) blue square and finally reaches b, without traversing the same square more than once.

Phantom Thnook, Nuske's eternal rival, gives Q queries to Nuske. The i-th query consists of four integers x_{i,1}, y_{i,1}, x_{i,2} and y_{i,2} and asks him the following: when the rectangular region of the grid bounded by (and including) the x_{i,1}-th row, x_{i,2}-th row, y_{i,1}-th column and y_{i,2}-th column is cut out, how many connected components consisting of blue squares there are in the region?

Process all the queries.

Constraints

1 ≤ N,M ≤ 2000

1 ≤ Q ≤ 200000

S_{i,j} is either 0 or 1.

S_{i,j} satisfies the condition explained in the statement.

1 ≤ x_{i,1} ≤ x_{i,2} ≤ N(1 ≤ i ≤ Q)

1 ≤ y_{i,1} ≤ y_{i,2} ≤ M(1 ≤ i ≤ Q)

Input

The input is given from Standard Input in the following format:

N M Q
S_{1,1}..S_{1,M}
:
S_{N,1}..S_{N,M}
x_{1,1} y_{i,1} x_{i,2} y_{i,2}
:
x_{Q,1} y_{Q,1} x_{Q,2} y_{Q,2}

Output

For each query, print the number of the connected components consisting of blue squares in the region.

Sample Input 1

3 4 4
1101
0110
1101
1 1 3 4
1 1 3 1
2 2 3 4
1 2 2 4

Sample Output 1

3
2
2
2

In the first query, the whole grid is specified. There are three components consisting of blue squares, and thus 3 should be printed.

In the second query, the region within the red frame is specified. There are two components consisting of blue squares, and thus 2 should be printed.
Note that squares that belong to the same component in the original grid may belong to different components.

Sample Input 2

5 5 6
11010
01110
10101
11101
01010
1 1 5 5
1 2 4 5
2 3 3 4
3 3 3 3
3 1 3 5
1 1 3 4

Sample Output 2

3
2
1
1
3
2

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

(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)
#-swank (disable-debugger) ; for CS Academy

;; BEGIN_INSERTED_CONTENTS
(declaim (inline get-2dcumul))
(defun get-2dcumul (cumul-table i0 j0 i1 j1)
  "Returns the cumulative sum of the region given by the rectangle [i0, i1)*[j0,
j1). CUMUL-TABLE must be appropriately initialized beforehand:
i.e. CUMUL-TABLE[i][j] = sum of the region given by the regtangle [0, i)*[0,
j)."
  (+ (- (aref cumul-table i1 j1)
        (aref cumul-table i0 j1)
        (aref cumul-table i1 j0))
     (aref cumul-table i0 j0)))

(declaim (inline initialize-2dcumul))
(defun initialize-2dcumul (cumul-table)
  (destructuring-bind (h+1 w+1) (array-dimensions cumul-table)
    (declare ((integer 0 #.most-positive-fixnum) h+1 w+1))
    (let ((h (- h+1 1))
          (w (- w+1 1)))
      (declare ((integer 0 #.most-positive-fixnum) h w))
      (dotimes (i h+1)
        (dotimes (j w)
          (incf (aref cumul-table i (+ j 1))
                (aref cumul-table i j))))
      (dotimes (j w+1)
        (dotimes (i h)
          (incf (aref cumul-table (+ i 1) j)
                (aref cumul-table i j)))))))

(declaim (ftype (function * (values fixnum &optional)) read-fixnum))
(defun read-fixnum (&optional (in *standard-input*))
  "NOTE: cannot read -2^62"
  (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))))))))

(declaim (inline read-schar))
(defun read-schar (&optional (stream *standard-input*))
  (declare #-swank (sb-kernel:ansi-stream stream)
           (inline read-byte))
  #+swank (read-char stream nil #\Newline) ; on SLIME
  #-swank (code-char (read-byte stream nil #.(char-code #\Newline))))

(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)))

(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
;;;

(defun main ()
  (let* ((n (read))
         (m (read))
         (q (read))
         (plan (make-array (list n m) :element-type 'bit :initial-element 0))
         (dp (make-array (list (+ n 1) (+ m 1)) :element-type 'uint31 :initial-element 0))
         (dp-h (make-array (list (+ n 1) (+ m 1)) :element-type 'uint31 :initial-element 0))
         (dp-v (make-array (list (+ n 1) (+ m 1)) :element-type 'uint31 :initial-element 0)))
    (dotimes (i n)
      (dotimes (j m (read-schar))
        (when (char= #\1 (read-schar))
          (setf (aref plan i j) 1))))
    (dotimes (i n)
      (dotimes (j m)
        (when (= 1 (aref plan i j))
          (incf (aref dp (+ i 1) (+ j 1)))
          (when (and (> j 0) (= 1 (aref plan i (- j 1))))
            (incf (aref dp-h (+ i 1) (+ j 1))))
          (when (and (> i 0) (= 1 (aref plan (- i 1) j)))
            (incf (aref dp-v (+ i 1) (+ j 1)))))))
    (initialize-2dcumul dp)
    (initialize-2dcumul dp-h)
    (initialize-2dcumul dp-v)
    #>dp
    #>dp-h
    #>dp-v
    (write-string
     (with-output-to-string (*standard-output* nil :element-type 'base-char)
       (dotimes (i q)
         (let ((y1 (- (read-fixnum) 1))
               (x1 (- (read-fixnum) 1))
               (y2 (read-fixnum))
               (x2 (read-fixnum)))
           (println (- #>(get-2dcumul dp y1 x1 y2 x2)
                       #>(get-2dcumul dp-h y1 (+ x1 1) y2 x2)
                       #>(get-2dcumul dp-v (+ y1 1) x1 y2 x2)))))))))

#-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 "3 4 4
1101
0110
1101
1 1 3 4
1 1 3 1
2 2 3 4
1 2 2 4
"
    "3
2
2
2
"))
  (it.bese.fiveam:is
   (common-lisp-user::io-equal "5 5 6
11010
01110
10101
11101
01010
1 1 5 5
1 2 4 5
2 3 3 4
3 3 3 3
3 1 3 5
1 1 3 4
"
    "3
2
1
1
3
2
")))