Problem:
Score : 400 points

Problem Statement

A maze is composed of a grid of H \times W squares - H vertical, W horizontal.

The square at the i-th row from the top and the j-th column from the left - (i,j) - is a wall if S_{ij} is # and a road if S_{ij} is ..

There is a magician in (C_h,C_w). He can do the following two kinds of moves:

Move A: Walk to a road square that is vertically or horizontally adjacent to the square he is currently in.

Move B: Use magic to warp himself to a road square in the 5\times 5 area centered at the square he is currently in.

In either case, he cannot go out of the maze.

At least how many times does he need to use the magic to reach (D_h, D_w)?

Constraints

1 \leq H,W \leq 10^3

1 \leq C_h,D_h \leq H

1 \leq C_w,D_w \leq W

S_{ij} is # or ..

S_{C_h C_w} and S_{D_h D_w} are ..

(C_h,C_w) \neq (D_h,D_w)

Input

Input is given from Standard Input in the following format:

H W
C_h C_w
D_h D_w
S_{11}\ldots S_{1W}
\vdots
S_{H1}\ldots S_{HW}

Output

Print the minimum number of times the magician needs to use the magic. If he cannot reach (D_h,D_w), print -1 instead.

Sample Input 1

4 4
1 1
4 4
..#.
..#.
.#..
.#..

Sample Output 1

1

For example, by walking to (2,2) and then using the magic to travel to (4,4), just one use of magic is enough.

Note that he cannot walk diagonally.

Sample Input 2

4 4
1 4
4 1
.##.
####
####
.##.

Sample Output 2

-1

He cannot move from there.

Sample Input 3

4 4
2 2
3 3
....
....
....
....

Sample Output 3

0

No use of magic is needed.

Sample Input 4

4 5
1 2
2 5
#.###
####.
#..##
#..##

Sample Output 4

2

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

(in-package :cl-user)
(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)

(defconstant +mod+ 1000000007)

(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))))

;; BEGIN_INSERTED_CONTENTS
;;;
;;; Queue with singly linked list
;;;

(defpackage :cp/queue
  (:use :cl)
  (:export #:queue #:make-queue #:enqueue #:dequeue #:queue-empty-p #:queue-peek #:enqueue-front))
(in-package :cp/queue)

(defstruct (queue (:constructor make-queue
                      (&optional list &aux (tail (last list))))
                  (:copier nil)
                  (:predicate nil))
  (list nil :type list)
  (tail nil :type list))

(declaim (inline enqueue))
(defun enqueue (obj queue)
  "Pushes OBJ to the end of QUEUE."
  (symbol-macrolet ((list (queue-list queue))
                    (tail (queue-tail queue)))
    (if (null list)
        (setf tail (list obj)
              list tail)
        (setf (cdr tail) (list obj)
              tail (cdr tail))))
  queue)

(declaim (inline dequeue))
(defun dequeue (queue)
  "Removes and returns the element at the front of QUEUE. Returns NIL if QUEUE
is empty."
  (pop (queue-list queue)))

(declaim (inline queue-empty-p))
(defun queue-empty-p (queue)
  (null (queue-list queue)))

(declaim (inline queue-peek))
(defun queue-peek (queue)
  (car (queue-list queue)))

(declaim (inline enqueue-front))
(defun enqueue-front (obj queue)
  "Pushes OBJ to the front of QUEUE."
  (symbol-macrolet ((list (queue-list queue))
                    (tail (queue-tail queue)))
    (if (null list)
        (setf tail (list obj)
              list tail)
        (push obj list))
    queue))

(defpackage :cp/read-schar
  (:use :cl)
  (:export #:read-schar))
(in-package :cp/read-schar)

(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))))

;; BEGIN_USE_PACKAGE
(eval-when (:compile-toplevel :load-toplevel :execute)
  (use-package :cp/read-schar :cl-user))
(eval-when (:compile-toplevel :load-toplevel :execute)
  (use-package :cp/queue :cl-user))
(in-package :cl-user)

;;;
;;; Body
;;;

(defconstant +inf+ #x7fffffff)
(defun main ()
  (declare #.opt
           (inline sort sb-impl::stable-sort-list))
  (let* ((h (read))
         (w (read))
         (ch (- (read) 1))
         (cw (- (read) 1))
         (dh (- (read) 1))
         (dw (- (read) 1))
         (plan (make-array (list h w) :element-type 'bit :initial-element 0))
         (dists (make-array (list h w) :element-type 'uint31 :initial-element +inf+))
         (que (make-queue)))
    (declare (uint16 h w ch cw dh dw))
    (dotimes (i h)
      (dotimes (j w (read-schar))
        (ecase (read-schar)
          (#\.)
          (#\# (setf (aref plan i j) 1)))))
    (setf (aref dists ch cw) 0)
    (enqueue (list* ch cw 0) que)
    (loop until (queue-empty-p que)
          for (i1 j1 . dist) of-type (uint31 uint31 . uint31) = (dequeue que)
          when (= dist (aref dists i1 j1))
          do (labels ((calc-cost (i2 j2)
                        (cond ((= 1 (aref plan i2 j2)) +inf+)
                              ((= 1 (+ (abs (- i1 i2)) (abs (- j1 j2)))) 0)
                              (t 1)))
                      (visit (i2 j2)
                        (when (and (<= 0 i2 (- h 1))
                                   (<= 0 j2 (- w 1)))
                          (let ((new-dist (+ dist (calc-cost i2 j2))))
                            (when (< new-dist (aref dists i2 j2))
                              (setf (aref dists i2 j2) new-dist)
                              (if (= dist new-dist)
                                  (enqueue-front (list* i2 j2 new-dist) que)
                                  (enqueue (list* i2 j2 new-dist) que)))))))
               (loop for i2 from (- i1 2) to (+ i1 2)
                     do (loop for j2 from (- j1 2) to (+ j1 2)
                              do (visit i2 j2)))))
    (println (if (= (aref dists dh dw) +inf+)
                 -1
                 (aref dists dh dw)))))

#-swank (main)

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

#+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* (or out (make-string-output-stream)))
         (res (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))))))
    (if out res (get-output-stream-string *standard-output*))))

#+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
   (equal "1
"
          (run "4 4
1 1
4 4
..#.
..#.
.#..
.#..
" nil)))
  (it.bese.fiveam:is
   (equal "-1
"
          (run "4 4
1 4
4 1
.##.
####
####
.##.
" nil)))
  (it.bese.fiveam:is
   (equal "0
"
          (run "4 4
2 2
3 3
....
....
....
....
" nil)))
  (it.bese.fiveam:is
   (equal "2
"
          (run "4 5
1 2
2 5
#.###
####.
#..##
#..##
" nil))))