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))))

;;;
;;; Disjoint set (union by size & path compression)
;;;

(defpackage :cp/disjoint-set
  (:use :cl)
  (:export #:disjoint-set #:make-disjoint-set #:disjoint-set-p
           #:ds-root #:ds-unite! #:ds-connected-p #:ds-size))
(in-package :cp/disjoint-set)

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

(declaim (inline ds-root))
(defun ds-root (disjoint-set x)
  "Returns the root of X."
  (declare ((mod #.array-total-size-limit) x))
  (let ((data (ds-data disjoint-set)))
    (labels ((recur (x)
               (if (< (aref data x) 0)
                   x
                   (setf (aref data x)
                         (recur (aref data x))))))
      (recur x))))

(declaim (inline ds-unite!))
(defun ds-unite! (disjoint-set x1 x2)
  "Destructively unites X1 and X2 and returns true iff X1 and X2 become
connected for the first time."
  (let ((root1 (ds-root disjoint-set x1))
        (root2 (ds-root disjoint-set x2)))
    (unless (= root1 root2)
      (let ((data (ds-data disjoint-set)))
        ;; NOTE: If you want X1 to always be root, just delete this form. (Time
        ;; complexity becomes worse, however.)
        (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 (disjoint-set x1 x2)
  "Returns true iff X1 and X2 have the same root."
  (= (ds-root disjoint-set x1) (ds-root disjoint-set x2)))

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

(defpackage :cp/adjacent-duplicates
  (:use :cl)
  (:export #:delete-adjacent-duplicates))
(in-package :cp/adjacent-duplicates)

(declaim (inline delete-adjacent-duplicates))
(defun delete-adjacent-duplicates (seq &key (test #'eql))
  "Destructively deletes adjacent duplicates of SEQ: e.g. #(1 1 1 2 2 1 3) ->
#(1 2 1 3)"
  (declare (sequence seq)
           (function test))
  (etypecase seq
    (vector
     (if (zerop (length seq))
         seq
         (let ((prev (aref seq 0))
               (end 1))
           (loop for pos from 1 below (length seq)
                 unless (funcall test prev (aref seq pos))
                 do (setf prev (aref seq pos)
                          (aref seq end) (aref seq pos)
                          end (+ 1 end)))
           ;; KLUDGE: Resorting to ADJUST-ARRAY is maybe substandard. 
           (if (array-has-fill-pointer-p seq)
               (adjust-array seq end :fill-pointer end)
               (adjust-array seq end)))))
    (list
     (loop for rest on seq
           unless (and (cdr rest)
                       (funcall test (first rest) (second rest)))
           collect (car rest)))))

;; BEGIN_USE_PACKAGE
(eval-when (:compile-toplevel :load-toplevel :execute)
  (use-package :cp/adjacent-duplicates :cl-user))
(eval-when (:compile-toplevel :load-toplevel :execute)
  (use-package :cp/disjoint-set :cl-user))
(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 (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))
         (graph (make-array (* h w) :element-type 'list :initial-element nil))
         (dset (make-disjoint-set (* h w))))
    (labels ((encode (y x) (+ (* y w) x))
             (unite (y1 x1 y2 x2)
               (when (and  (<= 0 y2 (- h 1))
                           (<= 0 x2 (- w 1))
                           (zerop (aref plan y2 x2)))
                 (assert (and (<= 0 y1 (- h 1))
                              (<= 0 x1 (- w 1))))
                 (assert (zerop (aref plan y1 x1)))
                 (ds-unite! dset (encode y1 x1) (encode y2 x2)))))
      (dotimes (i h)
        (dotimes (j w (read-schar))
          (ecase (read-schar)
            (#\.)
            (#\# (setf (aref plan i j) 1)))))
      (dotimes (i h)
        (dotimes (j w)
          (when (zerop (aref plan i j))
            (unite i j i (+ j 1))
            (unite i j i (- j 1))
            (unite i j (+ i 1) j)
            (unite i j (- i 1) j))))
      (dotimes (i1 h)
        (dotimes (j1 w)
          (when (zerop (aref plan i1 j1))
            (let ((code1 (ds-root dset (encode i1 j1))))
              (loop for i2 from (max 0 (- i1 2)) to (min (- h 1) (+ i1 2))
                    do (loop for j2 from (max 0 (- j1 2)) to (min (- w 1) (+ j1 2))
                             for code2 = (ds-root dset (encode i2 j2))
                             when (and (zerop (aref plan i2 j2))
                                       (/= code1 code2))
                             do (push code2 (aref graph code1))))))))
      (dotimes (v (* h w))
        (setf (aref graph v) (delete-adjacent-duplicates
                              (the list (sort (aref graph v) (lambda (x y)
                                                               (declare (fixnum x y))
                                                               (< x y)))))))
      (let ((que (make-queue))
            (dists (make-array (* h w) :element-type 'uint31 :initial-element +inf+))
            (s (ds-root dset (encode ch cw)))
            (g (ds-root dset (encode dh dw))))
        (enqueue s que)
        (setf (aref dists s) 0)
        (dbg s g)
        (loop until (queue-empty-p que)
              for v = (dequeue que)
              do (dolist (next (aref graph v))
                   (when (= +inf+ (aref dists next))
                     (setf (aref dists next) (+ (aref dists v) 1))
                     (enqueue next que))))
        (println (if (= (aref dists g) +inf+)
                     -1
                     (aref dists g)))))))

#-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))))