Problem:
Score : 100 points

Problem Statement

There is a simple directed graph G with N vertices, numbered 1, 2, \ldots, N.

For each i and j (1 \leq i, j \leq N), you are given an integer a_{i, j} that represents whether there is a directed edge from Vertex i to j.
If a_{i, j} = 1, there is a directed edge from Vertex i to j; if a_{i, j} = 0, there is not.

Find the number of different directed paths of length K in G, modulo 10^9 + 7.
We will also count a path that traverses the same edge multiple times.

Constraints

All values in input are integers.

1 \leq N \leq 50

1 \leq K \leq 10^{18}

a_{i, j} is 0 or 1.

a_{i, i} = 0

Input

Input is given from Standard Input in the following format:

N K
a_{1, 1} \ldots a_{1, N}
:
a_{N, 1} \ldots a_{N, N}

Output

Print the number of different directed paths of length K in G, modulo 10^9 + 7.

Sample Input 1

4 2
0 1 0 0
0 0 1 1
0 0 0 1
1 0 0 0

Sample Output 1

6

G is drawn in the figure below:

There are six directed paths of length 2:

1 → 2 → 3

1 → 2 → 4

2 → 3 → 4

2 → 4 → 1

3 → 4 → 1

4 → 1 → 2

Sample Input 2

3 3
0 1 0
1 0 1
0 0 0

Sample Output 2

3

G is drawn in the figure below:

There are three directed paths of length 3:

1 → 2 → 1 → 2

2 → 1 → 2 → 1

2 → 1 → 2 → 3

Sample Input 3

6 2
0 0 0 0 0 0
0 0 1 0 0 0
0 0 0 0 0 0
0 0 0 0 1 0
0 0 0 0 0 1
0 0 0 0 0 0

Sample Output 3

1

G is drawn in the figure below:

There is one directed path of length 2:

4 → 5 → 6

Sample Input 4

1 1
0

Sample Output 4

0

Sample Input 5

10 1000000000000000000
0 0 1 1 0 0 0 1 1 0
0 0 0 0 0 1 1 1 0 0
0 1 0 0 0 1 0 1 0 1
1 1 1 0 1 1 0 1 1 0
0 1 1 1 0 1 0 1 1 1
0 0 0 1 0 0 1 0 1 0
0 0 0 1 1 0 0 1 0 1
1 0 0 0 1 0 1 0 0 0
0 0 0 0 0 1 0 0 0 0
1 0 1 1 1 0 1 1 1 0

Sample Output 5

957538352

Be sure to print the count modulo 10^9 + 7.

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
;;;
;;; 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)))))


;;
;; Matrix multiplication over semiring
;;

;; NOTE: These funcions are slow on SBCL version earlier than 1.5.6 as the type
;; propagation of MAKE-ARRAY doesn't work. The following files are required to
;; enable the optimization.
;; version < 1.5.0: array-element-type.lisp, make-array-header.lisp
;; version < 1.5.6: make-array-header.lisp
(defun gemm! (a b c &key (op+ #'+) (op* #'*) (identity+ 0))
  "Calculates C := A*B. This function destructively modifies C. (OP+, OP*) must
comprise a semiring. IDENTITY+ is the identity element w.r.t. OP+."
  (declare ((simple-array * (* *)) a b c))
  (dotimes (row (array-dimension a 0))
    (dotimes (col (array-dimension b 1))
      (let ((res identity+))
        (dotimes (k (array-dimension a 1))
          (setf res
                (funcall op+ (funcall op* (aref a row k) (aref b k col)))))
        (setf (aref c row col) res))))
  c)

(declaim (inline gemm))
(defun gemm (a b &key (op+ #'+) (op* #'*) (identity+ 0))
  "Calculates A*B. (OP+, OP*) must comprise a semiring. IDENTITY+ is the
identity element w.r.t. OP+."
  (declare ((simple-array * (* *)) a b)
           (function op+ op*))
  (let ((c (make-array (list (array-dimension a 0) (array-dimension b 1))
                       :element-type '(unsigned-byte 31))))
    (dotimes (row (array-dimension a 0))
      (dotimes (col (array-dimension b 1))
        (let ((res identity+))
          (dotimes (k (array-dimension a 1))
            (setf res
                  (funcall op+ res (funcall op* (aref a row k) (aref b k col)))))
          (setf (aref c row col) res))))
    c))

(declaim (inline matrix-power))
(defun matrix-power (base power &key (op+ #'+) (op* #'*) (identity+ 0) (identity* 1))
  (declare ((simple-array * (* *)) base)
           (function op+ op*)
           ((integer 0 #.most-positive-fixnum) power))
  (let ((size (array-dimension base 0)))
    (assert (= size (array-dimension base 1)))
    (let ((iden (make-array (array-dimensions base)
                            :element-type '(unsigned-byte 31)
                            :initial-element identity+)))
      (dotimes (i size)
        (setf (aref iden i i) identity*))
      (labels ((recur (p)
                 (declare ((integer 0 #.most-positive-fixnum) p))
                 (cond ((zerop p) iden)
                       ((evenp p)
                        (let ((res (recur (ash p -1))))
                          (gemm res res :op+ op+ :op* op* :identity+ identity+)))
                       (t
                        (gemm base (recur (- p 1))
                              :op+ op+ :op* op* :identity+ identity+)))))
        (recur power)))))

(declaim (inline gemv))
(defun gemv (a x &key (op+ #'+) (op* #'*) (identity+ 0))
  "Calculates A*x for a matrix A and a vector x. (OP+, OP*) must form a
semiring. IDENTITY+ is the identity element w.r.t. OP+."
  (declare ((simple-array * (* *)) a)
           ((simple-array * (*)) x)
           (function op+ op*))
  (let ((y (make-array (array-dimension a 0) :element-type (array-element-type x))))
    (dotimes (i (length y))
      (let ((res identity+))
        (dotimes (j (length x))
          (setf res
                (funcall op+ res (funcall op* (aref a i j) (aref x j)))))
        (setf (aref y i) res)))
    y))

(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
;;;

(define-mod-operations +mod+)
(defun main ()
  (declare #.OPT)
  (let* ((n (read))
         (k (read))
         (mat (make-array (list n n) :element-type 'uint31 :initial-element 0)))
    (declare ((simple-array uint31 (* *)) mat))
    (dotimes (i n)
      (dotimes (j n)
        (setf (aref mat i j) (read))))
    (let ((mat (matrix-power mat k :op+ #'mod+ :op* #'mod*))
          (res 0))
      (dotimes (i n)
        (dotimes (j n)
          (incfmod res (aref mat i j))))
      (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)
    (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 "4 2
0 1 0 0
0 0 1 1
0 0 0 1
1 0 0 0
"
    "6
"))
  (it.bese.fiveam:is
   (common-lisp-user::io-equal "3 3
0 1 0
1 0 1
0 0 0
"
    "3
"))
  (it.bese.fiveam:is
   (common-lisp-user::io-equal "6 2
0 0 0 0 0 0
0 0 1 0 0 0
0 0 0 0 0 0
0 0 0 0 1 0
0 0 0 0 0 1
0 0 0 0 0 0
"
    "1
"))
  (it.bese.fiveam:is
   (common-lisp-user::io-equal "1 1
0
"
    "0
"))
  (it.bese.fiveam:is
   (common-lisp-user::io-equal "10 1000000000000000000
0 0 1 1 0 0 0 1 1 0
0 0 0 0 0 1 1 1 0 0
0 1 0 0 0 1 0 1 0 1
1 1 1 0 1 1 0 1 1 0
0 1 1 1 0 1 0 1 1 1
0 0 0 1 0 0 1 0 1 0
0 0 0 1 1 0 0 1 0 1
1 0 0 0 1 0 1 0 0 0
0 0 0 0 0 1 0 0 0 0
1 0 1 1 1 0 1 1 1 0
"
    "957538352
")))