Problem:
Score : 100 points

Problem Statement

You are given a grid of N rows and M columns. The square at the i-th row and j-th column will be denoted as (i,j).
A nonnegative integer A_{i,j} is written for each square (i,j).

You choose some of the squares so that each row and column contains at most K chosen squares.
Under this constraint, calculate the maximum value of the sum of the integers written on the chosen squares.
Additionally, calculate a way to choose squares that acheives the maximum.

Constraints

1 \leq N \leq 50

1 \leq K \leq N

0 \leq A_{i,j} \leq 10^9

All values in Input are integer.

Input

Input is given from Standard Input in the following format:

N K
A_{1,1} A_{1,2} \cdots A_{1,N}
A_{2,1} A_{2,2} \cdots A_{2,N}
\vdots
A_{N,1} A_{N,2} \cdots A_{N,N}

Output

On the first line, print the maximum value of the sum of the integers written on the chosen squares.

On the next N lines, print a way that achieves the maximum.

Precisely, output the strings t_1,t_2,\cdots,t_N, that satisfies t_{i,j}=X if you choose (i,j) and t_{i,j}=. otherwise.

You may print any way to choose squares that maximizes the sum.

Sample Input 1

3 1
5 3 2
1 4 8
7 6 9

Sample Output 1

19
X..
..X
.X.

Sample Input 2

3 2
10 10 1
10 10 1
1 1 10

Sample Output 2

50
XX.
XX.
..X

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 (set-dispatch-macro-character #\# #\> #'cl-debug-print:debug-print-reader)

(macrolet ((def-int (b)
             `(progn (deftype ,(intern (format nil "UINT~A" b)) () '(unsigned-byte ,b))
                     (deftype ,(intern (format nil "INT~A" b)) () '(signed-byte ,b))))
           (defs (&rest bits) `(progn ,@(mapcar (lambda (b) `(def-int ,b)) bits))))
  (defs 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
(defpackage :cp/max-flow
  (:use :cl)
  (:export #:edge #:add-edge #:reinitialize-flow-network #:max-flow-overflow
           #:edge-to #:edge-capacity #:edge-default-capacity #:edge-reversed))
(in-package :cp/max-flow)

(define-condition max-flow-overflow (error)
  ((graph :initarg :graph :reader max-flow-overflow-graph))
  (:report
   (lambda (condition stream)
     (format stream "MOST-POSITIVE-FIXNUM or more units can flow on graph ~W."
             (max-flow-overflow-graph condition)))))

(defstruct (edge (:constructor %make-edge
                     (to capacity reversed
                      &aux (default-capacity capacity))))
  (to nil :type (integer 0 #.most-positive-fixnum))
  (capacity 0 :type (integer 0 #.most-positive-fixnum))
  (default-capacity 0 :type (integer 0 #.most-positive-fixnum))
  (reversed nil :type (or null edge)))

(defmethod print-object ((edge edge) stream)
  (let ((*print-circle* t))
    (call-next-method)))

(defun add-edge (graph from-idx to-idx capacity &key bidirectional)
  "FROM-IDX, TO-IDX := index of vertex
GRAPH := vector of lists of all the edges that goes from each vertex

If BIDIRECTIONAL is true, ADD-EDGE adds the reversed edge of the same
capacity in addition."
  (declare (optimize (speed 3))
           ((simple-array list (*)) graph))
  (let* ((dep (%make-edge to-idx capacity nil))
         (ret (%make-edge from-idx
                          (if bidirectional capacity 0)
                          dep)))
    (setf (edge-reversed dep) ret)
    (push dep (aref graph from-idx))
    (push ret (aref graph to-idx))))

(declaim (inline reinitialize-flow-network))
(defun reinitialize-flow-network (graph)
  "Sets the current CAPACITY of every edge in GRAPH to the default
capacity. That is, this function reinitialize the graph network to the state
prior to sending flow."
  (loop for edges across graph
        do (dolist (edge edges)
             (setf (edge-capacity edge) (edge-default-capacity edge)))))

;;;
;;; Minimum cost flow (SSP)
;;;

(defpackage :cp/min-cost-flow
  (:use :cl :cp/max-flow)
  (:export #:cedge #:cedge-p #:copy-cedge #:add-cedge #:+inf-cost+ #:cost-type
           #:cedge-reversed #:cedge-cost #:cedge-capacity #:cedge-to #:cedge-default-capacity
           #:not-enough-capacity-error #:not-enough-capacity-error-graph
           #:not-enough-capacity-error-flow #:not-enough-capacity-error-score))
(in-package :cp/min-cost-flow)

;; COST-TYPE and +INF-COST+ may be changed. (A supposed use case is to adopt
;; bignum).

(deftype cost-type () 'fixnum)
(defconstant +inf-cost+ most-positive-fixnum)
(assert (and (typep +inf-cost+ 'cost-type)
             (subtypep 'cost-type 'integer)))

(defstruct (cedge (:constructor %make-cedge)
                  (:include edge))
  (cost 0 :type cost-type))

(define-condition not-enough-capacity-error (error)
  ((graph :initarg :graph :reader not-enough-capacity-error-graph)
   (flow :initarg :flow :reader not-enough-capacity-error-flow)
   (score :initarg :score :reader not-enough-capacity-error-score))
  (:report
   (lambda (c s)
     (format s "Cannot send ~A units of flow on graph ~A due to not enough capacity."
             (not-enough-capacity-error-flow c)
             (not-enough-capacity-error-graph c)))))

(defmethod print-object ((cedge cedge) stream)
  (let ((*print-circle* t))
    (call-next-method)))

(defun add-cedge (graph from-idx to-idx cost capacity)
  "FROM-IDX, TO-IDX := index of vertex
GRAPH := vector of list of all the edges that goes from the vertex"
  (declare ((simple-array list (*)) graph)
           (cost-type cost))
  (let* ((dep (%make-cedge :to to-idx :capacity capacity :cost cost))
         (ret (%make-cedge :to from-idx :capacity 0 :cost (- cost) :reversed dep)))
    (setf (cedge-reversed dep) ret)
    (push dep (aref graph from-idx))
    (push ret (aref graph to-idx))))

(defpackage :cp/ssp
  (:use :cl :cp/min-cost-flow)
  (:export #:min-cost-flow!))
(in-package :cp/ssp)

;; binary heap for Dijkstra's algorithm
(defstruct (heap (:constructor make-heap
                     (size
                      &aux (costs (make-array (1+ size) :element-type 'cost-type))
                           (vertices (make-array (1+ size) :element-type 'fixnum)))))
  (costs nil :type (simple-array cost-type (*)))
  (vertices nil :type (simple-array fixnum (*)))
  (position 1 :type (integer 1 #.most-positive-fixnum)))

(defun heap-push (cost vertex heap)
  (declare (optimize (speed 3)))
  (symbol-macrolet ((position (heap-position heap)))
    (when (>= position (length (heap-costs heap)))
      (setf (heap-costs heap)
            (adjust-array (heap-costs heap) (* position 2))
            (heap-vertices heap)
            (adjust-array (heap-vertices heap) (* position 2))))
    (let ((costs (heap-costs heap))
          (vertices (heap-vertices heap)))
      (labels ((update (pos)
                 (declare (optimize (safety 0)))
                 (unless (= pos 1)
                   (let ((parent-pos (ash pos -1)))
                     (when (< (aref costs pos) (aref costs parent-pos))
                       (rotatef (aref costs pos) (aref costs parent-pos))
                       (rotatef (aref vertices pos) (aref vertices parent-pos))
                       (update parent-pos))))))
        (setf (aref costs position) cost
              (aref vertices position) vertex)
        (update position)
        (incf position)
        heap))))

(defun heap-pop (heap)
  (declare (optimize (speed 3)))
  (symbol-macrolet ((position (heap-position heap)))
    (let ((costs (heap-costs heap))
          (vertices (heap-vertices heap)))
      (labels ((update (pos)
                 (declare (optimize (safety 0))
                          ((integer 1 #.most-positive-fixnum) pos))
                 (let* ((child-pos1 (+ pos pos))
                        (child-pos2 (1+ child-pos1)))
                   (when (<= child-pos1 position)
                     (if (<= child-pos2 position)
                         (if (< (aref costs child-pos1) (aref costs child-pos2))
                             (unless (< (aref costs pos) (aref costs child-pos1))
                               (rotatef (aref costs pos) (aref costs child-pos1))
                               (rotatef (aref vertices pos) (aref vertices child-pos1))
                               (update child-pos1))
                             (unless (< (aref costs pos) (aref costs child-pos2))
                               (rotatef (aref costs pos) (aref costs child-pos2))
                               (rotatef (aref vertices pos) (aref vertices child-pos2))
                               (update child-pos2)))
                         (unless (< (aref costs pos) (aref costs child-pos1))
                           (rotatef (aref costs pos) (aref costs child-pos1))
                           (rotatef (aref vertices pos) (aref vertices child-pos1))))))))
        (multiple-value-prog1 (values (aref costs 1) (aref vertices 1))
          (decf position)
          (setf (aref costs 1) (aref costs position)
                (aref vertices 1) (aref vertices position))
          (update 1))))))

(declaim (inline heap-empty-p))
(defun heap-empty-p (heap)
  (= (heap-position heap) 1))

(declaim (inline heap-reinitialize))
(defun heap-reinitialize (heap)
  (setf (heap-position heap) 1)
  heap)

(defun min-cost-flow! (graph src-idx dest-idx flow &key edge-count)
  "Returns the minimum cost to send FLOW units from SRC-IDX to DEST-IDX in
GRAPH. Destructively modifies GRAPH.

EDGE-COUNT := initial reserved size for heap (it should be the number of edges)"
  (declare (optimize (speed 3))
           ((integer 0 #.most-positive-fixnum) flow)
           ((simple-array list (*)) graph))
  (macrolet ((the-cost-type (form)
               (reduce (lambda (x y) `(,(car form) (the cost-type ,x) (the cost-type ,y)))
		       (cdr form))))
    (let* ((size (length graph))
           (edge-count (or edge-count (* size 2)))
           (prev-vertices (make-array size :element-type 'fixnum :initial-element 0))
           (prev-edges (locally
                           (declare (sb-ext:muffle-conditions style-warning))
                         (make-array size :element-type 'cedge)))
           (potential (make-array size :element-type 'cost-type :initial-element 0))
           (dist (make-array size :element-type 'cost-type))
           (pqueue (make-heap edge-count))
           (res 0))
      (declare (fixnum edge-count)
               (cost-type res))
      ;; FIXME: Actually we must do Bellman-Ford here to handle negative edges
      ;; properly. Currently this function returns a correct result also for a
      ;; graph containing negative edges, if no negative **cycles** are
      ;; contained. In this case, however, the worst-case time complexity is
      ;; exponential. If the input network is for a weighted bipartite matching
      ;; containing negative weights, this function completely works without any
      ;; problems.
      (loop while (> flow 0)
            do (fill dist +inf-cost+)
               (setf (aref dist src-idx) 0)
               (heap-reinitialize pqueue)
               (heap-push 0 src-idx pqueue)
               (loop until (heap-empty-p pqueue)
                     do (multiple-value-bind (cost v) (heap-pop pqueue)
                          (declare (cost-type cost)
                                   (fixnum v))
                          (when (<= cost (aref dist v))
                            (dolist (edge (aref graph v))
                              (let* ((next-v (cedge-to edge))
                                     (next-cost (the-cost-type
                                                 (+ (aref dist v)
                                                    (cedge-cost edge)
                                                    (aref potential v)
                                                    (- (aref potential next-v))))))
                                (when (and (> (cedge-capacity edge) 0)
                                           (> (aref dist next-v) next-cost))
                                  (setf (aref dist next-v) next-cost
                                        (aref prev-vertices next-v) v
                                        (aref prev-edges next-v) edge)
                                  (heap-push next-cost next-v pqueue)))))))
               (when (= (aref dist dest-idx) +inf-cost+)
                 (error 'not-enough-capacity-error :flow flow :graph graph :score res))
               (let ((max-flow flow))
                 (declare (fixnum max-flow))
                 (dotimes (v size)
                   (setf (aref potential v)
                         (min +inf-cost+
                              (+ (aref potential v) (aref dist v)))))
                 (when ())
                 (do ((v dest-idx (aref prev-vertices v)))
                     ((= v src-idx))
                   (setf max-flow (min max-flow (cedge-capacity (aref prev-edges v)))))
                 (decf flow max-flow)
                 (incf res (the cost-type (* max-flow (aref potential dest-idx))))
                 (do ((v dest-idx (aref prev-vertices v)))
                     ((= v src-idx))
                   (decf (cedge-capacity (aref prev-edges v)) max-flow)
                   (incf (cedge-capacity (cedge-reversed (aref prev-edges v))) max-flow))))
      res)))

(defpackage :cp/println-matrix
  (:use :cl)
  (:export #:println-matrix))
(in-package :cp/println-matrix)

(declaim (inline println-matrix))
(defun println-matrix (array &key (separator #\ ) (key #'identity) (writer #'write) (row-start 0) row-end (col-start 0) col-end)
  "Prints a 2-dimensional array."
  (declare ((array * (* *)) array)
           ((integer 0 #.most-positive-fixnum) row-start col-start))
  (let ((row-end (or row-end (array-dimension array 0)))
        (col-end (or col-end (array-dimension array 1))))
    (declare ((integer 0 #.most-positive-fixnum) row-end col-end))
    (loop for i from row-start below row-end
          do (loop for j from col-start below col-end
                   unless (= j col-start)
                   do (princ separator)
                   do (funcall writer (funcall key (aref array i j))))
             (terpri))))

;; BEGIN_USE_PACKAGE
(eval-when (:compile-toplevel :load-toplevel :execute)
  (use-package :cp/println-matrix :cl-user))
(eval-when (:compile-toplevel :load-toplevel :execute)
  (use-package :cp/ssp :cl-user))
(eval-when (:compile-toplevel :load-toplevel :execute)
  (use-package :cp/min-cost-flow :cl-user))
(in-package :cl-user)

;;;
;;; Body
;;;

(defconstant +max+ #.(expt 10 9))
(defun main ()
  (let* ((n (read))
         (k (read))
         (graph (make-array (+ 2 (* n (+ n 2))) :element-type 'list :initial-element nil))
         (src (* n (+ n 2)))
         (dest (+ 1 (* n (+ n 2)))))
    (labels ((encode (i j) (+ j (* i n)))
             (encode-row (i) (+ i (* n n)))
             (encode-col (j) (+ j (* n (+ n 1)))))
      (dotimes (i n)
        (dotimes (j n)
          (let ((a (read))
                (v (encode i j)))
            (add-cedge graph (encode-row i) v (- +max+ a) 1)
            (add-cedge graph v (encode-col j) 0 1))))
      (dotimes (i n)
        (add-cedge graph src (encode-row i) 0 k)
        (add-cedge graph (encode-col i) dest 0 k))
      (add-cedge graph src dest +max+ (* n k))
      (let ((score (min-cost-flow! graph src dest (* k n)))
            (res (make-array (list n n) :element-type 'base-char :initial-element #\.)))
        (dotimes (i n)
          (dolist (cedge (aref graph (encode-row i)))
            (let ((to (cedge-to cedge)))
              (when (and (<= 0 to (- (* n n) 1))
                         (zerop (cedge-capacity cedge)))
                (setf (aref res i (mod to n)) #\X)))))
        (println (- (* +max+ (* k n)) score))
        (println-matrix res :separator "" :writer #'write-char)))))

#-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)))

#-swank
(eval-when (:compile-toplevel)
  (when (and (boundp 'sb-c::*compiler-warning-count*)
             (> sb-c::*compiler-warning-count* 0))
    (sb-ext:quit :unix-status 1)))

;; To run: (5am:run! :sample)
#+swank
(it.bese.fiveam:test :sample
  (it.bese.fiveam:is
   (equal "19
X..
..X
.X.
"
          (run "3 1
5 3 2
1 4 8
7 6 9
" nil)))
  (it.bese.fiveam:is
   (equal "50
XX.
XX.
..X
"
          (run "3 2
10 10 1
10 10 1
1 1 10
" nil))))