Problem:
Score : 500 points

Problem Statement

Given are N integers A_1,\ldots,A_N.

We will choose exactly K of these elements. Find the maximum possible product of the chosen elements.

Then, print the maximum product modulo (10^9+7), using an integer between 0 and 10^9+6 (inclusive).

Constraints

1 \leq K \leq N \leq 2\times 10^5

|A_i| \leq 10^9

Input

Input is given from Standard Input in the following format:

N K
A_1 \ldots A_N

Output

Print the maximum product modulo (10^9+7), using an integer between 0 and 10^9+6 (inclusive).

Sample Input 1

4 2
1 2 -3 -4

Sample Output 1

12

The possible products of the two chosen elements are 2, -3, -4, -6, -8, and 12, so the maximum product is 12.

Sample Input 2

4 3
-1 -2 -3 -4

Sample Output 2

1000000001

The possible products of the three chosen elements are -24, -12, -8, and -6, so the maximum product is -6.

We print this value modulo (10^9+7), that is, 1000000001.

Sample Input 3

2 1
-1 1000000000

Sample Output 3

1000000000

The possible products of the one chosen element are -1 and 1000000000, so the maximum product is 1000000000.

Sample Input 4

10 10
1000000000 100000000 10000000 1000000 100000 10000 1000 100 10 1

Sample Output 4

999983200

Be sure to print the product modulo (10^9+7).

Solution in Common Lisp (SBCL 2.0.3) (judge verdict: Wrong Answer)

(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)))))


(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))))))))

(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 solve (as k)
  (declare ((simple-array int32 (*)) as))
  (let* ((n (length as))
         (n+ 0)
         (n- 0)
         (as+ (make-array n :element-type 'uint31 :initial-element 0))
         (as- (make-array n :element-type 'uint31 :initial-element 0)))
    (dotimes (i n)
      (let ((a (aref as i)))
        (cond ((> a 0)
               (setf (aref as+ n+) a)
               (incf n+))
              ((< a 0)
               (setf (aref as- n-) (- a))
               (incf n-)))))
    (setq as+ (sort as+ #'>)
          as- (sort as- #'>))
    (dbg as+ as-)
    (let ((cumuls+ (make-array (+ n+ 1) :element-type 'uint31 :initial-element 1))
          (cumuls- (make-array (+ n- 1) :element-type 'uint31 :initial-element 1))
          (logcumuls+ (make-array (+ n+ 1) :element-type 'double-float :initial-element 0d0))
          (logcumuls- (make-array (+ n- 1) :element-type 'double-float :initial-element 0d0))
          (as (delete 0 as)))
      (dotimes (i n+)
        (let ((a (aref as+ i)))
          (setf (aref cumuls+ (+ i 1)) (mod* (aref cumuls+ i) a)
                (aref logcumuls+ (+ i 1)) (+ (aref logcumuls+ i) (log (float a 1d0) 1.1d0)))))
      (dotimes (i n-)
        (let ((a (aref as- i)))
          (setf (aref cumuls- (+ i 1)) (mod* (aref cumuls- i) a)
                (aref logcumuls- (+ i 1)) (+ (aref logcumuls- i) (log (float a 1d0) 1.1d0)))))
      (setq as (sort as #'> :key #'abs))
      (let ((res1 0)
            (maxlog most-negative-double-float)
            fail)
        (loop for d- from 0 to k by 2
              for d+ = (- k d-)
              when (and (<= d- n-) (<= d+ n+))
              do (let ((log (+ (aref logcumuls+ d+) (aref logcumuls- d-))))
                   (when (< (abs (- log maxlog)) 1d-8)
                     (setq fail t)
                     (return))
                   (when (> log maxlog)
                     (when (or (> (abs (- log maxlog)) 1d-9)
                               (< (random 1d0) 0.5d0))
                       (setq maxlog log
                             res1 (mod* (aref cumuls+ d+) (aref cumuls- d-)))))))
        res1
        ;; (if fail
        ;;     (let ((minus 0)
        ;;           (res 1))
        ;;       #>as
        ;;       (dotimes (i (length as))
        ;;         (let ((a (aref as i)))
        ;;           (if (> a 0)
        ;;               (mulfmod res a)
        ;;               (if (= i (- k 1))
        ;;                   (if (evenp minus)
        ;;                       (let ((new-pos (position-if #'plusp as :start i)))
        ;;                         (mulfmod res (aref as new-pos)))
        ;;                       (mulfmod res a))
        ;;                   (progn
        ;;                     (mulfmod res a)
        ;;                     (incf minus))))
        ;;           (when (= i (- k 1))
        ;;             (return))))
        ;;       res)
        ;;     res1)
        ))))

(defun main ()
  (let* ((n (read))
         (k (read))
         (as (make-array n :element-type 'int32 :initial-element 0)))
    (dotimes (i n)
      (setf (aref as i) (read-fixnum)))
    (let ((num+ (count-if #'plusp as))
          (num0 (count 0 as))
          (num- (count-if #'minusp as)))
      (dbg num+ num0 num-)
      (println
       (if (loop for d- from 0 to k by 2
                 for d+ = (- k d-)
                 thereis (and (<= d- num-) (<= d+ num+)))
           (solve as k)
           (let ((as (sort as #'< :key #'abs)))
             (reduce #'mod* as :end k :initial-value 1)))))))

#-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)
    #+os-windows (run-program "powershell.exe" '("-Command" "Get-Clipboard") :output out :search t)
    #+os-unix (run-program "xsel" () :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
1 2 -3 -4
"
    "12
"))
  (it.bese.fiveam:is
   (common-lisp-user::io-equal "4 3
-1 -2 -3 -4
"
    "1000000001
"))
  (it.bese.fiveam:is
   (common-lisp-user::io-equal "2 1
-1 1000000000
"
    "1000000000
"))
  (it.bese.fiveam:is
   (common-lisp-user::io-equal "10 10
1000000000 100000000 10000000 1000000 100000 10000 1000 100 10 1
"
    "999983200
")))