結果
| 問題 | No.40 多項式の割り算 |
| コンテスト | |
| ユーザー |
 sansaqua
|
| 提出日時 | 2020-03-28 06:10:22 |
| 言語 | Common Lisp (sbcl 2.5.0) |
| 結果 |
AC
|
| 実行時間 | 13 ms / 5,000 ms |
| コード長 | 10,932 bytes |
| 記録 | |
| コンパイル時間 | 290 ms |
| コンパイル使用メモリ | 59,100 KB |
| 実行使用メモリ | 28,824 KB |
| 最終ジャッジ日時 | 2025-01-02 10:39:23 |
| 合計ジャッジ時間 | 1,911 ms |
|
ジャッジサーバーID （参考情報） |
judge3 / judge5 |
（要ログイン）
| ファイルパターン | 結果 |
|---|---|
| sample | AC * 3 |
| other | AC * 32 |
コンパイルメッセージ
; compiling file "/home/judge/data/code/Main.lisp" (written 02 JAN 2025 10:39:21 AM): ; wrote /home/judge/data/code/Main.fasl ; compilation finished in 0:00:00.181
ソースコード
(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
;; enclose the form with VALUES to avoid being captured by LOOP macro
#\# #\> (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
(declaim (inline println-sequence))
(defun println-sequence (sequence &key (out *standard-output*) (key #'identity))
(let ((init t))
(sequence:dosequence (x sequence)
(if init
(setq init nil)
(write-char #\ out))
(princ (funcall key x) out))
(terpri out)))
;; NOTE: These are poor man's utilities for polynomial arithmetic. NOT
;; sufficiently equipped in all senses.
(declaim (inline poly-value))
(defun poly-value (poly x modulus)
"Returns the value f(x)."
(declare (vector poly))
(let ((x^i 1)
(res 0))
(declare (fixnum x^i res))
(dotimes (i (length poly))
(setq res (mod (+ res (* x^i (aref poly i))) modulus))
(setq x^i (mod (* x^i x) modulus)))
res))
;; naive multiplication in O(n^2)
(declaim (inline poly-mult))
(defun poly-mult (u v modulus &optional result-vector)
"Multiplies u(x) and v(x) over Z/nZ in O(deg(u)deg(v)) time.
The result is stored in RESULT-VECTOR if it is given, otherwise a new vector is
created."
(declare (vector u v)
((or null vector) result-vector)
((integer 1 #.most-positive-fixnum) modulus))
(let* ((deg1 (loop for i from (- (length u) 1) downto 0
while (zerop (aref u i))
finally (return i)))
(deg2 (loop for i from (- (length v) 1) downto 0
while (zerop (aref v i))
finally (return i)))
(len (max 0 (+ deg1 deg2 1)))
(res (or result-vector (make-array len :element-type (array-element-type u)))))
(declare ((integer -1 (#.array-total-size-limit)) deg1 deg2 len))
(dotimes (d len res)
;; 0 <= i <= deg1, 0 <= j <= deg2
(loop with coef of-type (integer 0 #.most-positive-fixnum) = 0
for i from (max 0 (- d deg2)) to (min d deg1)
for j = (- d i)
do (setq coef (mod (+ coef (* (aref u i) (aref v j)))
modulus))
finally (setf (aref res d) coef)))))
(declaim (ftype (function * (values (mod #.most-positive-fixnum) &optional)) %mod-inverse))
(defun %mod-inverse (a modulus)
"Solves ax ≡ 1 mod m. A and M must be coprime."
(declare (optimize (speed 3))
(integer a)
((integer 1 #.most-positive-fixnum) modulus))
(labels ((%gcd (a b)
(declare (optimize (safety 0))
((integer 0 #.most-positive-fixnum) a b))
(if (zerop b)
(values 1 0)
(multiple-value-bind (p q) (floor a b) ; a = pb + q
(multiple-value-bind (v u) (%gcd b q)
(declare (fixnum u v))
(values u (the fixnum (- v (the fixnum (* p u))))))))))
(mod (%gcd (mod a modulus) modulus) modulus)))
;; naive division in O(n^2)
;; Reference: http://web.cs.iastate.edu/~cs577/handouts/polydivide.pdf
(declaim (inline poly-floor!))
(defun poly-floor! (u v modulus &optional quotient)
"Returns the quotient q(x) and the remainder r(x) over Z/nZ: u(x) = q(x)v(x) +
r(x), deg(r) < deg(v). This function destructively modifies U. The time
complexity is O((deg(u)-deg(v))deg(v)).
The quotient is stored in QUOTIENT if it is given, otherwise a new vector is
created.
Note that MODULUS and V[deg(V)] must be coprime."
(declare (vector u v)
((integer 1 #.most-positive-fixnum) modulus))
;; m := deg(u), n := deg(v)
(let* ((m (loop for i from (- (length u) 1) downto 0
while (zerop (aref u i))
finally (return i)))
(n (loop for i from (- (length v) 1) downto 0
unless (zerop (aref v i))
do (return i)
finally (error 'division-by-zero
:operation #'poly-floor!
:operands (list u v))))
(quot (or quotient
(make-array (max 0 (+ 1 (- m n)))
:element-type (array-element-type u))))
;; FIXME: Is it better to signal an error in non-coprime case?
(inv (%mod-inverse (aref v n) modulus)))
(declare ((integer -1 (#.array-total-size-limit)) m n))
(loop for k from (- m n) downto 0
do (setf (aref quot k)
(mod (* (aref u (+ n k)) inv) modulus))
(loop for j from (+ n k -1) downto k
do (setf (aref u j)
(mod (- (aref u j)
(* (aref quot k) (aref v (- j k))))
modulus))))
(loop for i from (- (length u) 1) downto n
do (setf (aref u i) 0)
finally (return (values quot u)))))
;; naive division in O(n^2)
(defun poly-mod! (poly divisor modulus)
"Returns the remainder of POLY divided by DIVISOR over Z/nZ. This function
destructively modifies POLY."
(declare (vector poly divisor)
((integer 1 #.most-positive-fixnum) modulus))
(let* ((m (loop for i from (- (length poly) 1) downto 0
while (zerop (aref poly i))
finally (return i)))
(n (loop for i from (- (length divisor) 1) downto 0
unless (zerop (aref divisor i))
do (return i)
finally (error 'division-by-zero
:operation #'poly-mod!
:operands (list poly divisor))))
(inv (%mod-inverse (aref divisor n) modulus)))
(declare ((integer -1 (#.array-total-size-limit)) m n))
(loop for pivot-deg from m downto n
for factor of-type (integer 0 #.most-positive-fixnum)
= (mod (* (aref poly pivot-deg) inv) modulus)
do (loop for delta from 0 to n
do (setf (aref poly (- pivot-deg delta))
(mod (- (aref poly (- pivot-deg delta))
(* factor (aref divisor (- n delta))))
modulus))))
poly))
(defun poly-power (poly exponent divisor modulus)
"Returns POLY to the power of EXPONENT modulo DIVISOR over Z/nZ."
(declare (vector poly divisor)
((integer 0 #.most-positive-fixnum) exponent)
((integer 1 #.most-positive-fixnum) modulus))
(labels
((recur (power)
(declare ((integer 0 #.most-positive-fixnum) power))
(cond ((zerop power)
(make-array 1 :element-type (array-element-type poly) :initial-element 1))
((oddp power)
(poly-mod! (poly-mult poly (recur (- power 1)) modulus)
divisor modulus))
((let ((res (recur (floor power 2))))
(poly-mod! (poly-mult res res modulus)
divisor modulus))))))
(recur exponent)))
(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
;;;
(defun main ()
(let* ((d (read))
(as (make-array (+ d 1) :element-type 'int32)))
(dotimes (i (+ d 1))
(setf (aref as i) (read)))
(let* ((res (poly-mod! as #(0 -1 0 1) +mod+))
(max-d (position 0 res :from-end t :test-not #'=)))
(if max-d
(progn
(println max-d)
(println-sequence (subseq res 0 (+ max-d 1))
:key (lambda (x) (if (> x 500000000) (- x +mod+) x))))
(format t "0~%0~%")))))
#-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 "5
1 1 1 0 0 1
"
"2
1 2 1
"))
(it.bese.fiveam:is
(common-lisp-user::io-equal "8
0 -5 0 4 0 1 -1 0 1
"
"0
0
"))
(it.bese.fiveam:is
(common-lisp-user::io-equal "5
-5 0 -1 1 1 1
"
"1
-5 2
")))