結果
| 問題 | No.995 タピオカオイシクナーレ |
| ユーザー |
 sansaqua
|
| 提出日時 | 2020-02-21 21:50:21 |
| 言語 | Common Lisp (sbcl 2.3.8) |
| 結果 |
AC
|
| 実行時間 | 33 ms / 2,000 ms |
| コード長 | 20,381 bytes |
| コンパイル時間 | 719 ms |
| コンパイル使用メモリ | 87,296 KB |
| 実行使用メモリ | 23,040 KB |
| 最終ジャッジ日時 | 2024-04-17 08:04:50 |
| 合計ジャッジ時間 | 2,330 ms |
|
ジャッジサーバーID （参考情報） |
judge3 / judge2 |
（要ログイン）
テストケース
テストケース表示| 入力 | 結果 | 実行時間 実行使用メモリ |
|---|---|---|
| testcase_00 | AC | 12 ms
23,040 KB |
| testcase_01 | AC | 11 ms
22,912 KB |
| testcase_02 | AC | 10 ms
22,912 KB |
| testcase_03 | AC | 11 ms
23,040 KB |
| testcase_04 | AC | 10 ms
22,912 KB |
| testcase_05 | AC | 11 ms
23,040 KB |
| testcase_06 | AC | 10 ms
22,912 KB |
| testcase_07 | AC | 10 ms
22,912 KB |
| testcase_08 | AC | 10 ms
22,784 KB |
| testcase_09 | AC | 11 ms
22,912 KB |
| testcase_10 | AC | 11 ms
22,912 KB |
| testcase_11 | AC | 11 ms
22,912 KB |
| testcase_12 | AC | 11 ms
22,912 KB |
| testcase_13 | AC | 10 ms
23,040 KB |
| testcase_14 | AC | 11 ms
23,040 KB |
| testcase_15 | AC | 10 ms
22,912 KB |
| testcase_16 | AC | 33 ms
22,912 KB |
| testcase_17 | AC | 33 ms
22,912 KB |
| testcase_18 | AC | 32 ms
22,912 KB |
| testcase_19 | AC | 32 ms
22,912 KB |
| testcase_20 | AC | 32 ms
22,912 KB |
| testcase_21 | AC | 32 ms
22,912 KB |
| testcase_22 | AC | 33 ms
22,912 KB |
| testcase_23 | AC | 33 ms
22,912 KB |
| testcase_24 | AC | 30 ms
22,912 KB |
| testcase_25 | AC | 31 ms
22,912 KB |
コンパイルメッセージ
; compiling file "/home/judge/data/code/Main.lisp" (written 17 APR 2024 08:04:47 AM): ; file: /home/judge/data/code/Main.lisp ; in: DEFUN MOD-INVERSE ; (+ U MODULUS) ; ; note: deleting unreachable code ; in: DEFUN MOD-LOG ; (GCD X MODULUS) ; ; note: unable to ; optimize ; due to type uncertainty: ; The first argument is a INTEGER, not a (OR (INTEGER -4611686018427387904 -1) ; (INTEGER 1 4611686018427387903)). ; (* RES X) ; ; note: forced to do */UNSIGNED=>INTEGER (cost 10) ; unable to do inline fixnum arithmetic (cost 2) because: ; The result is a (VALUES (MOD 21267647932558653948014168890775961605) ; &OPTIONAL), not a (VALUES FIXNUM &OPTIONAL). ; unable to do inline (signed-byte 64) arithmetic (cost 4) because: ; The result is a (VALUES (MOD 21267647932558653948014168890775961605) ; &OPTIONAL), not a (VALUES (SIGNED-BYTE 64) ; &OPTIONAL). ; (* X^M^I X^M) ; ; note: forced to do */UNSIGNED=>INTEGER (cost 10) ; unable to do inline fixnum arithmetic (cost 2) because: ; The result is a (VALUES (MOD 21267647932558653948014168890775961605) ; &OPTIONAL), not a (VALUES FIXNUM &OPTIONAL). ; unable to do inline (signed-byte 64) arithmetic (cost 4) because: ; The result is a (VALUES (MOD 21267647932558653948014168890775961605) ; &OPTIONAL), not a (VALUES (SIGNED-BYTE 64) ; &OPTIONAL). ; (* RES X) ; ; note: forced to do */UNSIGNED=>INTEGER (cost 10) ; unable to do inline fixnum arithmetic (cost 2) because: ; The result is a (VALUES (MOD 21267647932558653948014168890775961605) ; &OPTIONAL), not a (VALUES FIXNUM &OPTIONAL). ; unable to do inline (signed-byte 64) arithmetic (cost 4
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 (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 #\-))
(setf 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))))))))
;;;
;;; 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))
(sb-c:define-source-transform mod* (&rest args)
(if (null args)
1
(reduce (lambda (x y) `(mod (* ,x ,y) ,',divisor)) args)))
(defun mod+ (&rest args)
(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)))
(defun mod- (&rest args)
(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)))))
;;;
;;; Modular arithmetic
;;;
;; Blankinship algorithm
;; Reference: https://topcoder.g.hatena.ne.jp/spaghetti_source/20130126/1359171466 (Japanese)
(declaim (ftype (function * (values fixnum fixnum &optional)) %ext-gcd))
(defun %ext-gcd (a b)
(declare (optimize (speed 3) (safety 0))
(fixnum a b))
(let ((y 1)
(x 0)
(u 1)
(v 0))
(declare (fixnum y x u v))
(loop (when (zerop a)
(return (values x y)))
(let ((q (floor b a)))
(decf x (the fixnum (* q u)))
(rotatef x u)
(decf y (the fixnum (* q v)))
(rotatef y v)
(decf b (the fixnum (* q a)))
(rotatef b a)))))
;; recursive version
;; (defun %ext-gcd (a b)
;; (declare (optimize (speed 3) (safety 0))
;; (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) (%ext-gcd b q)
;; (declare (fixnum u v))
;; (values u (the fixnum (- v (the fixnum (* p u)))))))))
;; TODO: deal with bignums
(declaim (inline ext-gcd))
(defun ext-gcd (a b)
"Returns two integers X and Y which satisfy AX + BY = gcd(A, B)."
(declare ((integer #.(- most-positive-fixnum) #.most-positive-fixnum) a b))
(if (>= a 0)
(if (>= b 0)
(%ext-gcd a b)
(multiple-value-bind (x y) (%ext-gcd a (- b))
(declare (fixnum x y))
(values x (- y))))
(if (>= b 0)
(multiple-value-bind (x y) (%ext-gcd (- a) b)
(declare (fixnum x y))
(values (- x) y))
(multiple-value-bind (x y) (%ext-gcd (- a) (- b))
(declare (fixnum x y))
(values (- x) (- y))))))
(declaim (inline mod-inverse)
(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 (integer a)
;; ((integer 1 #.most-positive-fixnum) modulus))
;; (mod (%ext-gcd (mod a modulus) modulus) modulus))
(defun mod-inverse (a modulus)
(declare ((integer 1 #.most-positive-fixnum) modulus))
(let ((a (mod a modulus))
(b modulus)
(u 1)
(v 0))
(declare (fixnum a b u v))
(loop until (zerop b)
for quot = (floor a b)
do (decf a (the fixnum (* quot b)))
(rotatef a b)
(decf u (the fixnum (* quot v)))
(rotatef u v))
(setq u (mod u modulus))
(if (< u 0)
(+ u modulus)
u)))
;; not tested
;; TODO: move to another file
(declaim (inline binomial))
(defun mod-binomial (n k modulus)
(declare ((integer 0 #.most-positive-fixnum) modulus))
(if (or (< n k) (< n 0) (< k 0))
0
(let ((k (if (< k (- n k)) k (- n k)))
(num 1)
(denom 1))
(declare ((integer 0) k num denom))
(loop for x from n above (- n k)
do (setq num (mod (* num x) modulus)))
(loop for x from 1 to k
do (setq denom (mod (* denom x) modulus)))
(mod (* num (mod-inverse denom modulus)) modulus))))
(declaim (ftype (function * (values (or null (integer 1 #.most-positive-fixnum)) &optional)) mod-log))
(defun mod-log (x y modulus)
"Returns the smallest positive integer k that satiefies x^k ≡ y mod p.
Returns NIL if it is infeasible."
(declare (optimize (speed 3))
(integer x y)
((integer 1 #.most-positive-fixnum) modulus))
(let ((x (mod x modulus))
(y (mod y modulus))
(g (gcd x modulus)))
(declare (optimize (safety 0))
((mod #.most-positive-fixnum) x y g))
(if (= g 1)
;; coprime case
(let* ((m (+ 1 (isqrt (- modulus 1)))) ; smallest integer equal to or
; larger than sqrt(p)
(x^m (loop for i below m
for res of-type (integer 0 #.most-positive-fixnum) = x
then (mod (* res x) modulus)
finally (return res)))
(table (make-hash-table :size m :test 'eq)))
;; Constructs TABLE: yx^j |-> j (j = 0, ..., m-1)
(loop for j from 0 below m
for res of-type (integer 0 #.most-positive-fixnum) = y
then (mod (* res x) modulus)
do (setf (gethash res table) j))
;; Finds i and j that satisfy (x^m)^i = yx^j and returns m*i-j
(loop for i from 1 to m
for x^m^i of-type (integer 0 #.most-positive-fixnum) = x^m
then (mod (* x^m^i x^m) modulus)
for j = (gethash x^m^i table)
when j
do (locally
(declare ((integer 0 #.most-positive-fixnum) j))
(return (- (* i m) j)))
finally (return nil)))
;; If x and p are not coprime, let g := gcd(x, p), x := gx', y := gy', p
;; := gp' and solve x^(k-1) ≡ y'x'^(-1) mod p' instead. See
;; https://math.stackexchange.com/questions/131127/ for the detail.
(if (= x y)
;; This is tha special treatment for the case x ≡ y. Without this
;; (mod-log 4 0 4) returns not 1 but 2.
1
(multiple-value-bind (y-prime rem) (floor y g)
(if (zerop rem)
(let* ((x-prime (floor x g))
(p-prime (floor modulus g))
(next-rhs (mod (* y-prime (mod-inverse x-prime p-prime)) p-prime))
(res (mod-log x next-rhs p-prime)))
(declare ((integer 0 #.most-positive-fixnum) x-prime p-prime next-rhs))
(if res (+ 1 res) nil))
nil))))))
(declaim (inline %calc-min-factor))
(defun %calc-min-factor (x alpha)
"Returns k, so that x+k*alpha is the smallest non-negative number."
(if (plusp alpha)
(ceiling (- x) alpha)
(floor (- x) alpha)))
(declaim (inline %calc-max-factor))
(defun %calc-max-factor (x alpha)
"Returns k, so that x+k*alpha is the largest non-positive number."
(if (plusp alpha)
(floor (- x) alpha)
(ceiling (- x) alpha)))
(defun solve-bezout (a b c &optional min max)
"Returns an integer solution of a*x+b*y = c if it exists, otherwise
returns (VALUES NIL NIL).
If MIN is specified and MAX is null, the returned x is the smallest integer
equal to or larger than MIN. If MAX is specified and MIN is null, x is the
largest integer equal to or smaller than MAX. If the both are specified, x is an
integer in [MIN, MAX]. This function returns NIL when no x that satisfies the
given condition exists."
(declare (fixnum a b c)
((or null fixnum) min max))
(let ((gcd-ab (gcd a b)))
(if (zerop (mod c gcd-ab))
(multiple-value-bind (init-x init-y) (ext-gcd a b)
(let* ((factor (floor c gcd-ab))
;; m*x0 + n*y0 = d
(x0 (* init-x factor))
(y0 (* init-y factor)))
(if (and (null min) (null max))
(values x0 y0)
(let (;; general solution: x = x0 + kΔx, y = y0 - kΔy
(deltax (floor b gcd-ab))
(deltay (floor a gcd-ab)))
(if min
(let* ((k-min (%calc-min-factor (- x0 min) deltax))
(x (+ x0 (* k-min deltax)))
(y (- y0 (* k-min deltay))))
(if (and max (> x max))
(values nil nil)
(values x y)))
(let* ((k-max (%calc-max-factor (- x0 max) deltax))
(x (+ x0 (* k-max deltax)))
(y (- y0 (* k-max deltay))))
(if (<= x max)
(values x y)
(values nil nil))))))))
(values nil nil))))
;; Reference: http://drken1215.hatenablog.com/entry/2019/03/20/202800
(declaim (inline mod-echelon!))
(defun mod-echelon! (matrix modulus &optional extended)
"Returns the row echelon form of MATRIX by gaussian elimination and returns
the rank as the second value.
This function destructively modifies MATRIX."
(declare ((integer 1 #.most-positive-fixnum) modulus))
(destructuring-bind (m n) (array-dimensions matrix)
(declare ((integer 0 #.most-positive-fixnum) m n))
(dotimes (i m)
(dotimes (j n)
(setf (aref matrix i j) (mod (aref matrix i j) modulus))))
(let ((rank 0))
(dotimes (target-col (if extended (- n 1) n))
(let ((pivot-row (do ((i rank (+ 1 i)))
((= i m) -1)
(unless (zerop (aref matrix i target-col))
(return i)))))
(when (>= pivot-row 0)
;; swap rows
(loop for j from target-col below n
do (rotatef (aref matrix rank j) (aref matrix pivot-row j)))
(let ((inv (mod-inverse (aref matrix rank target-col) modulus)))
(dotimes (j n)
(setf (aref matrix rank j)
(mod (* inv (aref matrix rank j)) modulus)))
(dotimes (i m)
(unless (or (= i rank) (zerop (aref matrix i target-col)))
(let ((factor (aref matrix i target-col)))
(loop for j from target-col below n
do (setf (aref matrix i j)
(mod (- (aref matrix i j)
(mod (* (aref matrix rank j) factor) modulus))
modulus)))))))
(incf rank))))
(values matrix rank))))
(declaim (inline mod-inverse-matrix!))
(defun mod-inverse-matrix! (matrix modulus)
"Returns the inverse of MATRIX by gaussian elimination if it exists and
returns NIL otherwise. This function destructively modifies MATRIX."
(declare ((integer 1 #.most-positive-fixnum) modulus))
(destructuring-bind (m n) (array-dimensions matrix)
(declare ((integer 0 #.most-positive-fixnum) m n))
(assert (= m n))
(dotimes (i n)
(dotimes (j n)
(setf (aref matrix i j) (mod (aref matrix i j) modulus))))
(let ((result (make-array (list n n) :element-type (array-element-type matrix))))
(dotimes (i n) (setf (aref result i i) 1))
(dotimes (target n)
(let ((pivot-row (do ((i target (+ 1 i)))
((= i n) -1)
(unless (zerop (aref matrix i target))
(return i)))))
(when (= pivot-row -1) ; when singular
(return-from mod-inverse-matrix! nil))
(loop for j from target below n
do (rotatef (aref matrix target j) (aref matrix pivot-row j))
(rotatef (aref result target j) (aref result pivot-row j)))
(let ((inv (mod-inverse (aref matrix target target) modulus)))
;; process the pivot row
(dotimes (j n)
(setf (aref matrix target j)
(mod (* inv (aref matrix target j)) modulus))
(setf (aref result target j)
(mod (* inv (aref result target j)) modulus)))
;; eliminate the column
(dotimes (i n)
(unless (or (= i target) (zerop (aref matrix i target)))
(let ((factor (aref matrix i target)))
(dotimes (j n)
(setf (aref matrix i j)
(mod (- (aref matrix i j)
(mod (* (aref matrix target j) factor) modulus))
modulus))
(setf (aref result i j)
(mod (- (aref result i j)
(mod (* (aref result target j) factor) modulus))
modulus)))))))))
result)))
(declaim (inline mod-solve-linear-system))
(defun mod-solve-linear-system (matrix vector modulus)
"Solves Ax ≡ b and returns a root vector if it exists. Otherwise it returns
NIL. In addition, this function returns the rank of A as the second value."
(destructuring-bind (m n) (array-dimensions matrix)
(declare ((integer 0 #.most-positive-fixnum) m n))
(assert (= n (length vector)))
(let ((extended (make-array (list m (+ n 1)) :element-type (array-element-type matrix))))
(dotimes (i m)
(dotimes (j n) (setf (aref extended i j) (aref matrix i j)))
(setf (aref extended i n) (aref vector i)))
(let ((rank (nth-value 1 (mod-echelon! extended modulus t))))
(if (loop for i from rank below m
always (zerop (aref extended i n)))
(let ((result (make-array m
:element-type (array-element-type matrix)
:initial-element 0)))
(dotimes (i rank)
(setf (aref result i) (aref extended i n)))
(values result rank))
(values nil rank))))))
(declaim (inline power-mod))
(defun power-mod (base power modulus)
"BASE := integer
POWER, MODULUS := non-negative fixnum"
(declare ((integer 0 #.most-positive-fixnum) modulus power)
(integer base))
(labels ((recur (x p)
(declare ((integer 0 #.most-positive-fixnum) x p)
(values (integer 0 #.most-positive-fixnum)))
(cond ((zerop p) 1)
((evenp p) (recur (mod (* x x) modulus) (ash p -1)))
(t (mod (* x (recur x (- p 1))) modulus)))))
(recur (mod base modulus) power)))
(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 ()
(let* ((n (read))
(m (read))
(k (read))
(p (read))
(q (read))
(p/q (mod* p (mod-inverse q +mod+)))
(even-factor (mod* 500000004 (mod+ 1
(power-mod (mod (- 1 (* 2 p/q)) +mod+)
k
+mod+))))
(odd-factor (mod* 500000004 (mod- 1 (power-mod (mod (- 1 (* 2 p/q)) +mod+)
k
+mod+))))
(b1 0)
(b2 0))
(loop for i below n
for b = (read-fixnum)
do (if (< i m)
(incf b1 b)
(incf b2 b)))
(println (mod+ (mod* even-factor b1)
(mod* odd-factor b2)))))
#-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)))