結果
| 問題 | No.1875 Flip Cards |
| ユーザー |
 MtSaka
|
| 提出日時 | 2022-02-27 11:52:26 |
| 言語 | PyPy3 (7.3.15) |
| 結果 |
AC
|
| 実行時間 | 5,714 ms / 10,000 ms |
| コード長 | 23,142 bytes |
| コンパイル時間 | 351 ms |
| コンパイル使用メモリ | 86,904 KB |
| 実行使用メモリ | 267,732 KB |
| 最終ジャッジ日時 | 2023-09-22 13:06:18 |
| 合計ジャッジ時間 | 26,940 ms |
|
ジャッジサーバーID （参考情報） |
judge12 / judge15 |
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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 | 79 ms
71,548 KB |
| testcase_01 | AC | 85 ms
71,340 KB |
| testcase_02 | AC | 82 ms
71,404 KB |
| testcase_03 | AC | 1,602 ms
165,104 KB |
| testcase_04 | AC | 2,952 ms
251,996 KB |
| testcase_05 | AC | 3,481 ms
264,064 KB |
| testcase_06 | AC | 5,714 ms
266,756 KB |
| testcase_07 | AC | 5,608 ms
267,732 KB |
| testcase_08 | AC | 5,620 ms
267,276 KB |
| testcase_09 | AC | 91 ms
76,188 KB |
ソースコード
#Eki1009さんのものをお借りしました。(https://judge.yosupo.jp/submission/35450)
mod = 998244353
sum_e = (911660635, 509520358, 369330050, 332049552, 983190778, 123842337, 238493703, 975955924, 603855026, 856644456, 131300601, 842657263, 730768835, 942482514, 806263778, 151565301, 510815449, 503497456, 743006876, 741047443, 56250497)
sum_ie = (86583718, 372528824, 373294451, 645684063, 112220581, 692852209, 155456985, 797128860, 90816748, 860285882, 927414960, 354738543, 109331171, 293255632, 535113200, 308540755, 121186627, 608385704, 438932459, 359477183, 824071951)
def sqrt_mod(a):
a %= mod
if a < 2:
return a
k = (mod - 1) // 2
if pow(a, k, mod) != 1:
return -1
b = 1
while pow(b, k, mod) == 1:
b += 1
m, e = mod - 1, 0
while m % 2 == 0:
m >>= 1
e += 1
x = pow(a, (m - 1) // 2, mod)
y = a * x * x % mod
x *= a
x %= mod
z = pow(b, m, mod)
while y != 1:
j, t = 0, y
while t != 1:
j += 1
t *= t
t %= mod
z = pow(z, 1 << (e - j - 1), mod)
x *= z
x %= mod
z *= z
z %= mod
y *= z
y %= mod
e = j
return x
def inv_mod(a):
a %= mod
if a == 0:
return 0
s, t = mod, a
m0, m1 = 0, 1
while t:
u = s // t
s -= t * u
m0 -= m1 * u
s, t = t, s
m0, m1 = m1, m0
if m0 < 0:
m0 += mod // s
return m0
fac_ = [1, 1]
finv_ = [1, 1]
inv_ = [1, 1]
def fac(i):
while i >= len(fac_):
fac_.append(fac_[-1] * len(fac_) % mod)
return fac_[i]
def finv(i):
while i >= len(inv_):
inv_.append((-inv_[mod % len(inv_)]) * (mod // len(inv_)) % mod)
while i >= len(finv_):
finv_.append(finv_[-1] * inv_[len(finv_)] % mod)
return finv_[i]
def inv(i):
while i >= len(inv_):
inv_.append((-inv_[mod % len(inv_)]) * (mod // len(inv_)) % mod)
return inv_[i]
def butterfly(A):
n = len(A)
h = (n - 1).bit_length()
for ph in range(1, h + 1):
w = 1 << (ph - 1)
p = 1 << (h - ph)
now = 1
for s in range(w):
offset = s << (h - ph + 1)
for i in range(p):
l = A[i + offset]
r = A[i + offset + p] * now
A[i + offset] = (l + r) % mod
A[i + offset + p] = (l - r) % mod
now *= sum_e[(~s & -~s).bit_length() - 1]
now %= mod
def butterfly_inv(A):
n = len(A)
h = (n - 1).bit_length()
for ph in range(h, 0, -1):
w = 1 << (ph - 1)
p = 1 << (h - ph)
inow = 1
for s in range(w):
offset = s << (h - ph + 1)
for i in range(p):
l = A[i + offset]
r = A[i + offset + p]
A[i + offset] = (l + r) % mod
A[i + offset + p] = (mod + l - r) * inow % mod
inow *= sum_ie[(~s & -~s).bit_length() - 1]
inow %= mod
iz = inv_mod(n)
for i in range(n):
A[i] *= iz
A[i] %= mod
def convolution(_A, _B):
A = _A.copy()
B = _B.copy()
n = len(A)
m = len(B)
if not n or not m:
return []
if min(n, m) <= 60:
if n < m:
n, m = m, n
A, B = B, A
res = [0] * (n + m - 1)
for i in range(n):
for j in range(m):
res[i + j] += A[i] * B[j]
res[i + j] %= mod
return res
z = 1 << (n + m - 2).bit_length()
A += [0] * (z - n)
B += [0] * (z - m)
butterfly(A)
butterfly(B)
for i in range(z):
A[i] *= B[i]
A[i] %= mod
butterfly_inv(A)
return A[:n + m - 1]
def autocorrelation(_A):
A = _A.copy()
n = len(A)
if not n:
return []
if n <= 60:
res = [0] * (n + n - 1)
for i in range(n):
for j in range(n):
res[i + j] += A[i] * A[j]
res[i + j] %= mod
return res
z = 1 << (n + n - 2).bit_length()
A += [0] * (z - n)
butterfly(A)
for i in range(z):
A[i] *= A[i]
A[i] %= mod
butterfly_inv(A)
return A[:n + n - 1]
class FormalPowerSeries:
def __init__(self, poly=[]):
self.poly = [p % mod for p in poly]
def __getitem__(self, key):
if isinstance(key, slice):
res = self.poly[key]
return FormalPowerSeries(res)
else:
if key < 0:
raise IndexError("list index out of range")
if key >= len(self.poly):
return 0
else:
return self.poly[key]
def __setitem__(self, key, value):
if key < 0:
raise IndexError("list index out of range")
if key >= len(self.poly):
self.poly += [0] * (key - len(self.poly) + 1)
self.poly[key] = value % mod
def __len__(self):
return len(self.poly)
def resize(self, size):
if len(self) >= size:
return self[:size]
else:
return FormalPowerSeries(self.poly + [0] * (size - len(self)))
def shrink(self):
while self.poly and self.poly[-1] == 0:
self.poly.pop()
def times(self, n):
n %= mod
res = [p * n for p in self.poly]
return FormalPowerSeries(res)
def __pos__(self):
return self
def __neg__(self):
return self.times(-1)
def __add__(self, other):
if other.__class__ == FormalPowerSeries:
s = len(self)
t = len(other)
n = min(s, t)
res = [self[i] + other[i] for i in range(n)]
if s >= t:
res += self.poly[t:]
else:
res += other.poly[s:]
return FormalPowerSeries(res)
else:
return self + FormalPowerSeries([other])
def __radd__(self, other):
return self + other
def __sub__(self, other):
return self + (-other)
def __rsub__(self, other):
return (-self) + other
def __mul__(self, other):
if other.__class__ == FormalPowerSeries:
res = convolution(self.poly, other.poly)
return FormalPowerSeries(res)
else:
return self.times(other)
def __rmul__(self, other):
return self.times(other)
def __lshift__(self, other):
return FormalPowerSeries(([0] * other) + self.poly)
def __rshift__(self, other):
return self[other:]
def square(self):
res = autocorrelation(self.poly)
return FormalPowerSeries(res)
def inv(self, deg=-1):
if deg == -1:
deg = len(self) - 1
r = inv_mod(self[0])
m = 1
T = [0] * (deg + 1)
T[0] = r
res = FormalPowerSeries(T)
while m <= deg:
F = [0] * (2 * m)
G = [0] * (2 * m)
for j in range(min(len(self), 2 * m)):
F[j] = self[j]
for j in range(m):
G[j] = res[j]
butterfly(F)
butterfly(G)
for j in range(2 * m):
F[j] *= G[j]
F[j] %= mod
butterfly_inv(F)
for j in range(m):
F[j] = 0
butterfly(F)
for j in range(2 * m):
F[j] *= G[j]
F[j] %= mod
butterfly_inv(F)
for j in range(m, min(2 * m, deg + 1)):
res[j] = -F[j]
m <<= 1
return res
#P/Q
def __truediv__(self, other):
if other.__class__ == FormalPowerSeries:
return (self * other.inv())
else:
return self * inv_mod(other)
def __rtruediv__(self, other):
return other * self.inv()
#P,Qを多項式として見たときのPをQで割った商を求める
def __floordiv__(self, other):
if other.__class__ == FormalPowerSeries:
if len(self) < len(other):
return FormalPowerSeries()
else:
m = len(self) - len(other) + 1
res = (self[-1:-m-1:-1] * other[::-1].inv(m))[m-1::-1]
return res
else:
return self * inv_mod(other)
def __rfloordiv__(self, other):
return other * self.inv()
def __mod__(self, other):
if len(self) < len(other):
return self[:]
else:
res = self[:len(other) - 1] - ((self // other) * other)[:len(other) - 1]
return res
def differentiate(self, deg=-1):
if deg == -1:
deg = len(self) - 2
res = FormalPowerSeries([0] * (deg + 1))
for i in range(1, min(len(self), deg + 2)):
res[i - 1] = self[i] * i
return res
def integrate(self, deg=-1):
if deg == -1:
deg = len(self)
res = FormalPowerSeries([0] * (deg + 1))
for i in range(min(len(self), deg)):
res[i + 1] = self[i] * inv(i + 1)
return res
def log(self, deg=-1):
if deg == -1:
deg = len(self) - 1
return (self.differentiate() * self.inv(deg - 1))[:deg].integrate()
def exp(self, deg=-1):
if deg == -1:
deg = len(self) - 1
T = [0] * (deg + 1)
T[0] = 1 #T:res^{-1}
res = FormalPowerSeries(T)
m = 1
F = [1]
while m <= deg:
G = T[:m]
butterfly(G)
FG = [F[i] * G[i] % mod for i in range(m)]
butterfly_inv(FG)
FG[0] -= 1
delta = [0] * (2 * m)
for i in range(m):
delta[i + m] = FG[i]
eps = [0] * (2 * m)
if m == 1:
DF = []
else:
DF = res.differentiate(m - 2).poly
DF.append(0)
butterfly(DF)
for i in range(m):
DF[i] *= G[i]
DF[i] %= mod
butterfly_inv(DF)
for i in range(m - 1):
eps[i] = self[i + 1] * (i + 1) % mod
eps[i + m] = DF[i] - eps[i]
eps[m - 1] = DF[m - 1]
butterfly(delta)
DH = [0] * (2 * m)
for i in range(m - 1):
DH[i] = eps[i]
butterfly(DH)
for i in range(2 * m):
delta[i] *= DH[i]
delta[i] %= mod
butterfly_inv(delta)
for i in range(m, 2 * m):
eps[i] -= delta[i]
eps[i] %= mod
for i in range(2 * m - 1, 0, -1):
eps[i] = (eps[i - 1] * inv(i) - self[i]) % mod
eps[0] = -self[0]
butterfly(eps)
for i in range(m):
DH[i] = res[i]
DH[i + m] = 0
butterfly(DH)
for i in range(2 * m):
eps[i] *= DH[i]
eps[i] %= mod
butterfly_inv(eps)
for i in range(m, min(2 * m, deg + 1)):
res[i] = -eps[i]
if 2 * m > deg:
break
F = [0] * (2 * m)
G = [0] * (2 * m)
for i in range(2 * m):
F[i] = res[i]
for i in range(m):
G[i] = T[i]
butterfly(F)
butterfly(G)
P = [F[i] * G[i] % mod for i in range(2 * m)]
butterfly_inv(P)
for i in range(m):
P[i] = 0
butterfly(P)
for i in range(2 * m):
P[i] *= G[i]
P[i] %= mod
butterfly_inv(P)
for i in range(m, 2 * m):
T[i] = -P[i]
m <<= 1
return res
def __pow__(self, n, deg=-1):
if deg == -1:
deg = len(self) - 1
m = abs(n)
for d, p in enumerate(self.poly):
if p:
break
else:
return FormalPowerSeries()
if d * m >= len(self):
return FormalPowerSeries()
G = self[d:]
G = ((G * inv_mod(p)).log() * m).exp() * pow(p, m, mod)
res = FormalPowerSeries([0] * (d * m) + G.poly)
if n >= 0:
return res[:deg + 1]
else:
return res.inv()[:deg + 1]
def sqrt(self, deg=-1):
if deg == -1:
deg = len(self) - 1
if len(self) == 0:
return FormalPowerSeries()
if self[0] == 0:
for d in range(1, len(self)):
if self[d]:
if d & 1:
return -1
if deg < d // 2:
break
res = self[d:].sqrt(deg - d // 2)
if res == -1:
return -1
res = res << (d // 2)
return res
return FormalPowerSeries()
sqr = sqrt_mod(self[0])
if sqr == -1:
return -1
T = [0] * (deg + 1)
T[0] = sqr
res = FormalPowerSeries(T)
T[0] = inv_mod(sqr) #T:res^{-1}
m = 1
two_inv = (mod + 1) // 2
F = [sqr]
while m <= deg:
for i in range(m):
F[i] *= F[i]
F[i] %= mod
butterfly_inv(F)
delta = [0] * (2 * m)
for i in range(m):
delta[i + m] = F[i] - self[i] - self[i + m]
butterfly(delta)
G = [0] * (2 * m)
for i in range(m):
G[i] = T[i]
butterfly(G)
for i in range(2 * m):
delta[i] *= G[i]
delta[i] %= mod
butterfly_inv(delta)
for i in range(m, min(2 * m, deg + 1)):
res[i] = -delta[i] * two_inv
if 2 * m > deg:
break
F = res.poly[:2 * m]
butterfly(F)
eps = [F[i] * G[i] % mod for i in range(2 * m)]
butterfly_inv(eps)
for i in range(m):
eps[i] = 0
butterfly(eps)
for i in range(2 * m):
eps[i] *= G[i]
eps[i] %= mod
butterfly_inv(eps)
for i in range(m, 2 * m):
T[i] = -eps[i]
m <<= 1
return res
#各p_i(0<=i<m)についてf(p_i)を求める
def multipoint_evaluation(self, P):
m = len(P)
size = 1 << (m - 1).bit_length()
G = [FormalPowerSeries([1]) for _ in range(2 * size)]
for i in range(m):
G[size + i] = FormalPowerSeries([-P[i], 1])
for i in range(size - 1, 0, -1):
G[i] = G[2 * i] * G[2 * i + 1]
G[1] = self % G[1]
for i in range(2, size + m):
G[i] = G[i >> 1] % G[i]
res = [G[i][0] for i in range(size, size + m)]
return res
#f(x+a)
def taylor_shift(self, a):
a %= mod
n = len(self)
t = 1
F = self[:]
G = FormalPowerSeries([0] * n)
for i in range(n):
F[i] *= fac(i)
for i in range(n):
G[i] = t * finv(i)
t = t * a % mod
res = (F * G[::-1])[n - 1:]
for i in range(n):
res[i] *= finv(i)
return res
#Q(P)
def composition(self, P, deg=-1):
if deg == -1:
deg = len(self) - 1
k = int(deg ** 0.5 + 1)
d = (deg + k) // k
X = [FormalPowerSeries([1])]
for i in range(k):
X.append((X[i] * P)[:deg + 1])
Y = [FormalPowerSeries([0] * (deg + 1)) for _ in range(k)]
for i in range(k):
for j in range(d):
if i * d + j > deg:
break
for t in range(deg + 1):
if t >= len(X[j]):
break
Y[i][t] += X[j][t] * self[i * d + j]
res = FormalPowerSeries([0] * (deg + 1))
Z = FormalPowerSeries([1])
for i in range(k):
Y[i] = (Y[i] * Z)[:deg + 1]
for j in range(len(Y[i])):
res[j] += Y[i][j]
Z = (Z * X[d])[:deg + 1]
return res
#[x^n]P/Qを求める
def poly_coef(Q, P, n):
while n:
R = [0] * len(Q.poly)
for i, q in enumerate(Q.poly):
if i & 1:
R[i] = -q
else:
R[i] = q
R = FormalPowerSeries(R)
P = P * R
Q = Q * R
if n & 1:
P = P[1::2]
else:
P = P[::2]
Q = Q[::2]
n >>= 1
return P[0]
def subset_sum(A, limit):
C = [0] * (limit + 1)
for a in A:
C[a] += 1
res = FormalPowerSeries([0] * (limit + 1))
for i in range(1, limit + 1):
for k in range(1, limit // i + 1):
j = i * k
res[j] += ((k & 1) * 2 - 1) * C[i] * inv(k)
return res.exp(limit).poly
def partition_function(n):
res = FormalPowerSeries([0] * (n + 1))
res[0] = 1
for k in range(1, n+1):
k1 = k * (3 * k + 1 )// 2
k2 = k * (3 * k - 1) // 2
if k2 > n:
break
if k1 <= n:
res[k1] += 1 - (k & 1) * 2
if k2 <= n:
res[k2] += 1 - (k & 1) * 2
return res.inv().poly
def bernoulli_number(n):
n += 1
Q = FormalPowerSeries([finv(i + 1) for i in range(n)]).inv(n - 1)
res = [v * fac(i) % mod for i, v in enumerate(Q.poly)]
return res
def stirling_first(n):
P = []
a = n
while a:
if a & 1:
P.append(1)
P.append(0)
a >>= 1
res = FormalPowerSeries([1])
t = 0
for x in P[::-1]:
if x:
res *= FormalPowerSeries([-t, 1])
t += 1
else:
res *= res.taylor_shift(-t)
t *= 2
return res.poly
def stirling_second(n):
F = FormalPowerSeries([0] * (n + 1))
G = FormalPowerSeries([0] * (n + 1))
for i in range(n + 1):
F[i] = pow(i, n, mod) * finv(i)
G[i] = (1 - (i & 1) * 2) * finv(i)
return (F * G)[:n + 1].poly
def polynominal_interpolation(X, Y):
n = len(X)
size = 1 << (n - 1).bit_length()
M = [FormalPowerSeries([1]) for _ in range(2 * size)]
G = [0] * (2 * size)
for i in range(n):
M[size + i] = FormalPowerSeries([-X[i], 1])
for i in range(size - 1, 0, -1):
M[i] = M[2 * i] * M[2 * i + 1]
G[1] = M[1].differentiate() % M[1]
for i in range(2, size + n):
G[i] = G[i >> 1] % M[i]
for i in range(n):
G[size + i] = FormalPowerSeries([Y[i] * inv_mod(G[size + i][0])])
for i in range(size - 1, 0, -1):
G[i] = G[2 * i] * M[2 * i + 1] + G[2 * i + 1] * M[2 * i]
return G[1][:n]
class Mat2:
def __init__(self, a00=FormalPowerSeries([1]), a01=FormalPowerSeries(),
a10=FormalPowerSeries(), a11=FormalPowerSeries([1])):
self.a00 = a00
self.a01 = a01
self.a10 = a10
self.a11 = a11
def __mul__(self, other):
if type(other) == Mat2:
A00 = self.a00 * other.a00 + self.a01 * other.a10
A01 = self.a00 * other.a01 + self.a01 * other.a11
A10 = self.a10 * other.a00 + self.a11 * other.a10
A11 = self.a10 * other.a01 + self.a11 * other.a11
A00.shrink()
A01.shrink()
A10.shrink()
A11.shrink()
return Mat2(A00, A01, A10, A11)
else:
b0 = self.a00 * other[0] + self.a01 * other[1]
b1 = self.a10 * other[0] + self.a11 * other[1]
b0.shrink()
b1.shrink()
return (b0, b1)
def __imul__(self, other):
A00 = self.a00 * other.a00 + self.a01 * other.a10
A01 = self.a00 * other.a01 + self.a01 * other.a11
A10 = self.a10 * other.a00 + self.a11 * other.a10
A11 = self.a10 * other.a01 + self.a11 * other.a11
A00.shrink()
A01.shrink()
A10.shrink()
A11.shrink()
self.a0 = A0
self.a0 = A0
self.a0 = A0
self.a0 = A0
def _inner_naive_gcd(m, p):
quo = p[0] // p[1]
rem = p[0] - p[1] * quo
b10 = m.a00 - m.a10 * quo
b11 = m.a01 - m.a11 * quo
rem.shrink()
b10.shrink()
b11.shrink()
b10, m.a10 = m.a10, b10
b11, m.a11 = m.a11, b11
b10, m.a00 = m.a00, b10
b11, m.a01 = m.a01, b11
return (p[1], rem)
def _inner_half_gcd(p):
n, m = len(p[0]), len(p[1])
k = (n + 1) // 2
if m <= k:
return Mat2()
m1 = _inner_half_gcd((p[0] >> k, p[1] >> k))
p = m1 * p
if len(p[1]) <= k:
return m1
p = _inner_naive_gcd(m1, p)
if len(p[1]) <= k:
return m1
l = len(p[0]) - 1
j = 2 * k - 1
p = (p[0] >> j, p[1] >> j)
return _inner_half_gcd(p) * m1
def _inner_poly_gcd(a, b):
p = (a, b)
p[0].shrink()
p[1].shrink()
n, m = len(p[0]), len(p[1])
if n < m:
mat = _inner_poly_gcd(p[1], p[0])
mat.a00, mat.a01 = mat.a01, mat.a00
mat.a10, mat.a11 = mat.a11, mat.a10
return mat
res = Mat2()
while 1:
m1 = _inner_half_gcd(p)
p = m1 * p
if len(p[1]) == 0:
return m1 * res
p = _inner_naive_gcd(m1, p)
if len(p[1]) == 0:
return m1 * res
res = m1 * res
def poly_gcd(a, b):
p = (a, b)
m = _inner_poly_gcd(a, b)
p = m * p
if len(p[0]):
coef = inv_mod(p[0][-1])
p[0] *= coef
return p[0]
def poly_inv(f, g):
p = (f, g)
m = _inner_poly_gcd(f, g)
_gcd = (m * p)[0]
if len(_gcd) != 1:
return -1
x = (FormalPowerSeries([1]), g)
res = ((m * x)[0] % g) * inv_mod(_gcd[0])
res.shrink()
return res
import sys
input=sys.stdin.readline
write=sys.stdout.write
def inv_sum(a,b,m):
sz=1
while sz<len(a):
sz<<=1
num=[FormalPowerSeries()]*(2*sz-1)
den=[FormalPowerSeries()]*(2*sz-1)
for i in range(sz-1,2*sz-1):
if(i-sz+1<len(a)):
num[i]=a[i-sz+1]
den[i]=b[i-sz+1]
else:
num[i]=FormalPowerSeries([0])
den[i]=FormalPowerSeries([1])
for i in range(sz-2,-1,-1):
den[i]=den[2*i+1]*den[2*i+2]
num[i]=num[2*i+1]*den[2*i+2]+num[2*i+2]*den[2*i+1]
num[0]=num[0]*(den[0].inv(m))
num[0]=num[0].resize(m+1)
return num[0]
n,m=map(int,input().split())
prod=1
f=[0]*(n)
g=[0]*(n)
for i in range(n):
a,b,c=map(int,input().split())
prod*=pow(a,c,mod)
prod%=mod
t=b*inv_mod(a)%mod
f[i]=FormalPowerSeries([t*c%mod])
g[i]=FormalPowerSeries([1,t])
h=inv_sum(f,g,m)
h=h.integrate().resize(m+1).exp().taylor_shift(-1)
f=[0]*(m+1)
g=[0]*(m+1)
for i in range(m+1):
f[i]=FormalPowerSeries([h[i]])
g[i]=FormalPowerSeries([1,(mod-i)%mod])
h=inv_sum(f,g,m)
for i in range(1,m+1):
write(str(h[i]*prod%mod)+'\n')