結果
| 問題 |
No.189 SUPER HAPPY DAY
|
| ユーザー |
 ntuda
|
| 提出日時 | 2025-06-25 20:17:54 |
| 言語 | PyPy3 (7.3.15) |
| 結果 |
AC
|
| 実行時間 | 216 ms / 5,000 ms |
| コード長 | 11,090 bytes |
| コンパイル時間 | 307 ms |
| コンパイル使用メモリ | 82,612 KB |
| 実行使用メモリ | 81,196 KB |
| 最終ジャッジ日時 | 2025-06-25 20:18:00 |
| 合計ジャッジ時間 | 4,711 ms |
|
ジャッジサーバーID （参考情報） |
judge1 / judge2 |
（要ログイン）
| ファイルパターン | 結果 |
|---|---|
| sample | AC * 3 |
| other | AC * 23 |
ソースコード
MOD = 10 ** 9 + 9
import typing
def _ceil_pow2(n: int) -> int:
x = 0
while (1 << x) < n:
x += 1
return x
def _bsf(n: int) -> int:
x = 0
while n % 2 == 0:
x += 1
n //= 2
return x
def _is_prime(n: int) -> bool:
'''
Reference:
M. Forisek and J. Jancina,
Fast Primality Testing for Integers That Fit into a Machine Word
'''
if n <= 1:
return False
if n == 2 or n == 7 or n == 61:
return True
if n % 2 == 0:
return False
d = n - 1
while d % 2 == 0:
d //= 2
for a in (2, 7, 61):
t = d
y = pow(a, t, n)
while t != n - 1 and y != 1 and y != n - 1:
y = y * y % n
t <<= 1
if y != n - 1 and t % 2 == 0:
return False
return True
def _inv_gcd(a: int, b: int) -> typing.Tuple[int, int]:
a %= b
if a == 0:
return (b, 0)
# Contracts:
# [1] s - m0 * a = 0 (mod b)
# [2] t - m1 * a = 0 (mod b)
# [3] s * |m1| + t * |m0| <= b
s = b
t = a
m0 = 0
m1 = 1
while t:
u = s // t
s -= t * u
m0 -= m1 * u # |m1 * u| <= |m1| * s <= b
# [3]:
# (s - t * u) * |m1| + t * |m0 - m1 * u|
# <= s * |m1| - t * u * |m1| + t * (|m0| + |m1| * u)
# = s * |m1| + t * |m0| <= b
s, t = t, s
m0, m1 = m1, m0
# by [3]: |m0| <= b/g
# by g != b: |m0| < b/g
if m0 < 0:
m0 += b // s
return (s, m0)
def _primitive_root(m: int) -> int:
if m == 2:
return 1
if m == 167772161:
return 3
if m == 469762049:
return 3
if m == 754974721:
return 11
if m == 998244353:
return 3
divs = [2] + [0] * 19
cnt = 1
x = (m - 1) // 2
while x % 2 == 0:
x //= 2
i = 3
while i * i <= x:
if x % i == 0:
divs[cnt] = i
cnt += 1
while x % i == 0:
x //= i
i += 2
if x > 1:
divs[cnt] = x
cnt += 1
g = 2
while True:
for i in range(cnt):
if pow(g, (m - 1) // divs[i], m) == 1:
break
else:
return g
g += 1
class ModContext:
context: typing.List[int] = []
def __init__(self, mod: int) -> None:
assert 1 <= mod
self.mod = mod
def __enter__(self) -> None:
self.context.append(self.mod)
def __exit__(self, exc_type: typing.Any, exc_value: typing.Any,
traceback: typing.Any) -> None:
self.context.pop()
@classmethod
def get_mod(cls) -> int:
return cls.context[-1]
class Modint:
def __init__(self, v: int = 0) -> None:
self._mod = ModContext.get_mod()
if v == 0:
self._v = 0
else:
self._v = v % self._mod
def mod(self) -> int:
return self._mod
def val(self) -> int:
return self._v
def __iadd__(self, rhs: typing.Union['Modint', int]) -> 'Modint':
if isinstance(rhs, Modint):
self._v += rhs._v
else:
self._v += rhs
if self._v >= self._mod:
self._v -= self._mod
return self
def __isub__(self, rhs: typing.Union['Modint', int]) -> 'Modint':
if isinstance(rhs, Modint):
self._v -= rhs._v
else:
self._v -= rhs
if self._v < 0:
self._v += self._mod
return self
def __imul__(self, rhs: typing.Union['Modint', int]) -> 'Modint':
if isinstance(rhs, Modint):
self._v = self._v * rhs._v % self._mod
else:
self._v = self._v * rhs % self._mod
return self
def __ifloordiv__(self, rhs: typing.Union['Modint', int]) -> 'Modint':
if isinstance(rhs, Modint):
inv = rhs.inv()._v
else:
inv = _inv_gcd(rhs, self._mod)[1]
self._v = self._v * inv % self._mod
return self
def __pos__(self) -> 'Modint':
return self
def __neg__(self) -> 'Modint':
return Modint() - self
def __pow__(self, n: int) -> 'Modint':
assert 0 <= n
return Modint(pow(self._v, n, self._mod))
def inv(self) -> 'Modint':
eg = _inv_gcd(self._v, self._mod)
assert eg[0] == 1
return Modint(eg[1])
def __add__(self, rhs: typing.Union['Modint', int]) -> 'Modint':
if isinstance(rhs, Modint):
result = self._v + rhs._v
if result >= self._mod:
result -= self._mod
return raw(result)
else:
return Modint(self._v + rhs)
def __sub__(self, rhs: typing.Union['Modint', int]) -> 'Modint':
if isinstance(rhs, Modint):
result = self._v - rhs._v
if result < 0:
result += self._mod
return raw(result)
else:
return Modint(self._v - rhs)
def __mul__(self, rhs: typing.Union['Modint', int]) -> 'Modint':
if isinstance(rhs, Modint):
return Modint(self._v * rhs._v)
else:
return Modint(self._v * rhs)
def __floordiv__(self, rhs: typing.Union['Modint', int]) -> 'Modint':
if isinstance(rhs, Modint):
inv = rhs.inv()._v
else:
inv = _inv_gcd(rhs, self._mod)[1]
return Modint(self._v * inv)
def __eq__(self, rhs: typing.Union['Modint', int]) -> bool: # type: ignore
if isinstance(rhs, Modint):
return self._v == rhs._v
else:
return self._v == rhs
def __ne__(self, rhs: typing.Union['Modint', int]) -> bool: # type: ignore
if isinstance(rhs, Modint):
return self._v != rhs._v
else:
return self._v != rhs
def raw(v: int) -> Modint:
x = Modint()
x._v = v
return x
_sum_e = {} # _sum_e[i] = ies[0] * ... * ies[i - 1] * es[i]
def _butterfly(a: typing.List[Modint]) -> None:
g = _primitive_root(a[0].mod())
n = len(a)
h = _ceil_pow2(n)
if a[0].mod() not in _sum_e:
es = [Modint(0)] * 30 # es[i]^(2^(2+i)) == 1
ies = [Modint(0)] * 30
cnt2 = _bsf(a[0].mod() - 1)
e = Modint(g) ** ((a[0].mod() - 1) >> cnt2)
ie = e.inv()
for i in range(cnt2, 1, -1):
# e^(2^i) == 1
es[i - 2] = e
ies[i - 2] = ie
e = e * e
ie = ie * ie
sum_e = [Modint(0)] * 30
now = Modint(1)
for i in range(cnt2 - 2):
sum_e[i] = es[i] * now
now *= ies[i]
_sum_e[a[0].mod()] = sum_e
else:
sum_e = _sum_e[a[0].mod()]
for ph in range(1, h + 1):
w = 1 << (ph - 1)
p = 1 << (h - ph)
now = Modint(1)
for s in range(w):
offset = s << (h - ph + 1)
for i in range(p):
left = a[i + offset]
right = a[i + offset + p] * now
a[i + offset] = left + right
a[i + offset + p] = left - right
now *= sum_e[_bsf(~s)]
_sum_ie = {} # _sum_ie[i] = es[0] * ... * es[i - 1] * ies[i]
def _butterfly_inv(a: typing.List[Modint]) -> None:
g = _primitive_root(a[0].mod())
n = len(a)
h = _ceil_pow2(n)
if a[0].mod() not in _sum_ie:
es = [Modint(0)] * 30 # es[i]^(2^(2+i)) == 1
ies = [Modint(0)] * 30
cnt2 = _bsf(a[0].mod() - 1)
e = Modint(g) ** ((a[0].mod() - 1) >> cnt2)
ie = e.inv()
for i in range(cnt2, 1, -1):
# e^(2^i) == 1
es[i - 2] = e
ies[i - 2] = ie
e = e * e
ie = ie * ie
sum_ie = [Modint(0)] * 30
now = Modint(1)
for i in range(cnt2 - 2):
sum_ie[i] = ies[i] * now
now *= es[i]
_sum_ie[a[0].mod()] = sum_ie
else:
sum_ie = _sum_ie[a[0].mod()]
for ph in range(h, 0, -1):
w = 1 << (ph - 1)
p = 1 << (h - ph)
inow = Modint(1)
for s in range(w):
offset = s << (h - ph + 1)
for i in range(p):
left = a[i + offset]
right = a[i + offset + p]
a[i + offset] = left + right
a[i + offset + p] = Modint(
(a[0].mod() + left.val() - right.val()) * inow.val())
inow *= sum_ie[_bsf(~s)]
def convolution_mod(a: typing.List[Modint],
b: typing.List[Modint]) -> typing.List[Modint]:
n = len(a)
m = len(b)
if n == 0 or m == 0:
return []
if min(n, m) <= 60:
if n < m:
n, m = m, n
a, b = b, a
ans = [Modint(0) for _ in range(n + m - 1)]
for i in range(n):
for j in range(m):
ans[i + j] += a[i] * b[j]
return ans
z = 1 << _ceil_pow2(n + m - 1)
while len(a) < z:
a.append(Modint(0))
_butterfly(a)
while len(b) < z:
b.append(Modint(0))
_butterfly(b)
for i in range(z):
a[i] *= b[i]
_butterfly_inv(a)
a = a[:n + m - 1]
iz = Modint(z).inv()
for i in range(n + m - 1):
a[i] *= iz
return a
def convolution(mod: int, a: typing.List[typing.Any],
b: typing.List[typing.Any]) -> typing.List[typing.Any]:
n = len(a)
m = len(b)
if n == 0 or m == 0:
return []
with ModContext(mod):
a2 = list(map(Modint, a))
b2 = list(map(Modint, b))
return list(map(lambda c: c.val(), convolution_mod(a2, b2)))
def convolution_int(
a: typing.List[int], b: typing.List[int]) -> typing.List[int]:
n = len(a)
m = len(b)
if n == 0 or m == 0:
return []
mod1 = 754974721 # 2^24
mod2 = 167772161 # 2^25
mod3 = 469762049 # 2^26
m2m3 = mod2 * mod3
m1m3 = mod1 * mod3
m1m2 = mod1 * mod2
m1m2m3 = mod1 * mod2 * mod3
i1 = _inv_gcd(mod2 * mod3, mod1)[1]
i2 = _inv_gcd(mod1 * mod3, mod2)[1]
i3 = _inv_gcd(mod1 * mod2, mod3)[1]
c1 = convolution(mod1, a, b)
c2 = convolution(mod2, a, b)
c3 = convolution(mod3, a, b)
c = [0] * (n + m - 1)
for i in range(n + m - 1):
c[i] += (c1[i] * i1) % mod1 * m2m3
c[i] += (c2[i] * i2) % mod2 * m1m3
c[i] += (c3[i] * i3) % mod3 * m1m2
c[i] %= m1m2m3
return c
MD = input().split()
N = max(len(MD[0]), len(MD[1]))
A = [1] * 10
B = [1]
C = [B]
for i in range(N):
B = convolution(MOD, A, B)
C.append(B)
XM = []
for l in range(2):
M = MD[l]
NM = len(M)
X = [0] * (9 * NM + 1)
tmp = 0
for i in range(NM):
a = int(M[i])
ri = NM - i -1
b = len(C[ri])
for j in range(a):
for k in range(b):
X[tmp + j + k] += C[ri][k]
X[tmp + j + k] %= MOD
tmp += a
tmp %= MOD
X[tmp] += 1
X[tmp] %= MOD
X[0] = 0
XM.append(X)
ans = 0
for i in range(min(len(XM[0]),len(XM[1]))):
ans += XM[0][i] * XM[1][i]
ans %= MOD
print(ans)