結果
| 問題 |
No.1321 塗るめた
|
| コンテスト | |
| ユーザー |
 vwxyz
|
| 提出日時 | 2023-04-27 02:08:28 |
| 言語 | PyPy3 (7.3.15) |
| 結果 |
AC
|
| 実行時間 | 650 ms / 2,000 ms |
| コード長 | 10,418 bytes |
| コンパイル時間 | 150 ms |
| コンパイル使用メモリ | 82,308 KB |
| 実行使用メモリ | 129,704 KB |
| 最終ジャッジ日時 | 2024-11-16 09:47:08 |
| 合計ジャッジ時間 | 16,163 ms |
|
ジャッジサーバーID （参考情報） |
judge5 / judge1 |
（要ログイン）
| ファイルパターン | 結果 |
|---|---|
| sample | AC * 2 |
| other | AC * 45 |
ソースコード
from collections import deque
import math
import sys
readline=sys.stdin.readline
_fft_mod = 998244353
_fft_sum_e = (911660635, 509520358, 369330050, 332049552, 983190778, 123842337, 238493703, 975955924, 603855026, 856644456, 131300601,
842657263, 730768835, 942482514, 806263778, 151565301, 510815449, 503497456, 743006876, 741047443, 56250497, 867605899, 0, 0, 0, 0, 0, 0, 0, 0)
_fft_sum_ie = (86583718, 372528824, 373294451, 645684063, 112220581, 692852209, 155456985, 797128860, 90816748, 860285882, 927414960, 354738543,
109331171, 293255632, 535113200, 308540755, 121186627, 608385704, 438932459, 359477183, 824071951, 103369235, 0, 0, 0, 0, 0, 0, 0, 0)
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 % _fft_mod
a[i + offset] = (l + r) % _fft_mod
a[i + offset + p] = (l - r) % _fft_mod
now *= _fft_sum_e[(~s & -~s).bit_length() - 1]
now %= _fft_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) % _fft_mod
a[i + offset + p] = (l - r) * inow % _fft_mod
inow *= _fft_sum_ie[(~s & -~s).bit_length() - 1]
inow %= _fft_mod
def _convolution_naive(a, b):
n = len(a)
m = len(b)
ans = [0] * (n + m - 1)
if n < m:
for j in range(m):
for i in range(n):
ans[i + j] = (ans[i + j] + a[i] * b[j]) % _fft_mod
else:
for i in range(n):
for j in range(m):
ans[i + j] = (ans[i + j] + a[i] * b[j]) % _fft_mod
return ans
def _convolution_fft(a, b):
a = a.copy()
b = b.copy()
n = len(a)
m = len(b)
z = 1 << (n + m - 2).bit_length()
a += [0] * (z - n)
_butterfly(a)
b += [0] * (z - m)
_butterfly(b)
for i in range(z):
a[i] = a[i] * b[i] % _fft_mod
_butterfly_inv(a)
a = a[:n + m - 1]
iz = pow(z, _fft_mod - 2, _fft_mod)
for i in range(n + m - 1):
a[i] = a[i] * iz % _fft_mod
return a
def _convolution_square(a):
a = a.copy()
n = len(a)
z = 1 << (2 * n - 2).bit_length()
a += [0] * (z - n)
_butterfly(a)
for i in range(z):
a[i] = a[i] * a[i] % _fft_mod
_butterfly_inv(a)
a = a[:2 * n - 1]
iz = pow(z, _fft_mod - 2, _fft_mod)
for i in range(2 * n - 1):
a[i] = a[i] * iz % _fft_mod
return a
def convolution(a, b):
"""It calculates (+, x) convolution in mod 998244353.
Given two arrays a[0], a[1], ..., a[n - 1] and b[0], b[1], ..., b[m - 1],
it calculates the array c of length n + m - 1, defined by
> c[i] = sum(a[j] * b[i - j] for j in range(i + 1)) % 998244353.
It returns an empty list if at least one of a and b are empty.
Complexity
----------
> O(n log n), where n = len(a) + len(b).
"""
n = len(a)
m = len(b)
if n == 0 or m == 0:
return []
if min(n, m) <= 100:
return _convolution_naive(a, b)
if a is b:
return _convolution_square(a)
return _convolution_fft(a, b)
# Reference: https://opt-cp.com/fps-fast-algorithms/
def inv(a):
"""It calculates the inverse of formal power series in O(n log n) time, where n = len(a).
"""
n = len(a)
assert n > 0 and a[0] != 0
res = [pow(a[0], _fft_mod - 2, _fft_mod)]
m = 1
while m < n:
f = a[:min(n, 2 * m)]
g = res.copy()
f += [0] * (2 * m - len(f))
_butterfly(f)
g += [0] * (2 * m - len(g))
_butterfly(g)
for i in range(2 * m):
f[i] = f[i] * g[i] % _fft_mod
_butterfly_inv(f)
f = f[m:] + [0] * m
_butterfly(f)
for i in range(2 * m):
f[i] = f[i] * g[i] % _fft_mod
_butterfly_inv(f)
f = f[:m]
iz = pow(2 * m, _fft_mod - 2, _fft_mod)
iz *= -iz
iz %= _fft_mod
for i in range(m):
f[i] = f[i] * iz % _fft_mod
res.extend(f)
m *= 2
res = res[:n]
return res
def integ_inplace(a):
n = len(a)
assert n > 0
if n == 1:
return []
a.pop()
a.insert(0, 0)
inv = [1, 1]
for i in range(2, n):
inv.append(-inv[_fft_mod%i] * (_fft_mod//i) % _fft_mod)
a[i] = a[i] * inv[i] % _fft_mod
def deriv_inplace(a):
n = len(a)
assert n > 0
for i in range(2, n):
a[i] = a[i] * i % _fft_mod
a.pop(0)
a.append(0)
def log(a):
a = a.copy()
n = len(a)
assert n > 0 and a[0] == 1
a_inv = inv(a)
deriv_inplace(a)
a = convolution(a, a_inv)[:n]
integ_inplace(a)
return a
def exp(a):
a = a.copy()
n = len(a)
assert n > 0 and a[0] == 0
g = [1]
a[0] = 1
h_drv = a.copy()
deriv_inplace(h_drv)
m = 1
while m < n:
f_fft = a[:m] + [0] * m
_butterfly(f_fft)
if m > 1:
_f = [f_fft[i] * g_fft[i] % _fft_mod for i in range(m)]
_butterfly_inv(_f)
_f = _f[m // 2:] + [0] * (m // 2)
_butterfly(_f)
for i in range(m):
_f[i] = _f[i] * g_fft[i] % _fft_mod
_butterfly_inv(_f)
_f = _f[:m//2]
iz = pow(m, _fft_mod - 2, _fft_mod)
iz *= -iz
iz %= _fft_mod
for i in range(m//2):
_f[i] = _f[i] * iz % _fft_mod
g.extend(_f)
t = a[:m]
deriv_inplace(t)
r = h_drv[:m - 1]
r.append(0)
_butterfly(r)
for i in range(m):
r[i] = r[i] * f_fft[i] % _fft_mod
_butterfly_inv(r)
im = pow(-m, _fft_mod - 2, _fft_mod)
for i in range(m):
r[i] = r[i] * im % _fft_mod
for i in range(m):
t[i] = (t[i] + r[i]) % _fft_mod
t = [t[-1]] + t[:-1]
t += [0] * m
_butterfly(t)
g_fft = g + [0] * (2 * m - len(g))
_butterfly(g_fft)
for i in range(2 * m):
t[i] = t[i] * g_fft[i] % _fft_mod
_butterfly_inv(t)
t = t[:m]
i2m = pow(2 * m, _fft_mod - 2, _fft_mod)
for i in range(m):
t[i] = t[i] * i2m % _fft_mod
v = a[m:min(n, 2 * m)]
v += [0] * (m - len(v))
t = [0] * (m - 1) + t + [0]
integ_inplace(t)
for i in range(m):
v[i] = (v[i] - t[m + i]) % _fft_mod
v += [0] * m
_butterfly(v)
for i in range(2 * m):
v[i] = v[i] * f_fft[i] % _fft_mod
_butterfly_inv(v)
v = v[:m]
i2m = pow(2 * m, _fft_mod - 2, _fft_mod)
for i in range(m):
v[i] = v[i] * i2m % _fft_mod
for i in range(min(n - m, m)):
a[m + i] = v[i]
m *= 2
return a
def pow_fps(a, k):
a = a.copy()
n = len(a)
l = 0
while l < len(a) and not a[l]:
l += 1
if l * k >= n:
return [0] * n
ic = pow(a[l], _fft_mod - 2, _fft_mod)
pc = pow(a[l], k, _fft_mod)
a = log([a[i] * ic % _fft_mod for i in range(l, len(a))])
for i in range(len(a)):
a[i] = a[i] * k % _fft_mod
a = exp(a)
for i in range(len(a)):
a[i] = a[i] * pc % _fft_mod
a = [0] * (l * k) + a[:n - l * k]
return a
def Extended_Euclid(n,m):
stack=[]
while m:
stack.append((n,m))
n,m=m,n%m
if n>=0:
x,y=1,0
else:
x,y=-1,0
for i in range(len(stack)-1,-1,-1):
n,m=stack[i]
x,y=y,x-(n//m)*y
return x,y
class MOD:
def __init__(self,p,e=None):
self.p=p
self.e=e
if self.e==None:
self.mod=self.p
else:
self.mod=self.p**self.e
def Pow(self,a,n):
a%=self.mod
if n>=0:
return pow(a,n,self.mod)
else:
assert math.gcd(a,self.mod)==1
x=Extended_Euclid(a,self.mod)[0]
return pow(x,-n,self.mod)
def Build_Fact(self,N):
assert N>=0
self.factorial=[1]
if self.e==None:
for i in range(1,N+1):
self.factorial.append(self.factorial[-1]*i%self.mod)
else:
self.cnt=[0]*(N+1)
for i in range(1,N+1):
self.cnt[i]=self.cnt[i-1]
ii=i
while ii%self.p==0:
ii//=self.p
self.cnt[i]+=1
self.factorial.append(self.factorial[-1]*ii%self.mod)
self.factorial_inve=[None]*(N+1)
self.factorial_inve[-1]=self.Pow(self.factorial[-1],-1)
for i in range(N-1,-1,-1):
ii=i+1
while ii%self.p==0:
ii//=self.p
self.factorial_inve[i]=(self.factorial_inve[i+1]*ii)%self.mod
def Fact(self,N):
if N<0:
return 0
retu=self.factorial[N]
if self.e!=None and self.cnt[N]:
retu*=pow(self.p,self.cnt[N],self.mod)%self.mod
retu%=self.mod
return retu
def Fact_Inve(self,N):
if self.e!=None and self.cnt[N]:
return None
return self.factorial_inve[N]
def Comb(self,N,K,divisible_count=False):
if K<0 or K>N:
return 0
retu=self.factorial[N]*self.factorial_inve[K]%self.mod*self.factorial_inve[N-K]%self.mod
if self.e!=None:
cnt=self.cnt[N]-self.cnt[N-K]-self.cnt[K]
if divisible_count:
return retu,cnt
else:
retu*=pow(self.p,cnt,self.mod)
retu%=self.mod
return retu
N,M,K=map(int,readline().split())
mod=998244353
MD=MOD(mod)
MD.Build_Fact(N)
P=[None]*(N-K+1)
for i in range(N-K+1):
P[i]=MD.Fact_Inve(i+1)
P=pow_fps(P,K)
ans=0
for n in range(K,N+1):
ans+=P[n-K]*MD.Pow(M,N-n)%mod*MD.Fact_Inve(N-n)%mod
ans*=MD.Comb(M,K)*MD.Fact(N)%mod
ans%=mod
print(ans)