結果
| 問題 | No.1321 塗るめた |
| ユーザー |
 vwxyz
|
| 提出日時 | 2023-04-27 01:39:35 |
| 言語 | PyPy3 (7.3.15) |
| 結果 |
TLE
|
| 実行時間 | - |
| コード長 | 32,015 bytes |
| コンパイル時間 | 276 ms |
| コンパイル使用メモリ | 82,456 KB |
| 実行使用メモリ | 300,784 KB |
| 最終ジャッジ日時 | 2024-04-28 01:31:24 |
| 合計ジャッジ時間 | 6,730 ms |
|
ジャッジサーバーID （参考情報） |
judge2 / judge4 |
（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 | 54 ms
65,280 KB |
| testcase_01 | AC | 133 ms
78,816 KB |
| testcase_02 | AC | 51 ms
59,904 KB |
| testcase_03 | AC | 47 ms
59,648 KB |
| testcase_04 | AC | 50 ms
60,032 KB |
| testcase_05 | AC | 50 ms
59,648 KB |
| testcase_06 | AC | 53 ms
59,884 KB |
| testcase_07 | AC | 54 ms
60,544 KB |
| testcase_08 | AC | 72 ms
73,088 KB |
| testcase_09 | AC | 47 ms
59,520 KB |
| testcase_10 | AC | 48 ms
59,520 KB |
| testcase_11 | AC | 76 ms
76,084 KB |
| testcase_12 | TLE | - |
| testcase_13 | TLE | - |
| testcase_14 | -- | - |
| testcase_15 | -- | - |
| testcase_16 | -- | - |
| testcase_17 | -- | - |
| testcase_18 | -- | - |
| testcase_19 | -- | - |
| testcase_20 | -- | - |
| testcase_21 | -- | - |
| testcase_22 | -- | - |
| testcase_23 | -- | - |
| testcase_24 | -- | - |
| testcase_25 | -- | - |
| testcase_26 | -- | - |
| testcase_27 | -- | - |
| testcase_28 | -- | - |
| testcase_29 | -- | - |
| testcase_30 | -- | - |
| testcase_31 | -- | - |
| testcase_32 | -- | - |
| testcase_33 | -- | - |
| testcase_34 | -- | - |
| testcase_35 | -- | - |
| testcase_36 | -- | - |
| testcase_37 | -- | - |
| testcase_38 | -- | - |
| testcase_39 | -- | - |
| testcase_40 | -- | - |
| testcase_41 | -- | - |
| testcase_42 | -- | - |
| testcase_43 | -- | - |
| testcase_44 | -- | - |
| testcase_45 | -- | - |
| testcase_46 | -- | - |
ソースコード
import math
import sys
readline=sys.stdin.readline
mod=998244353
def NTT(polynomial0,polynomial1):
"""
if len(polynomial0)*len(polynomial1)<=50:
poly=[0]*(len(polynomial0)+len(polynomial1)-1)
for i in range(len(polynomial0)):
for j in range(len(polynomial1)):
poly[i+j]+=polynomial0[i]*polynomial1[j]%mod
poly[i+j]%=mod
return poly
"""
if mod==998244353:
prim_root=3
prim_root_inve=332748118
else:
prim_root=Primitive_Root(mod)
prim_root_inve=MOD(mod).Pow(prim_root,-1)
def DFT(polynomial,n,inverse=False):
if inverse:
for bit in range(1,n+1):
a=1<<bit-1
x=pow(prim_root,mod-1>>bit,mod)
U=[1]
for _ in range(a):
U.append(U[-1]*x%mod)
for i in range(1<<n-bit):
for j in range(a):
s=i*2*a+j
t=s+a
polynomial[s],polynomial[t]=(polynomial[s]+polynomial[t]*U[j])%mod,(polynomial[s]-polynomial[t]*U[j])%mod
x=pow((mod+1)//2,n,mod)
for i in range(1<<n):
polynomial[i]*=x
polynomial[i]%=mod
else:
for bit in range(n,0,-1):
a=1<<bit-1
x=pow(prim_root_inve,mod-1>>bit,mod)
U=[1]
for _ in range(a):
U.append(U[-1]*x%mod)
for i in range(1<<n-bit):
for j in range(a):
s=i*2*a+j
t=s+a
polynomial[s],polynomial[t]=(polynomial[s]+polynomial[t])%mod,U[j]*(polynomial[s]-polynomial[t])%mod
l=len(polynomial0)+len(polynomial1)-1
n=(len(polynomial0)+len(polynomial1)-2).bit_length()
polynomial0=polynomial0+[0]*((1<<n)-len(polynomial0))
polynomial1=polynomial1+[0]*((1<<n)-len(polynomial1))
DFT(polynomial0,n)
DFT(polynomial1,n)
ntt=[x*y%mod for x,y in zip(polynomial0,polynomial1)]
DFT(ntt,n,inverse=True)
ntt=ntt[:l]
return ntt
def NTT_Pow(polynomial,N):
if N==0:
return [1]
if N==1:
return [x for x in polynomial]
if mod==998244353:
prim_root=3
prim_root_inve=332748118
else:
prim_root=Primitive_Root(mod)
prim_root_inve=MOD(mod).Pow(prim_root,-1)
def DFT(polynomial,n,inverse=False):
if inverse:
for bit in range(1,n+1):
a=1<<bit-1
x=pow(prim_root,mod-1>>bit,mod)
U=[1]
for _ in range(a):
U.append(U[-1]*x%mod)
for i in range(1<<n-bit):
for j in range(a):
s=i*2*a+j
t=s+a
polynomial[s],polynomial[t]=(polynomial[s]+polynomial[t]*U[j])%mod,(polynomial[s]-polynomial[t]*U[j])%mod
x=pow((mod+1)//2,n,mod)
for i in range(1<<n):
polynomial[i]*=x
polynomial[i]%=mod
else:
for bit in range(n,0,-1):
a=1<<bit-1
x=pow(prim_root_inve,mod-1>>bit,mod)
U=[1]
for _ in range(a):
U.append(U[-1]*x%mod)
for i in range(1<<n-bit):
for j in range(a):
s=i*2*a+j
t=s+a
polynomial[s],polynomial[t]=(polynomial[s]+polynomial[t])%mod,U[j]*(polynomial[s]-polynomial[t])%mod
n=((len(polynomial)-1)*N).bit_length()
ntt=polynomial+[0]*((1<<n)-len(polynomial))
DFT(ntt,n)
for i in range(1<<n):
ntt[i]=pow(ntt[i],N,mod)
DFT(ntt,n,inverse=True)
ntt=ntt[:(len(polynomial)-1)*N+1]
return ntt
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:
mod=self.p
else:
mod=self.p**self.e
def Pow(self,a,n):
a%=mod
if n>=0:
return pow(a,n,mod)
else:
assert math.gcd(a,mod)==1
x=Extended_Euclid(a,mod)[0]
return pow(x,-n,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%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%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)%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],mod)%mod
retu%=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]%mod*self.factorial_inve[N-K]%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,mod)
retu%=mod
return retu
def Tonelli_Shanks(N,p):
if pow(N,p>>1,p)==p-1:
retu=None
elif p%4==3:
retu=pow(N,(p+1)//4,p)
else:
for nonresidue in range(1,p):
if pow(nonresidue,p>>1,p)==p-1:
break
pp=p-1
cnt=0
while pp%2==0:
pp//=2
cnt+=1
s=pow(N,pp,p)
retu=pow(N,(pp+1)//2,p)
for i in range(cnt-2,-1,-1):
if pow(s,1<<i,p)==p-1:
s*=pow(nonresidue,p>>1+i,p)
s%=p
retu*=pow(nonresidue,p>>2+i,p)
retu%=p
return retu
def FFT(polynomial0,polynomial1,digit=10**5):
def DFT(polynomial,n,inverse=False):
if inverse:
primitive_root=[math.cos(-i*2*math.pi/(1<<n))+math.sin(-i*2*math.pi/(1<<n))*1j for i in range(1<<n)]
else:
primitive_root=[math.cos(i*2*math.pi/(1<<n))+math.sin(i*2*math.pi/(1<<n))*1j for i in range(1<<n)]
if inverse:
for bit in range(1,n+1):
a=1<<bit-1
for i in range(1<<n-bit):
for j in range(a):
s=i*2*a+j
t=s+a
polynomial[s],polynomial[t]=polynomial[s]+polynomial[t]*primitive_root[j<<n-bit],polynomial[s]-polynomial[t]*primitive_root[j<<n-bit]
else:
for bit in range(n,0,-1):
a=1<<bit-1
for i in range(1<<n-bit):
for j in range(a):
s=i*2*a+j
t=s+a
polynomial[s],polynomial[t]=polynomial[s]+polynomial[t],primitive_root[j<<n-bit]*(polynomial[s]-polynomial[t])
def FFT_(polynomial0,polynomial1):
N0=len(polynomial0)
N1=len(polynomial1)
N=N0+N1-1
n=(N-1).bit_length()
polynomial0=polynomial0+[0]*((1<<n)-N0)
polynomial1=polynomial1+[0]*((1<<n)-N1)
DFT(polynomial0,n)
DFT(polynomial1,n)
fft=[x*y for x,y in zip(polynomial0,polynomial1)]
DFT(fft,n,inverse=True)
fft=[round((fft[i]/(1<<n)).real) for i in range(N)]
return fft
N0=len(polynomial0)
N1=len(polynomial1)
N=N0+N1-1
polynomial00,polynomial01=[None]*N0,[None]*N0
polynomial10,polynomial11=[None]*N1,[None]*N1
for i in range(N0):
polynomial00[i],polynomial01[i]=divmod(polynomial0[i],digit)
for i in range(N1):
polynomial10[i],polynomial11[i]=divmod(polynomial1[i],digit)
polynomial=[0]*(N)
a=digit**2-digit
for i,x in enumerate(FFT_(polynomial00,polynomial10)):
polynomial[i]+=x*a
a=digit-1
for i,x in enumerate(FFT_(polynomial01,polynomial11)):
polynomial[i]-=x*a
for i,x in enumerate(FFT_([x1+x2 for x1,x2 in zip(polynomial00,polynomial01)],[x1+x2 for x1,x2 in zip(polynomial10,polynomial11)])):
polynomial[i]+=x*digit
return polynomial
def FFT_Pow(polynomial,N):
if N==0:
return [1]
if N==1:
return [x for x in polynomial]
def DFT(polynomial,n,inverse=False):
if inverse:
primitive_root=[math.cos(-i*2*math.pi/(1<<n))+math.sin(-i*2*math.pi/(1<<n))*1j for i in range(1<<n)]
else:
primitive_root=[math.cos(i*2*math.pi/(1<<n))+math.sin(i*2*math.pi/(1<<n))*1j for i in range(1<<n)]
if inverse:
for bit in range(1,n+1):
a=1<<bit-1
for i in range(1<<n-bit):
for j in range(a):
s=i*2*a+j
t=s+a
polynomial[s],polynomial[t]=polynomial[s]+polynomial[t]*primitive_root[j<<n-bit],polynomial[s]-polynomial[t]*primitive_root[j<<n-bit]
else:
for bit in range(n,0,-1):
a=1<<bit-1
for i in range(1<<n-bit):
for j in range(a):
s=i*2*a+j
t=s+a
polynomial[s],polynomial[t]=polynomial[s]+polynomial[t],primitive_root[j<<n-bit]*(polynomial[s]-polynomial[t])
n=((len(polynomial)-1)*N).bit_length()
fft=polynomial+[0]*((1<<n)-len(polynomial))
DFT(fft,n)
for i in range(1<<n):
fft[i]=pow(fft[i],N)
DFT(fft,n,inverse=True)
fft=[round((fft[i]/(1<<n)).real) for i in range((len(polynomial)-1)*N+1)]
return fft
class Polynomial:
def __init__(self,polynomial,max_degree=-1,eps=0,mod=0):
self.max_degree=max_degree
if self.max_degree!=-1 and len(polynomial)>self.max_degree+1:
self.polynomial=polynomial[:self.max_degree+1]
else:
self.polynomial=polynomial
mod=mod
self.eps=eps
def __eq__(self,other):
if type(other)!=Polynomial:
return False
if len(self.polynomial)!=len(other.polynomial):
return False
for i in range(len(self.polynomial)):
if self.eps<abs(self.polynomial[i]-other.polynomial[i]):
return False
return True
def __ne__(self,other):
if type(other)!=Polynomial:
return True
if len(self.polynomial)!=len(other.polynomial):
return True
for i in range(len(self.polynomial)):
if self.eps<abs(self.polynomial[i]-other.polynomial[i]):
return True
return False
def __add__(self,other):
if type(other)==Polynomial:
summ=[0]*max(len(self.polynomial),len(other.polynomial))
for i in range(len(self.polynomial)):
summ[i]+=self.polynomial[i]
for i in range(len(other.polynomial)):
summ[i]+=other.polynomial[i]
if mod:
for i in range(len(summ)):
summ[i]%=mod
else:
summ=[x for x in self.polynomial] if self.polynomial else [0]
summ[0]+=other
if mod:
summ[0]%=mod
while summ and abs(summ[-1])<=self.eps:
summ.pop()
summ=Polynomial(summ,max_degree=self.max_degree,eps=self.eps,mod=mod)
return summ
def __sub__(self,other):
if type(other)==Polynomial:
diff=[0]*max(len(self.polynomial),len(other.polynomial))
for i in range(len(self.polynomial)):
diff[i]+=self.polynomial[i]
for i in range(len(other.polynomial)):
diff[i]-=other.polynomial[i]
if mod:
for i in range(len(diff)):
diff[i]%=mod
else:
diff=[x for x in self.polynomial] if self.polynomial else [0]
diff[0]-=other
if mod:
diff[0]%=mod
while diff and abs(diff[-1])<=self.eps:
diff.pop()
diff=Polynomial(diff,max_degree=self.max_degree,eps=self.eps,mod=mod)
return diff
def __mul__(self,other):
if type(other)==Polynomial:
if self.max_degree==-1:
prod=[0]*(len(self.polynomial)+len(other.polynomial)-1)
for i in range(len(self.polynomial)):
for j in range(len(other.polynomial)):
prod[i+j]+=self.polynomial[i]*other.polynomial[j]
else:
prod=[0]*min(len(self.polynomial)+len(other.polynomial)-1,self.max_degree+1)
for i in range(len(self.polynomial)):
for j in range(min(len(other.polynomial),self.max_degree+1-i)):
prod[i+j]+=self.polynomial[i]*other.polynomial[j]
if mod:
for i in range(len(prod)):
prod[i]%=mod
else:
if mod:
prod=[x*other%mod for x in self.polynomial]
else:
prod=[x*other for x in self.polynomial]
while prod and abs(prod[-1])<=self.eps:
prod.pop()
prod=Polynomial(prod,max_degree=self.max_degree,eps=self.eps,mod=mod)
return prod
def __matmul__(self,other):
assert type(other)==Polynomial
if mod:
prod=NTT(self.polynomial,other.polynomial)
else:
prod=FFT(self.polynomial,other.polynomial)
if self.max_degree!=-1 and len(prod)>self.max_degree+1:
prod=prod[:self.max_degree+1]
while prod and abs(prod[-1])<=self.eps:
prod.pop()
prod=Polynomial(prod,max_degree=self.max_degree,eps=self.eps,mod=mod)
return prod
def __pow__(self,other):
if other==0:
prod=Polynomial([1],max_degree=self.max_degree,eps=self.eps,mod=mod)
elif other==1:
prod=Polynomial([x for x in self.polynomial],max_degree=self.max_degree,eps=self.eps,mod=mod)
else:
prod=[1]
doub=self.polynomial
if mod:
convolve=NTT
convolve_Pow=NTT_Pow
else:
convolve=FFT
convolve_Pow=FFT_Pow
while other>=2:
if other&1:
prod=convolve(prod,doub)
if self.max_degree!=-1:
prod=prod[:self.max_degree+1]
doub=convolve_Pow(doub,2)
if self.max_degree!=-1:
doub=doub[:self.max_degree+1]
other>>=1
prod=convolve(prod,doub)
if self.max_degree!=-1:
prod=prod[:self.max_degree+1]
prod=Polynomial(prod,max_degree=self.max_degree,eps=self.eps,mod=mod)
return prod
def __truediv__(self,other):
if type(other)==Polynomial:
assert other.polynomial
for n in range(len(other.polynomial)):
if self.eps<abs(other.polynomial[n]):
break
assert len(self.polynomial)>n
for i in range(n):
assert abs(self.polynomial[i])<=self.eps
self_polynomial=self.polynomial[n:]
other_polynomial=other.polynomial[n:]
if mod:
inve=MOD(mod).Pow(other_polynomial[0],-1)
else:
inve=1/other_polynomial[0]
quot=[]
for i in range(len(self_polynomial)-len(other_polynomial)+1):
if mod:
quot.append(self_polynomial[i]*inve%mod)
else:
quot.append(self_polynomial[i]*inve)
for j in range(len(other_polynomial)):
self_polynomial[i+j]-=other_polynomial[j]*quot[-1]
if mod:
self_polynomial[i+j]%=mod
for i in range(max(0,len(self_polynomial)-len(other_polynomial)+1),len(self_polynomial)):
if self.eps<abs(self_polynomial[i]):
assert self.max_degree!=-1
self_polynomial=self_polynomial[-len(other_polynomial)+1:]+[0]*(len(other_polynomial)-1-len(self_polynomial))
while len(quot)<=self.max_degree:
self_polynomial.append(0)
if mod:
quot.append(self_polynomial[0]*inve%mod)
self_polynomial=[(self_polynomial[i]-other_polynomial[i]*quot[-1])%mod for i in range(1,len(self_polynomial))]
else:
quot.append(self_polynomial[0]*inve)
self_polynomial=[(self_polynomial[i]-other_polynomial[i]*quot[-1]) for i in range(1,len(self_polynomial))]
break
quot=Polynomial(quot,max_degree=self.max_degree,eps=self.eps,mod=mod)
else:
assert self.eps<abs(other)
if mod:
inve=MOD(mod).Pow(other,-1)
quot=Polynomial([x*inve%mod for x in self.polynomial],max_degree=self.max_degree,eps=self.eps,mod=mod)
else:
quot=Polynomial([x/other for x in self.polynomial],max_degree=self.max_degree,eps=self.eps,mod=mod)
return quot
def __rtruediv__(self,other):
assert self.polynomial and self.eps<self.polynomial[0]
assert self.max_degree!=-1
if mod:
quot=[MOD(mod).Pow(self.polynomial[0],-1)]
if mod==998244353:
prim_root=3
prim_root_inve=332748118
else:
prim_root=Primitive_Root(mod)
prim_root_inve=MOD(mod).Pow(prim_root,-1)
def DFT(polynomial,n,inverse=False):
polynomial=polynomial+[0]*((1<<n)-len(polynomial))
if inverse:
for bit in range(1,n+1):
a=1<<bit-1
x=pow(prim_root,mod-1>>bit,mod)
U=[1]
for _ in range(a):
U.append(U[-1]*x%mod)
for i in range(1<<n-bit):
for j in range(a):
s=i*2*a+j
t=s+a
polynomial[s],polynomial[t]=(polynomial[s]+polynomial[t]*U[j])%mod,(polynomial[s]-polynomial[t]*U[j])%mod
x=pow((mod+1)//2,n,mod)
for i in range(1<<n):
polynomial[i]*=x
polynomial[i]%=mod
else:
for bit in range(n,0,-1):
a=1<<bit-1
x=pow(prim_root_inve,mod-1>>bit,mod)
U=[1]
for _ in range(a):
U.append(U[-1]*x%mod)
for i in range(1<<n-bit):
for j in range(a):
s=i*2*a+j
t=s+a
polynomial[s],polynomial[t]=(polynomial[s]+polynomial[t])%mod,U[j]*(polynomial[s]-polynomial[t])%mod
return polynomial
else:
quot=[1/self.polynomial[0]]
def DFT(polynomial,n,inverse=False):
N=len(polynomial)
if inverse:
primitive_root=[math.cos(-i*2*math.pi/(1<<n))+math.sin(-i*2*math.pi/(1<<n))*1j for i in range(1<<n)]
else:
primitive_root=[math.cos(i*2*math.pi/(1<<n))+math.sin(i*2*math.pi/(1<<n))*1j for i in range(1<<n)]
polynomial=polynomial+[0]*((1<<n)-N)
if inverse:
for bit in range(1,n+1):
a=1<<bit-1
for i in range(1<<n-bit):
for j in range(a):
s=i*2*a+j
t=s+a
polynomial[s],polynomial[t]=polynomial[s]+polynomial[t]*primitive_root[j<<n-bit],polynomial[s]-polynomial[t]*primitive_root[j<<n-bit]
for i in range(1<<n):
polynomial[i]=round((polynomial[i]/(1<<n)).real)
else:
for bit in range(n,0,-1):
a=1<<bit-1
for i in range(1<<n-bit):
for j in range(a):
s=i*2*a+j
t=s+a
polynomial[s],polynomial[t]=polynomial[s]+polynomial[t],primitive_root[j<<n-bit]*(polynomial[s]-polynomial[t])
return polynomial
for n in range(self.max_degree.bit_length()):
prev=quot
if mod:
quot=DFT([x*y%mod*y%mod for x,y in zip(DFT(self.polynomial[:1<<n+1],n+2),DFT(prev,n+2))],n+2,inverse=True)[:1<<n+1]
else:
quot=DFT([x*y*y for x,y in zip(DFT(self.polynomial[:1<<n+1],n+2),DFT(prev,n+2))],n+2,inverse=True)[:1<<n+1]
for i in range(1<<n):
quot[i]=2*prev[i]-quot[i]
if mod:
quot[i]%=mod
for i in range(1<<n,1<<n+1):
quot[i]=-quot[i]
if mod:
quot[i]%=mod
quot=quot[:self.max_degree+1]
if abs(other-1)>self.eps:
for i in range(len(quot)):
quot[i]*=other
if mod:
quot[i]%=mod
quot=Polynomial(quot,max_degree=self.max_degree,eps=self.eps,mod=mod)
return quot
def __floordiv__(self,other):
assert type(other)==Polynomial
quot=[0]*(len(self.polynomial)-len(other.polynomial)+1)
rema=[x for x in self.polynomial]
if mod:
inve=MOD(mod).Pow(other.polynomial[-1],-1)
for i in range(len(self.polynomial)-len(other.polynomial),-1,-1):
quot[i]=rema[i+len(other.polynomial)-1]*inve%mod
for j in range(len(other.polynomial)):
rema[i+j]-=quot[i]*other.polynomial[j]
rema[i+j]%=mod
else:
inve=1/other.polynomial[-1]
for i in range(len(self.polynomial)-len(other.polynomial),-1,-1):
quot[i]=rema[i+len(other.polynomial)-1]*inve
for j in range(len(other.polynomial)):
rema[i+j]-=quot[i]*other.polynomial[j]
quot=Polynomial(quot,max_degree=self.max_degree,eps=self.eps,mod=mod)
return quot
def __mod__(self,other):
assert type(other)==Polynomial
quot=[0]*(len(self.polynomial)-len(other.polynomial)+1)
rema=[x for x in self.polynomial]
if mod:
inve=MOD(mod).Pow(other.polynomial[-1],-1)
for i in range(len(self.polynomial)-len(other.polynomial),-1,-1):
quot[i]=rema[i+len(other.polynomial)-1]*inve%mod
for j in range(len(other.polynomial)):
rema[i+j]-=quot[i]*other.polynomial[j]
rema[i+j]%=mod
else:
inve=1/other.polynomial[-1]
for i in range(len(self.polynomial)-len(other.polynomial),-1,-1):
quot[i]=rema[i+len(other.polynomial)-1]*inve
for j in range(len(other.polynomial)):
rema[i+j]-=quot[i]*other.polynomial[j]
while rema and abs(rema[-1])<=self.eps:
rema.pop()
rema=Polynomial(rema,max_degree=self.max_degree,eps=self.eps,mod=mod)
return rema
def __divmod__(self,other):
assert type(other)==Polynomial
quot=[0]*(len(self.polynomial)-len(other.polynomial)+1)
rema=[x for x in self.polynomial]
if mod:
inve=MOD(mod).Pow(other.polynomial[-1],-1)
for i in range(len(self.polynomial)-len(other.polynomial),-1,-1):
quot[i]=rema[i+len(other.polynomial)-1]*inve%mod
for j in range(len(other.polynomial)):
rema[i+j]-=quot[i]*other.polynomial[j]
rema[i+j]%=mod
else:
inve=1/other.polynomial[-1]
for i in range(len(self.polynomial)-len(other.polynomial),-1,-1):
quot[i]=rema[i+len(other.polynomial)-1]*inve
for j in range(len(other.polynomial)):
rema[i+j]-=quot[i]*other.polynomial[j]
while rema and abs(rema[-1])<=self.eps:
rema.pop()
quot=Polynomial(quot,max_degree=self.max_degree,eps=self.eps,mod=mod)
rema=Polynomial(rema,max_degree=self.max_degree,eps=self.eps,mod=mod)
return quot,rema
def __neg__(self):
if mod:
nega=Polynomial([(-x)%mod for x in self.polynomial],max_degree=self.max_degree,eps=self.eps,mod=mod)
else:
nega=Polynomial([-x for x in self.polynomial],max_degree=self.max_degree,eps=self.eps,mod=mod)
return nega
def __pos__(self):
posi=Polynomial([x for x in self.polynomial],max_degree=self.max_degree,eps=self.eps,mod=mod)
return posi
def __bool__(self):
return self.polynomial
def __getitem__(self,n):
if type(n)==int:
if n<=len(self.polynomial)-1:
return self.polynomial[n]
else:
return 0
else:
return Polynomial(polynomial=self.polynomial[n],max_degree=self.max_degree,eps=self.eps,mod=mod)
def __setitem__(self,n,a):
if mod:
a%=mod
if self.max_degree==-1 or n<=self.max_degree:
if n<=len(self.polynomial)-1:
self.polynomial[n]=a
elif self.eps<abs(a):
self.polynomial+=[0]*(n-len(self.polynomial))+[a]
def __iter__(self):
for x in self.polynomial:
yield x
def __call__(self,x):
retu=0
pow_x=1
for i in range(len(self.polynomial)):
retu+=pow_x*self.polynomial[i]
pow_x*=x
if mod:
retu%=mod
pow_x%=mod
return retu
def __str__(self):
return "["+", ".join(map(str,self.polynomial))+"]"
def __len__(self):
return len(self.polynomial)
def differential(self):
if mod:
differential=[x*i%mod for i,x in enumerate(self.polynomial[1:],1)]
else:
differential=[x*i for i,x in enumerate(self.polynomial[1:],1)]
return Polynomial(differential,max_degree=self.max_degree,eps=self.eps,mod=mod)
def integral(self):
if mod:
integral=[0]+[x*MOD(mod).Pow(i+1,-1)%mod for i,x in enumerate(self.polynomial)]
else:
integral=[0]+[x/(i+1) for i,x in enumerate(self.polynomial)]
while integral and abs(integral[-1])<=self.eps:
integral.pop()
return Polynomial(integral,max_degree=self.max_degree,eps=self.eps,mod=mod)
def log(self):
assert self.max_degree!=-1
assert self.polynomial and abs(self.polynomial[0]-1)<=self.eps
log=(1/self)
if mod:
log=Polynomial(NTT(self.differential().polynomial,log.polynomial),max_degree=self.max_degree,eps=self.eps,mod=mod)
else:
log=Polynomial(FFT(self.differential().polynomial,log.polynomial),max_degree=self.max_degree,eps=self.eps,mod=mod)
log=log.integral()
return log
def Newton(self,n0,f,differentiated_f=None):
newton=[n0]
while len(newton)<self.max_degree+1:
prev=newton
if differentiated_f==None:
newton=f(prev,self.polynomial)
else:
newton=f(prev)
for i in range(min(len(self.polynomial),len(newton))):
newton[i]-=self.polynomial[i]
newton[i]%=mod
if mod:
newton=NTT(newton,(1/Polynomial(differentiated_f(prev),max_degree=len(newton)-1,eps=self.eps,mod=mod)).polynomial)[:len(newton)]
else:
newton=FFT(newton,(1/Polynomial(differentiated_f(prev),max_degree=len(newton)-1,eps=self.eps,mod=mod)).polynomial)[:len(newton)]
for i in range(len(newton)):
newton[i]=-newton[i]
newton[i]%=mod
for i in range(len(prev)):
newton[i]+=prev[i]
newton[i]%=mod
newton=newton[:self.max_degree+1]
while newton and newton[-1]<=self.eps:
newton.pop()
return Polynomial(newton,max_degree=self.max_degree,eps=self.eps,mod=mod)
def sqrt(self):
if self.polynomial:
for cnt0 in range(len(self.polynomial)):
if self.polynomial[cnt0]:
break
if cnt0%2:
sqrt=None
else:
if mod:
n0=Tonelli_Shanks(self.polynomial[cnt0],mod)
else:
if self.polynomial[cnt0]>=self.eps:
n0=self.polynomial[cnt0]**.5
if n0==None:
sqrt=None
else:
def f(prev):
if mod:
return NTT_Pow(prev,2)+[0]
else:
return FFT_Pow(prev,2)+[0]
def differentiated_f(prev):
retu=[0]*(2*len(prev)-1)
for i in range(len(prev)):
retu[i]+=2*prev[i]
if mod:
retu[i]%mod
return retu
sqrt=[0]*(cnt0//2)+Polynomial(self.polynomial[cnt0:],max_degree=self.max_degree-cnt0//2,mod=mod).Newton(n0,f,differentiated_f).polynomial
sqrt=Polynomial(sqrt,max_degree=self.max_degree,eps=self.eps,mod=mod)
else:
sqrt=Polynomial([],max_degree=self.max_degree,eps=self.eps,mod=mod)
return sqrt
def exp(self):
assert not self.polynomial or abs(self.polynomial[0])<=self.eps
def f(prev,poly):
newton=Polynomial(prev,max_degree=2*len(prev)-1,eps=self.eps,mod=mod).log().polynomial
newton+=[0]*(2*len(prev)-len(newton))
for i in range(min(len(poly),len(newton))):
newton[i]-=poly[i]
if mod:
for i in range(len(newton)):
newton[i]%=mod
if mod:
return NTT(prev,newton)[:2*len(prev)]
else:
return FFT(prev,newton)[:2*len(prev)]
return Polynomial(self.polynomial,max_degree=self.max_degree,mod=mod).Newton(1,f)
def Degree(self):
return len(self.polynomial)-1
N,M,K=map(int,readline().split())
mod=998244353
MD=MOD(mod)
MD.Build_Fact(N)
poly=[None]*(N+1)
poly[0]=0
for i in range(1,N+1):
poly[i]=MD.Fact_Inve(i)
P=Polynomial(poly,max_degree=N,mod=mod)
P**=K
ans=0
for n in range(N+1):
ans+=P[n]*MD.Pow(M,N-n)%mod*MD.Fact_Inve(N-n)%mod
ans*=MD.Comb(M,K)*MD.Fact(N)
ans%=mod
print(ans)