結果
| 問題 | No.1145 Sums of Powers |
| ユーザー |
 vwxyz
|
| 提出日時 | 2023-04-22 18:10:39 |
| 言語 | PyPy3 (7.3.15) |
| 結果 |
AC
|
| 実行時間 | 1,789 ms / 2,000 ms |
| コード長 | 18,851 bytes |
| コンパイル時間 | 386 ms |
| コンパイル使用メモリ | 82,176 KB |
| 実行使用メモリ | 192,312 KB |
| 最終ジャッジ日時 | 2024-11-07 06:18:06 |
| 合計ジャッジ時間 | 6,732 ms |
|
ジャッジサーバーID （参考情報） |
judge2 / judge1 |
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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 | 56 ms
58,112 KB |
| testcase_01 | AC | 55 ms
58,112 KB |
| testcase_02 | AC | 166 ms
78,224 KB |
| testcase_03 | AC | 1,774 ms
192,152 KB |
| testcase_04 | AC | 1,789 ms
190,820 KB |
| testcase_05 | AC | 1,764 ms
192,312 KB |
ソースコード
import sys
from collections import deque
readline=sys.stdin.readline
def NTT(polynomial0,polynomial1):
if len(polynomial0)*len(polynomial1)<=60:
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
prim_root=3
prim_root_inve=332748118
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
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 __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=[pow(self.polynomial[0],mod-2,mod)]
prim_root=3
prim_root_inve=332748118
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:
polynomial=[x*y*y for x,y in zip(DFT(self.polynomial[:1<<n+1],n+2),DFT(prev,n+2))]
quot=DFT(polynomial,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]
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*pow(i+1,mod-2,mod)%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)
log=Polynomial(NTT(self.differential().polynomial,log.polynomial),max_degree=self.max_degree,mod=mod)
log=log.integral()
return log
def Degree(self):
return len(self.polynomial)-1
N,M=map(int,readline().split())
A=list(map(int,readline().split()))
mod=998244353
queue=deque([[1,-a] for a in A])
while len(queue)>=2:
f0=queue.pop()
f1=queue.pop()
f=NTT(f0,f1)[:M+1]
queue.appendleft(f)
ans=0
f=Polynomial(queue[0],max_degree=M,mod=mod)
f=-(f.log())
ans_lst=[f[i]*i%mod for i in range(1,M+1)]
print(*ans_lst)