結果
| 問題 | No.1938 Lagrange Sum |
| ユーザー |
 vwxyz
|
| 提出日時 | 2023-05-09 11:52:08 |
| 言語 | PyPy3 (7.3.15) |
| 結果 |
AC
|
| 実行時間 | 2,800 ms / 3,000 ms |
| コード長 | 44,395 bytes |
| コンパイル時間 | 496 ms |
| コンパイル使用メモリ | 86,772 KB |
| 実行使用メモリ | 77,688 KB |
| 最終ジャッジ日時 | 2024-11-25 23:17:59 |
| 合計ジャッジ時間 | 37,853 ms |
|
ジャッジサーバーID （参考情報） |
judge2 / judge3 |
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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 | 2,236 ms
77,124 KB |
| testcase_01 | AC | 2,234 ms
77,148 KB |
| testcase_02 | AC | 2,233 ms
77,296 KB |
| testcase_03 | AC | 55 ms
64,164 KB |
| testcase_04 | AC | 53 ms
64,236 KB |
| testcase_05 | AC | 1,895 ms
77,100 KB |
| testcase_06 | AC | 2,068 ms
77,244 KB |
| testcase_07 | AC | 78 ms
76,400 KB |
| testcase_08 | AC | 488 ms
77,472 KB |
| testcase_09 | AC | 211 ms
76,652 KB |
| testcase_10 | AC | 2,744 ms
77,508 KB |
| testcase_11 | AC | 527 ms
77,240 KB |
| testcase_12 | AC | 2,768 ms
76,852 KB |
| testcase_13 | AC | 2,800 ms
77,688 KB |
| testcase_14 | AC | 1,012 ms
77,300 KB |
| testcase_15 | AC | 161 ms
76,964 KB |
| testcase_16 | AC | 2,574 ms
77,512 KB |
| testcase_17 | AC | 967 ms
76,848 KB |
| testcase_18 | AC | 982 ms
77,316 KB |
| testcase_19 | AC | 2,679 ms
77,112 KB |
| testcase_20 | AC | 2,661 ms
76,996 KB |
| testcase_21 | AC | 115 ms
76,472 KB |
| testcase_22 | AC | 65 ms
72,924 KB |
| testcase_23 | AC | 103 ms
76,920 KB |
| testcase_24 | AC | 2,595 ms
76,944 KB |
| testcase_25 | AC | 54 ms
64,180 KB |
| testcase_26 | AC | 54 ms
62,952 KB |
| testcase_27 | AC | 54 ms
65,156 KB |
ソースコード
import sys
readline=sys.stdin.readline
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 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 Build_Inverse(self,N):
self.inverse=[None]*(N+1)
assert self.p>N
self.inverse[1]=1
for n in range(2,N+1):
if n%self.p==0:
continue
a,b=divmod(self.mod,n)
self.inverse[n]=(-a*self.inverse[b])%self.mod
def Inverse(self,n):
return self.inverse[n]
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
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
self.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 self.mod:
for i in range(len(summ)):
summ[i]%=self.mod
else:
summ=[x for x in self.polynomial] if self.polynomial else [0]
summ[0]+=other
if self.mod:
summ[0]%=self.mod
while summ and abs(summ[-1])<=self.eps:
summ.pop()
summ=Polynomial(summ,max_degree=self.max_degree,eps=self.eps,mod=self.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 self.mod:
for i in range(len(diff)):
diff[i]%=self.mod
else:
diff=[x for x in self.polynomial] if self.polynomial else [0]
diff[0]-=other
if self.mod:
diff[0]%=self.mod
while diff and abs(diff[-1])<=self.eps:
diff.pop()
diff=Polynomial(diff,max_degree=self.max_degree,eps=self.eps,mod=self.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 self.mod:
for i in range(len(prod)):
prod[i]%=self.mod
else:
if self.mod:
prod=[x*other%self.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=self.mod)
return prod
def __matmul__(self,other):
assert type(other)==Polynomial
if self.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=self.mod)
return prod
def __pow__(self,other):
if other==0:
prod=Polynomial([1],max_degree=self.max_degree,eps=self.eps,mod=self.mod)
elif other==1:
prod=Polynomial([x for x in self.polynomial],max_degree=self.max_degree,eps=self.eps,mod=self.mod)
else:
prod=[1]
doub=self.polynomial
if self.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=self.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 self.mod:
inve=MOD(self.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 self.mod:
quot.append(self_polynomial[i]*inve%self.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 self.mod:
self_polynomial[i+j]%=self.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 self.mod:
quot.append(self_polynomial[0]*inve%self.mod)
self_polynomial=[(self_polynomial[i]-other_polynomial[i]*quot[-1])%self.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=self.mod)
else:
assert self.eps<abs(other)
if self.mod:
inve=MOD(self.mod).Pow(other,-1)
quot=Polynomial([x*inve%self.mod for x in self.polynomial],max_degree=self.max_degree,eps=self.eps,mod=self.mod)
else:
quot=Polynomial([x/other for x in self.polynomial],max_degree=self.max_degree,eps=self.eps,mod=self.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 self.mod:
inve=MOD(self.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%self.mod
for j in range(len(other.polynomial)):
rema[i+j]-=quot[i]*other.polynomial[j]
rema[i+j]%=self.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=self.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 self.mod:
inve=MOD(self.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%self.mod
for j in range(len(other.polynomial)):
rema[i+j]-=quot[i]*other.polynomial[j]
rema[i+j]%=self.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=self.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 self.mod:
inve=MOD(self.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%self.mod
for j in range(len(other.polynomial)):
rema[i+j]-=quot[i]*other.polynomial[j]
rema[i+j]%=self.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=self.mod)
rema=Polynomial(rema,max_degree=self.max_degree,eps=self.eps,mod=self.mod)
return quot,rema
def __neg__(self):
if self.mod:
nega=Polynomial([(-x)%self.mod for x in self.polynomial],max_degree=self.max_degree,eps=self.eps,mod=self.mod)
else:
nega=Polynomial([-x for x in self.polynomial],max_degree=self.max_degree,eps=self.eps,mod=self.mod)
return nega
def __pos__(self):
posi=Polynomial([x for x in self.polynomial],max_degree=self.max_degree,eps=self.eps,mod=self.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=self.mod)
def __setitem__(self,n,a):
if self.mod:
a%=self.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 self.mod:
retu%=self.mod
pow_x%=self.mod
return retu
def __str__(self):
return "["+", ".join(map(str,self.polynomial))+"]"
def __len__(self):
return len(self.polynomial)
def differentiate(self):
if self.mod:
differential=[x*i%self.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=self.mod)
def integrate(self):
if self.mod:
integral=[0]+[x*MOD(mod).Pow(i+1,-1)%self.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=self.mod)
def inverse(self):
assert self.polynomial and self.eps<self.polynomial[0]
assert self.max_degree!=-1
if self.mod:
quot=[MOD(self.mod).Pow(self.polynomial[0],-1)]
if self.mod==998244353:
prim_root=3
prim_root_inve=332748118
else:
prim_root=Primitive_Root(self.mod)
prim_root_inve=MOD(self.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,self.mod-1>>bit,self.mod)
U=[1]
for _ in range(a):
U.append(U[-1]*x%self.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])%self.mod,(polynomial[s]-polynomial[t]*U[j])%self.mod
x=pow((self.mod+1)//2,n,self.mod)
for i in range(1<<n):
polynomial[i]*=x
polynomial[i]%=self.mod
else:
for bit in range(n,0,-1):
a=1<<bit-1
x=pow(prim_root_inve,self.mod-1>>bit,self.mod)
U=[1]
for _ in range(a):
U.append(U[-1]*x%self.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])%self.mod,U[j]*(polynomial[s]-polynomial[t])%self.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
DFT_prev=DFT(prev,n+1)
if self.mod:
quot=[x*y%self.mod for x,y in zip(DFT_prev,DFT(self.polynomial[:1<<n+1],n+1))]
else:
quot=[x*y for x,y in zip(DFT_prev,DFT(self.polynomial[:1<<n+1],n+1))]
quot=DFT([0]*(1<<n)+DFT(quot,n+1,inverse=True)[1<<n:],n+1)
if self.mod:
quot=[(-x*y)%self.mod for x,y in zip(DFT_prev,quot)]
else:
quot=[-x*y for x,y in zip(DFT_prev,quot)]
quot=prev+DFT(quot,n+1,inverse=True)[1<<n:]
quot=quot[:self.max_degree+1]
quot=Polynomial(quot,max_degree=self.max_degree,eps=self.eps,mod=self.mod)
return quot
def log(self):
assert self.max_degree!=-1
assert self.polynomial and abs(self.polynomial[0]-1)<=self.eps
log=self.inverse()
if self.mod:
log=Polynomial(NTT(self.differentiate().polynomial,log.polynomial),max_degree=self.max_degree,eps=self.eps,mod=self.mod)
else:
log=Polynomial(FFT(self.differentiate().polynomial,log.polynomial),max_degree=self.max_degree,eps=self.eps,mod=self.mod)
log=log.integrate()
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]%=self.mod
if self.mod:
newton=NTT(newton,Polynomial(differentiated_f(prev),max_degree=len(newton)-1,eps=self.eps,mod=self.mod).inverse().polynomial)[:len(newton)]
else:
newton=FFT(newton,Polynomial(differentiated_f(prev),max_degree=len(newton)-1,eps=self.eps,mod=self.mod).inverse().polynomial)[:len(newton)]
for i in range(len(newton)):
newton[i]=-newton[i]
newton[i]%=self.mod
for i in range(len(prev)):
newton[i]+=prev[i]
newton[i]%=self.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=self.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 self.mod:
n0=Tonelli_Shanks(self.polynomial[cnt0],self.mod)
else:
if self.polynomial[cnt0]>=self.eps:
n0=self.polynomial[cnt0]**.5
if n0==None:
sqrt=None
else:
def f(prev):
if self.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 self.mod:
retu[i]%self.mod
return retu
sqrt=[0]*(cnt0//2)+Polynomial(self.polynomial[cnt0:],max_degree=self.max_degree-cnt0//2,mod=self.mod).Newton(n0,f,differentiated_f).polynomial
sqrt=Polynomial(sqrt,max_degree=self.max_degree,eps=self.eps,mod=self.mod)
else:
sqrt=Polynomial([],max_degree=self.max_degree,eps=self.eps,mod=self.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=self.mod).log().polynomial
newton+=[0]*(2*len(prev)-len(newton))
for i in range(min(len(poly),len(newton))):
newton[i]-=poly[i]
if self.mod:
for i in range(len(newton)):
newton[i]%=self.mod
if self.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=self.mod).Newton(1,f)
def Degree(self):
return len(self.polynomial)-1
def Hadamard(polynomial,n,mod=0,inverse=False):
polynomial_=[x for x in polynomial]+[0]*((1<<n)-len(polynomial))
for bit in range(n):
for i in range(1<<n):
ii=i^(1<<bit)
if i>ii:
continue
polynomial_[i],polynomial_[ii]=polynomial_[i]+polynomial_[ii],polynomial_[i]-polynomial_[ii]
if mod:
polynomial_[i]%=mod
polynomial_[ii]%=mod
if inverse:
if mod:
inve_2=pow((mod+1)//2,n)
for i in range(1<<n):
polynomial_[i]*=inve_2
polynomial_[i]%=mod
else:
pow_2=pow(2,n)
for i in range(1<<n):
polynomial_[i]/=pow_2
return polynomial_
def XOR_Convolution(polynomial0,polynomial1,mod=0):
n=(max(len(polynomial0),len(polynomial1))-1).bit_length()
Hadamard_polynomial0=Hadamard(polynomial0,n,mod=mod)
Hadamard_polynomial1=Hadamard(polynomial1,n,mod=mod)
if mod:
convolution=[x*y%mod for x,y in zip(Hadamard_polynomial0,Hadamard_polynomial1)]
else:
convolution=[x*y for x,y in zip(Hadamard_polynomial0,Hadamard_polynomial1)]
convolution=Hadamard(convolution,n,mod=mod,inverse=True)
return convolution
def Bostan_Mori(poly_nume,poly_deno,N,mod=0,convolve=None):
if type(poly_nume)==Polynomial:
poly_nume=poly_nume.polynomial
if type(poly_deno)==Polynomial:
poly_deno=poly_deno.polynomial
if convolve==None:
def convolve(poly_nume,poly_deno):
conv=[0]*(len(poly_nume)+len(poly_deno)-1)
for i in range(len(poly_nume)):
for j in range(len(poly_deno)):
x=poly_nume[i]*poly_deno[j]
if mod:
x%=mod
conv[i+j]+=x
if mod:
for i in range(len(conv)):
conv[i]%=mod
return conv
while N:
poly_deno_=[-x if i%2 else x for i,x in enumerate(poly_deno)]
if N%2:
poly_nume=convolve(poly_nume,poly_deno_)[1::2]
else:
poly_nume=convolve(poly_nume,poly_deno_)[::2]
poly_deno=convolve(poly_deno,poly_deno_)[::2]
if mod:
for i in range(len(poly_nume)):
poly_nume[i]%=mod
for i in range(len(poly_deno)):
poly_deno[i]%=mod
N//=2
return poly_nume[0]
#mod = 998244353
imag = 911660635
iimag = 86583718
rate2 = (911660635, 509520358, 369330050, 332049552, 983190778, 123842337, 238493703, 975955924, 603855026, 856644456, 131300601,
842657263, 730768835, 942482514, 806263778, 151565301, 510815449, 503497456, 743006876, 741047443, 56250497, 867605899)
irate2 = (86583718, 372528824, 373294451, 645684063, 112220581, 692852209, 155456985, 797128860, 90816748, 860285882, 927414960,
354738543, 109331171, 293255632, 535113200, 308540755, 121186627, 608385704, 438932459, 359477183, 824071951, 103369235)
rate3 = (372528824, 337190230, 454590761, 816400692, 578227951, 180142363, 83780245, 6597683, 70046822, 623238099,
183021267, 402682409, 631680428, 344509872, 689220186, 365017329, 774342554, 729444058, 102986190, 128751033, 395565204)
irate3 = (509520358, 929031873, 170256584, 839780419, 282974284, 395914482, 444904435, 72135471, 638914820, 66769500,
771127074, 985925487, 262319669, 262341272, 625870173, 768022760, 859816005, 914661783, 430819711, 272774365, 530924681)
def butterfly(a):
n = len(a)
h = (n - 1).bit_length()
len_ = 0
while len_ < h:
if h - len_ == 1:
p = 1 << (h - len_ - 1)
rot = 1
for s in range(1 << len_):
offset = s << (h - len_)
for i in range(p):
l = a[i + offset]
r = a[i + offset + p] * rot % mod
a[i + offset] = (l + r) % mod
a[i + offset + p] = (l - r) % mod
if s + 1 != 1 << len_:
rot *= rate2[(~s & -~s).bit_length() - 1]
rot %= mod
len_ += 1
else:
p = 1 << (h - len_ - 2)
rot = 1
for s in range(1 << len_):
rot2 = rot * rot % mod
rot3 = rot2 * rot % mod
offset = s << (h - len_)
for i in range(p):
a0 = a[i + offset]
a1 = a[i + offset + p] * rot
a2 = a[i + offset + p * 2] * rot2
a3 = a[i + offset + p * 3] * rot3
a1na3imag = (a1 - a3) % mod * imag
a[i + offset] = (a0 + a2 + a1 + a3) % mod
a[i + offset + p] = (a0 + a2 - a1 - a3) % mod
a[i + offset + p * 2] = (a0 - a2 + a1na3imag) % mod
a[i + offset + p * 3] = (a0 - a2 - a1na3imag) % mod
if s + 1 != 1 << len_:
rot *= rate3[(~s & -~s).bit_length() - 1]
rot %= mod
len_ += 2
def butterfly_inv(a):
n = len(a)
h = (n - 1).bit_length()
len_ = h
while len_:
if len_ == 1:
p = 1 << (h - len_)
irot = 1
for s in range(1 << (len_ - 1)):
offset = s << (h - len_ + 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] = (l - r) * irot % mod
if s + 1 != (1 << (len_ - 1)):
irot *= irate2[(~s & -~s).bit_length() - 1]
irot %= mod
len_ -= 1
else:
p = 1 << (h - len_)
irot = 1
for s in range(1 << (len_ - 2)):
irot2 = irot * irot % mod
irot3 = irot2 * irot % mod
offset = s << (h - len_ + 2)
for i in range(p):
a0 = a[i + offset]
a1 = a[i + offset + p]
a2 = a[i + offset + p * 2]
a3 = a[i + offset + p * 3]
a2na3iimag = (a2 - a3) * iimag % mod
a[i + offset] = (a0 + a1 + a2 + a3) % mod
a[i + offset + p] = (a0 - a1 + a2na3iimag) * irot % mod
a[i + offset + p * 2] = (a0 + a1 - a2 - a3) * irot2 % mod
a[i + offset + p * 3] = (a0 - a1 - a2na3iimag) * irot3 % mod
if s + 1 != (1 << (len_ - 2)):
irot *= irate3[(~s & -~s).bit_length() - 1]
irot %= mod
len_ -= 2
def integrate(a):
a=a.copy()
n = len(a)
assert n > 0
a.pop()
a.insert(0, 0)
inv = [1, 1]
for i in range(2, n):
inv.append(-inv[mod%i] * (mod//i) % mod)
a[i] = a[i] * inv[i] % mod
return a
def differentiate(a):
n = len(a)
assert n > 0
for i in range(2, n):
a[i] = a[i] * i % mod
a.pop(0)
a.append(0)
return a
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]) % mod
else:
for i in range(n):
for j in range(m):
ans[i + j] = (ans[i + j] + a[i] * b[j]) % mod
return ans
def convolution_ntt(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] % mod
butterfly_inv(a)
a = a[:n + m - 1]
iz = pow(z, mod - 2, mod)
for i in range(n + m - 1):
a[i] = a[i] * iz % 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] % mod
butterfly_inv(a)
a = a[:2 * n - 1]
iz = pow(z, mod - 2, mod)
for i in range(2 * n - 1):
a[i] = a[i] * iz % 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) <= 60:
return convolution_naive(a, b)
if a is b:
return convolution_square(a)
return convolution_ntt(a, b)
def inverse(a):
n = len(a)
assert n > 0 and a[0] != 0
res = [pow(a[0], mod - 2, mod)]
m = 1
while m < n:
f = a[:min(n,2*m)] + [0]*(2*m-min(n,2*m))
g = res + [0]*m
butterfly(f)
butterfly(g)
for i in range(2*m):
f[i] = f[i] * g[i] % mod
butterfly_inv(f)
f = f[m:] + [0]*m
butterfly(f)
for i in range(2*m):
f[i] = f[i] * g[i] % mod
butterfly_inv(f)
iz = pow(2*m, mod-2, mod)
iz = (-iz*iz) % mod
for i in range(m):
f[i] = f[i] * iz % mod
res += f[:m]
m <<= 1
return res[:n]
def log(a):
a = a.copy()
n = len(a)
assert n > 0 and a[0] == 1
a_inv = inverse(a)
a=differentiate(a)
a = convolution(a, a_inv)[:n]
a=integrate(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()
h_drv=differentiate(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] % 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] % mod
butterfly_inv(_f)
_f = _f[:m//2]
iz = pow(m, mod - 2, mod)
iz *= -iz
iz %= mod
for i in range(m//2):
_f[i] = _f[i] * iz % mod
g.extend(_f)
t = a[:m]
t=differentiate(t)
r = h_drv[:m - 1]
r.append(0)
butterfly(r)
for i in range(m):
r[i] = r[i] * f_fft[i] % mod
butterfly_inv(r)
im = pow(-m, mod - 2, mod)
for i in range(m):
r[i] = r[i] * im % mod
for i in range(m):
t[i] = (t[i] + r[i]) % 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] % mod
butterfly_inv(t)
t = t[:m]
i2m = pow(2 * m, mod - 2, mod)
for i in range(m):
t[i] = t[i] * i2m % mod
v = a[m:min(n, 2 * m)]
v += [0] * (m - len(v))
t = [0] * (m - 1) + t + [0]
t=integrate(t)
for i in range(m):
v[i] = (v[i] - t[m + i]) % mod
v += [0] * m
butterfly(v)
for i in range(2 * m):
v[i] = v[i] * f_fft[i] % mod
butterfly_inv(v)
v = v[:m]
i2m = pow(2 * m, mod - 2, mod)
for i in range(m):
v[i] = v[i] * i2m % mod
for i in range(min(n - m, m)):
a[m + i] = v[i]
m *= 2
return a
def power(a,k):
n = len(a)
assert n>0
if k==0:
return [1]+[0]*(n-1)
l = 0
while l < len(a) and not a[l]:
l += 1
if l * k >= n:
return [0] * n
ic = pow(a[l], mod - 2, mod)
pc = pow(a[l], k, mod)
a = log([a[i] * ic % mod for i in range(l, len(a))])
for i in range(len(a)):
a[i] = a[i] * k % mod
a = exp(a)
for i in range(len(a)):
a[i] = a[i] * pc % mod
a = [0] * (l * k) + a[:n - l * k]
return a
def sqrt(a):
if len(a) == 0:
return []
if a[0] == 0:
for d in range(1, len(a)):
if a[d]:
if d & 1:
return None
if len(a) - 1 < d // 2:
break
res=sqrt(a[d:]+[0]*(d//2))
if res == None:
return None
res = [0]*(d//2)+res
return res
return [0]*len(a)
sqr = Tonelli_Shanks(a[0],mod)
if sqr == None:
return None
T = [0] * (len(a))
T[0] = sqr
res = T.copy()
T[0] = pow(sqr,mod-2,mod) #T:res^{-1}
m = 1
two_inv = (mod + 1) // 2
F = [sqr]
while m <= len(a) - 1:
for i in range(m):
F[i] *= F[i]
F[i] %= mod
butterfly_inv(F)
iz = pow(m, mod-2, mod)
for i in range(m):
F[i] = F[i] * iz % mod
delta = [0] * (2 * m)
for i in range(m):
delta[i + m] = F[i] - a[i] - (a[i + m] if i+m<len(a) else 0)
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)
iz = pow(2*m, mod-2, mod)
for i in range(2*m):
delta[i] = delta[i] * iz % mod
for i in range(m, min(2 * m, len(a))):
res[i] = -delta[i] * two_inv%mod
res[i]%=mod
if 2 * m > len(a) - 1:
break
F = res[: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
iz = pow(2*m, mod-2, mod)
for i in range(m,2*m):
eps[i] = eps[i] * iz % mod
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]*iz
T[i]%=mod
iz = iz*iz % mod
m <<= 1
return res
def taylor_shift(a,c):
a=a.copy()
n=len(a)
MD=MOD(mod)
MD.Build_Fact(n-1)
for i in range(n):
a[i]*=MD.Fact(i)
a[i]%=mod
C=[1]
for i in range(1,n):
C.append(C[-1]*c%mod)
for i in range(n):
C[i]*=MD.Fact_Inve(i)
C[i]%=mod
a=convolution(a,C[::-1])[n-1:]
for i in range(n):
a[i]*=MD.Fact_Inve(i)
a[i]%=mod
return a
def division_modulus(f,g):
n=len(f)
m=len(g)
while m and g[m-1]==0:
m-=1
assert m
if n>=m:
fR=f[::-1][:n-m+1]
gR=g[:m][::-1][:n-m+1]+[0]*max(0,n-m+1-m)
qR=convolution(fR,inverse(gR))[:n-m+1]
q=qR[::-1]
r=[(f[i]-x)%mod for i,x in enumerate(convolution(g,q)[:m-1])]
while r and r[-1]==0:
r.pop()
else:
q,r=[],f.copy()
return q,r
def multipoint_evaluation(f, x):
n = len(x)
sz = 1 << (n - 1).bit_length()
g = [[1] for _ in range(2 * sz)]
for i in range(n):
g[i + sz] = [-x[i], 1]
for i in range(1, sz)[::-1]:
g[i] = convolution(g[2 * i],g[2 * i + 1])
g[1] =division_modulus(f,g[1])[1]
for i in range(2, 2 * sz):
g[i]=division_modulus(g[i>>1],g[i])[1]
res = [g[i + sz][0] if g[i+sz] else 0 for i in range(n)]
return res
class Lagrange_Interpolation:
def __init__(self,X=None,Y=None,x0=None,xd=None,mod=0):
self.degree=len(Y)-1
self.mod=mod
assert self.mod==0 or self.degree<self.mod
if x0!=None and xd!=None:
assert xd>0
if self.mod:
self.X=[(x0+i*xd)%self.mod for i in range(self.degree+1)]
fact_inve=1
for i in range(1,self.degree+1):
fact_inve*=i*xd
fact_inve%=self.mod
fact_inve=MOD(self.mod).Pow(fact_inve,-1)
self.coefficient=[y for y in Y]
for i in range(self.degree-1,-1,-2):
self.coefficient[i]*=-1
for i in range(self.degree,-1,-1):
self.coefficient[i]*=fact_inve
self.coefficient[i]%=self.mod
self.coefficient[self.degree-i]*=fact_inve
self.coefficient[self.degree-i]%=self.mod
fact_inve*=i*xd
fact_inve%=self.mod
else:
self.X=[x0+i*xd for i in range(self.degree+1)]
self.coefficient=[y for y in Y]
for i in range(self.degree-1,-1,-2):
self.coefficient[i]*=-1
fact=1
for i in range(1,self.degree+2):
self.coefficient[i-1]/=fact
self.coefficient[self.degree-i+1]/=fact
fact*=i*xd
else:
self.X=X
assert len(self.X)==self.degree+1
self.coefficient=[1]*(self.degree+1)
for i in range(self.degree+1):
for j in range(self.degree+1):
if i==j:
continue
self.coefficient[i]*=X[i]-X[j]
if self.mod:
self.coefficient[i]%=self.mod
if self.mod:
for i in range(self.degree+1):
self.coefficient[i]=MOD(self.mod).Pow(self.coefficient[i],-1)*Y[i]%self.mod
else:
for i in range(self.degree+1):
self.coefficient[i]=Y[i]/self.coefficient[i]
def __call__(self,N):
if self.mod:
N%=self.mod
XX=[N-x for x in self.X]
XX_left=[1]*(self.degree+2)
for i in range(1,self.degree+2):
XX_left[i]=XX_left[i-1]*XX[i-1]
if self.mod:
XX_left[i]%=self.mod
XX_right=[1]*(self.degree+2)
for i in range(self.degree,-1,-1):
XX_right[i]=XX_right[i+1]*XX[i]
if self.mod:
XX_right[i]%=self.mod
if self.mod:
return sum(XX_left[i]*XX_right[i+1]%self.mod*self.coefficient[i]%self.mod for i in range(self.degree+1))%self.mod
else:
return sum(XX_left[i]*XX_right[i+1]*self.coefficient[i] for i in range(self.degree+1))
N,a=map(int,readline().split())
X,Y=[],[]
for _ in range(N):
x,y=map(int,readline().split())
X.append(x)
Y.append(y)
mod=998244353
if a in X:
i=X.index(a)
LI=Lagrange_Interpolation([X[j] for j in range(N) if i!=j],[Y[j] for j in range(N) if i!=j],mod=mod)
ans=(LI(a)+Y[i]*(N-1))%mod
else:
MD=MOD(mod)
XX=[1]*N
for i in range(N):
for j in range(N):
if i==j:
continue
XX[i]*=X[i]-X[j]
XX[i]%=mod
XX[i]=MD.Pow(XX[i],-1)
A=1
for x in X:
A*=a-x
A%=mod
inverse=[MD.Pow(a-x,-1) for x in X]
ans=0
for i in range(N):
for j in range(N):
if i==j:
continue
ans+=A*inverse[i]%mod*inverse[j]%mod*Y[j]%mod*XX[j]%mod*(X[j]-X[i])%mod
ans%=mod
print(ans)