結果
| 問題 | No.1559 Next Rational |
| ユーザー |
 vwxyz
|
| 提出日時 | 2022-10-20 17:16:21 |
| 言語 | Python3 (3.12.2 + numpy 1.26.4 + scipy 1.12.0) |
| 結果 |
AC
|
| 実行時間 | 43 ms / 2,000 ms |
| コード長 | 22,100 bytes |
| コンパイル時間 | 120 ms |
| コンパイル使用メモリ | 14,976 KB |
| 実行使用メモリ | 13,184 KB |
| 最終ジャッジ日時 | 2024-06-30 09:29:11 |
| 合計ジャッジ時間 | 1,784 ms |
|
ジャッジサーバーID （参考情報） |
judge1 / judge2 |
（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 | 42 ms
13,056 KB |
| testcase_01 | AC | 42 ms
13,056 KB |
| testcase_02 | AC | 40 ms
13,056 KB |
| testcase_03 | AC | 40 ms
13,056 KB |
| testcase_04 | AC | 40 ms
13,184 KB |
| testcase_05 | AC | 40 ms
13,184 KB |
| testcase_06 | AC | 40 ms
13,056 KB |
| testcase_07 | AC | 39 ms
13,184 KB |
| testcase_08 | AC | 40 ms
13,184 KB |
| testcase_09 | AC | 40 ms
13,056 KB |
| testcase_10 | AC | 43 ms
13,056 KB |
| testcase_11 | AC | 41 ms
13,056 KB |
| testcase_12 | AC | 40 ms
13,056 KB |
| testcase_13 | AC | 41 ms
13,056 KB |
| testcase_14 | AC | 40 ms
13,056 KB |
| testcase_15 | AC | 41 ms
13,184 KB |
| testcase_16 | AC | 39 ms
13,056 KB |
| testcase_17 | AC | 40 ms
13,056 KB |
ソースコード
import sys
readline=sys.stdin.readline
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
def Primitive_Root(p):
if p==2:
return 1
if p==167772161:
return 3
if p==469762049:
return 3
if p==754974721:
return 11
if p==998244353:
return 3
if p==10**9+7:
return 5
divisors=[2]
pp=(p-1)//2
while pp%2==0:
pp//=2
for d in range(3,pp+1,2):
if d**2>pp:
break
if pp%d==0:
divisors.append(d)
while pp%d==0:
pp//=d
if pp>1:
divisors.append(pp)
primitive_root=2
while True:
for d in divisors:
if pow(primitive_root,(p-1)//d,p)==1:
break
else:
return primitive_root
primitive_root+=1
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 __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 __rtruediv__(self,other):
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,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 self.mod:
polynomial=[x*y*y%self.mod 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]
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 self.mod:
quot[i]%=self.mod
for i in range(1<<n,1<<n+1):
quot[i]=-quot[i]
if self.mod:
quot[i]%=self.mod
quot=quot[:self.max_degree+1]
for i in range(len(quot)):
quot[i]*=other
if self.mod:
quot[i]%=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 Degree(self):
return len(self.polynomial)-1
def NTT(polynomial0,polynomial1):
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 Bostan_Mori(poly_nume,poly_deno,N,mod=0,fft=False,ntt=False):
if type(poly_nume)==Polynomial:
poly_nume=poly_nume.polynomial
if type(poly_deno)==Polynomial:
poly_deno=poly_deno.polynomial
if ntt:
convolve=NTT
elif fft:
convolve=FFT
else:
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)):
conv[i+j]+=poly_nume[i]*poly_deno[j]
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 fft and 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=10**9+7
N,A,B,K=map(int,readline().split())
x=(A**2%mod+B**2%mod+K)*pow(A*B%mod,mod-2,mod)%mod
ans=Bostan_Mori([A,B-x*A],[1,-x,1],N-1,mod)
print(ans)