結果
| 問題 | No.42 貯金箱の溜息 |
| ユーザー |
 vwxyz
|
| 提出日時 | 2021-08-21 07:40:42 |
| 言語 | PyPy3 (7.3.15) |
| 結果 |
AC
|
| 実行時間 | 335 ms / 5,000 ms |
| コード長 | 20,417 bytes |
| コンパイル時間 | 184 ms |
| コンパイル使用メモリ | 82,304 KB |
| 実行使用メモリ | 91,500 KB |
| 最終ジャッジ日時 | 2024-10-14 16:19:32 |
| 合計ジャッジ時間 | 2,197 ms |
|
ジャッジサーバーID （参考情報） |
judge4 / judge3 |
（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 | 335 ms
91,500 KB |
| testcase_01 | AC | 300 ms
91,280 KB |
| testcase_02 | AC | 295 ms
91,280 KB |
ソースコード
import bisect
import copy
import decimal
import fractions
import functools
import heapq
import itertools
import math
import random
import sys
from collections import Counter,deque,defaultdict
from functools import lru_cache,reduce
from heapq import heappush,heappop,heapify,heappushpop,_heappop_max,_heapify_max
def _heappush_max(heap,item):
heap.append(item)
heapq._siftdown_max(heap, 0, len(heap)-1)
def _heappushpop_max(heap, item):
if heap and item < heap[0]:
item, heap[0] = heap[0], item
heapq._siftup_max(heap, 0)
return item
from math import degrees, gcd as GCD
read=sys.stdin.read
readline=sys.stdin.readline
readlines=sys.stdin.readlines
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,mod):
self.mod=mod
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]
for i in range(1,N+1):
self.factorial.append((self.factorial[-1]*i)%self.mod)
self.factorial_inv=[None]*(N+1)
self.factorial_inv[-1]=self.Pow(self.factorial[-1],-1)
for i in range(N-1,-1,-1):
self.factorial_inv[i]=(self.factorial_inv[i+1]*(i+1))%self.mod
return self.factorial_inv
def Fact(self,N):
return self.factorial[N]
def Fact_Inv(self,N):
return self.factorial_inv[N]
def Comb(self,N,K):
if K<0 or K>N:
return 0
s=self.factorial[N]
s=(s*self.factorial_inv[K])%self.mod
s=(s*self.factorial_inv[N-K])%self.mod
return s
class Lagrange_Interpolation:
def __init__(self,X=False,Y=False,x0=None,xd=None,mod=0):
self.degree=len(Y)-1
self.mod=mod
assert self.degree<self.mod
if x0!=None and xd!=None:
assert xd>0
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=X
assert len(self.X)==self.degree+1
self.coefficient=[y for y in Y]
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]
self.coefficient%=self.mod
def __getitem__(self,N):
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]%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]%self.mod
return sum(XX_left[i]*XX_right[i+1]*self.coefficient[i] for i in range(self.degree+1))%self.mod
def NTT(polynomial1,polynomial2):
prim_root=3
prim_root_inve=MOD(mod).Pow(prim_root,-1)
def DFT(polynomial,inverse=False):
dft=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
dft[s],dft[t]=(dft[s]+dft[t]*U[j])%mod,(dft[s]-dft[t]*U[j])%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
dft[s],dft[t]=(dft[s]+dft[t])%mod,U[j]*(dft[s]-dft[t])%mod
return dft
N=len(polynomial1)+len(polynomial2)-1
n=(N-1).bit_length()
ntt=[x*y%mod for x,y in zip(DFT(polynomial1),DFT(polynomial2))]
ntt=DFT(ntt,inverse=True)
x=pow((mod+1)//2,n)
ntt=[ntt[i]*x%mod for i in range(N)]
return ntt
def FFT(polynomial1,polynomial2,digit=10**5):
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)]
dft=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
dft[s],dft[t]=dft[s]+dft[t]*primitive_root[j<<n-bit],dft[s]-dft[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
dft[s],dft[t]=dft[s]+dft[t],primitive_root[j<<n-bit]*(dft[s]-dft[t])
return dft
def FFT_(polynomial1,polynomial2):
N1=len(polynomial1)
N2=len(polynomial2)
N=N1+N2-1
n=(N-1).bit_length()
fft=[x*y for x,y in zip(DFT(polynomial1,n),DFT(polynomial2,n))]
fft=DFT(fft,n,inverse=True)
fft=[round((fft[i]/(1<<n)).real) for i in range(N)]
return fft
N1=len(polynomial1)
N2=len(polynomial2)
N=N1+N2-1
polynomial11,polynomial12=[None]*N1,[None]*N1
polynomial21,polynomial22=[None]*N2,[None]*N2
for i in range(N1):
polynomial11[i],polynomial12[i]=divmod(polynomial1[i],digit)
for i in range(N2):
polynomial21[i],polynomial22[i]=divmod(polynomial2[i],digit)
polynomial=[0]*(N)
a=digit**2-digit
for i,x in enumerate(FFT_(polynomial11,polynomial21)):
polynomial[i]+=x*a
a=digit-1
for i,x in enumerate(FFT_(polynomial12,polynomial22)):
polynomial[i]-=x*a
for i,x in enumerate(FFT_([x1+x2 for x1,x2 in zip(polynomial11,polynomial12)],[x1+x2 for x1,x2 in zip(polynomial21,polynomial22)])):
polynomial[i]+=x*digit
return polynomial
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=1e-12,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 abs(self.polynomial[i]-other.polynomial[i])>self.eps:
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 abs(self.polynomial[i]-other.polynomial[i])>self.eps:
return True
return False
def __add__(self,other):
assert 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
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):
assert 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
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
while prod and abs(prod[-1])<self.eps:
prod.pop()
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(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:]
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
rema.pop()
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]
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
rema.pop()
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]
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 n<=len(self.polynomial)-1:
return self.polynomial[n]
else:
return 0
def __setitem__(self,n,x):
if self.mod:
x%=self.mod
if self.max_degree==-1 or n<=self.max_degree:
if n<=len(self.polynomial)-1:
self.polynomial[n]=x
elif self.eps<abs(x):
self.polynomial+=[0]*(n-len(self.polynomial))+[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 Bostan_Mori(poly_deno,poly_nume,N,mod=0,fft=False,ntt=False):
if type(poly_deno)==Polynomial:
poly_deno=poly_deno.polynomial
if type(poly_nume)==Polynomial:
poly_nume=poly_nume.polynomial
if ntt:
convolve=NTT
elif fft:
convolve=FFT
else:
def convolve(poly_deno,poly_nume):
conv=[0]*(len(poly_deno)+len(poly_nume)-1)
for i in range(len(poly_deno)):
for j in range(len(poly_nume)):
conv[i+j]+=poly_deno[i]*poly_nume[j]
if mod:
for i in range(len(conv)):
conv[i]%=mod
return conv
while N:
poly_nume_=[-x if i%2 else x for i,x in enumerate(poly_nume)]
if N%2:
poly_deno=convolve(poly_deno,poly_nume_)[1::2]
else:
poly_deno=convolve(poly_deno,poly_nume_)[::2]
poly_nume=convolve(poly_nume,poly_nume_)[::2]
if fft and mod:
for i in range(len(poly_deno)):
poly_deno[i]%=mod
for i in range(len(poly_nume)):
poly_nume[i]%=mod
N//=2
return poly_deno[0]
mod=10**9+9
P=Polynomial([1],max_degree=3000-1,mod=mod)
for money in (1,5,10,50,100,500):
lst=[0]*(money+1)
lst[0]=1
lst[money]=-1
P/=Polynomial(lst)
T=int(readline())
for _ in range(T):
M=int(readline())
r=M%500
Y=[P[i] for i in range(r,3000,500)]
LI=Lagrange_Interpolation(Y=Y,x0=r,xd=500,mod=mod)
ans=LI[M]
print(ans)