結果
| 問題 |
No.2446 完全列
|
| コンテスト | |
| ユーザー |
👑  Kazun
|
| 提出日時 | 2023-08-25 23:18:20 |
| 言語 | PyPy3 (7.3.15) |
| 結果 |
WA
|
| 実行時間 | - |
| コード長 | 13,000 bytes |
| コンパイル時間 | 543 ms |
| コンパイル使用メモリ | 81,664 KB |
| 実行使用メモリ | 56,960 KB |
| 最終ジャッジ日時 | 2024-12-24 14:22:40 |
| 合計ジャッジ時間 | 3,071 ms |
|
ジャッジサーバーID （参考情報） |
judge2 / judge1 |
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| ファイルパターン | 結果 |
|---|---|
| other | AC * 18 WA * 12 |
ソースコード
from copy import deepcopy
class Modulo_Matrix():
__slots__=("ele","row","col","size")
#入力
def __init__(self,M):
""" 行列 M の定義
M: 行列
※ Mod: 法はグローバル変数から指定
"""
self.ele=[[x%Mod for x in X] for X in M]
R=len(M)
if R!=0:
C=len(M[0])
else:
C=0
self.row=R
self.col=C
self.size=(R,C)
#出力
def __str__(self):
return "["+"\n".join(map(str,self.ele))+"]"
def __repr__(self):
return str(self)
#+,-
def __pos__(self):
return self
def __neg__(self):
return self.__scale__(-1)
#加法
def __add__(self,other):
M=self.ele; N=other.ele
L=[[0]*self.col for _ in range(self.row)]
for i in range(self.row):
Li,Mi,Ni=L[i],M[i],N[i]
for j in range(self.col):
Li[j]=Mi[j]+Ni[j]
return Modulo_Matrix(L)
def __iadd__(self,other):
M=self.ele; N=other.ele
for i in range(self.row):
Mi,Ni=M[i],N[i]
for j in range(self.col):
Mi[j]+=Ni[j]
Mi[j]%=Mod
return self
#減法
def __sub__(self,other):
M=self.ele; N=other.ele
L=[[0]*self.col for _ in range(self.row)]
for i in range(self.row):
Li,Mi,Ni=L[i],M[i],N[i]
for j in range(self.col):
Li[j]=Mi[j]-Ni[j]
return Modulo_Matrix(L)
def __isub__(self,other):
M=self.ele; N=other.ele
for i in range(self.row):
Mi,Ni=M[i],N[i]
for j in range(self.col):
Mi[j]-=Ni[j]
Mi[j]%=Mod
return self
#乗法
def __mul__(self,other):
if isinstance(other,Modulo_Matrix):
assert self.col==other.row, "左側の列と右側の行が一致しません.({},{})".format(self.size,other.size)
M=self.ele; N=other.ele
E=[[0]*other.col for _ in range(self.row)]
for i in range(self.row):
Ei,Mi=E[i],M[i]
for k in range(self.col):
m_ik,Nk=Mi[k],N[k]
for j in range(other.col):
Ei[j]+=m_ik*Nk[j]
Ei[j]%=Mod
return Modulo_Matrix(E)
elif isinstance(other,int):
return self.__scale__(other)
def __rmul__(self,other):
if isinstance(other,int):
return self.__scale__(other)
def inverse(self):
assert self.row==self.col,"正方行列ではありません."
M=self
N=M.row
R=[[1 if i==j else 0 for j in range(N)] for i in range(N)]
T=deepcopy(M.ele)
for j in range(N):
if T[j][j]==0:
for i in range(j+1,N):
if T[i][j]:
break
else:
assert 0, "正則行列ではありません"
T[j],T[i]=T[i],T[j]
R[j],R[i]=R[i],R[j]
Tj,Rj=T[j],R[j]
inv=pow(Tj[j], Mod-2, Mod)
for k in range(N):
Tj[k]*=inv; Tj[k]%=Mod
Rj[k]*=inv; Rj[k]%=Mod
for i in range(N):
if i==j: continue
c=T[i][j]
Ti,Ri=T[i],R[i]
for k in range(N):
Ti[k]-=Tj[k]*c; Ti[k]%=Mod
Ri[k]-=Rj[k]*c; Ri[k]%=Mod
return Modulo_Matrix(R)
#スカラー倍
def __scale__(self,r):
M=self.ele
r%=Mod
L=[[(r*M[i][j])%Mod for j in range(self.col)] for i in range(self.row)]
return Modulo_Matrix(L)
#累乗
def __pow__(self,n):
assert self.row==self.col, "正方行列ではありません."
r=self.col
def __mat_mul(A,B):
E=[[0]*r for _ in range(r)]
for i in range(r):
a=A[i]; e=E[i]
for k in range(r):
b=B[k]
for j in range(r):
e[j]+=a[k]*b[j]
e[j]%=Mod
return E
X=deepcopy(self.ele)
E=[[1 if i==j else 0 for j in range(r)] for i in range(r)]
sgn=1 if n>=0 else -1
n=abs(n)
while True:
if n&1:
E=__mat_mul(E,X)
n>>=1
if n:
X=__mat_mul(X,X)
else:
break
if sgn==1:
return Modulo_Matrix(E)
else:
return Modulo_Matrix(E).inverse()
#等号
def __eq__(self,other):
return self.ele==other.ele
#不等号
def __neq__(self,other):
return not(self==other)
#転置
def transpose(self):
return Modulo_Matrix(list(map(list,zip(*self.ele))))
#行基本変形
def row_reduce(self):
M=self
(R,C)=M.size
T=[]
for i in range(R):
U=[]
for j in range(C):
U.append(M.ele[i][j])
T.append(U)
I=0
for J in range(C):
if T[I][J]==0:
for i in range(I+1,R):
if T[i][J]!=0:
T[i],T[I]=T[I],T[i]
break
if T[I][J]!=0:
u=T[I][J]
u_inv=pow(u, Mod-2, Mod)
for j in range(C):
T[I][j]*=u_inv
T[I][j]%=Mod
for i in range(R):
if i!=I:
v=T[i][J]
for j in range(C):
T[i][j]-=v*T[I][j]
T[i][j]%=Mod
I+=1
if I==R:
break
return Modulo_Matrix(T)
#列基本変形
def column_reduce(self):
M=self
(R,C)=M.size
T=[]
for i in range(R):
U=[]
for j in range(C):
U.append(M.ele[i][j])
T.append(U)
J=0
for I in range(R):
if T[I][J]==0:
for j in range(J+1,C):
if T[I][j]!=0:
for k in range(R):
T[k][j],T[k][J]=T[k][J],T[k][j]
break
if T[I][J]!=0:
u=T[I][J]
u_inv=pow(u, Mod-2, Mod)
for i in range(R):
T[i][J]*=u_inv
T[i][J]%=Mod
for j in range(C):
if j!=J:
v=T[I][j]
for i in range(R):
T[i][j]-=v*T[i][J]
T[i][j]%=Mod
J+=1
if J==C:
break
return Modulo_Matrix(T)
#行列の階数
def rank(self):
M=self.row_reduce()
(R,C)=M.size
T=M.ele
rnk=0
for i in range(R):
f=False
for j in range(C):
if T[i][j]!=0:
f=True
break
if f:
rnk+=1
else:
break
return rnk
#行の結合
def row_union(self,other):
return Modulo_Matrix(self.ele+other.ele)
#列の結合
def column_union(self,other):
E=[]
for i in range(self.row):
E.append(self.ele[i]+other.ele[i])
return Modulo_Matrix(E)
def __getitem__(self,index):
if isinstance(index, int):
return self.ele[index]
else:
return self.ele[index[0]][index[1]]
def __setitem__(self,index,val):
assert isinstance(index,tuple) and len(index)==2
self.ele[index[0]][index[1]]=val
#==================================================
class Modulo_Vector:
def __init__(self, vector):
self.vec = [vi % Mod for vi in vector]
self.size = len(vector)
#出力
def __str__(self):
return str(self.vec)
def __repr__(self):
return str(self)
def __bool__(self):
return any(self.vec)
def __iter__(self):
yield from self.vec
#+,-
def __pos__(self):
return self
def __neg__(self):
return self.__scale__(-1)
#加法
def __add__(self, other):
assert self.size == other.size, f"2つのベクトルのサイズが異なります. ({self.size}, {other.size})"
return Modulo_Vector([vi + wi for vi, wi in zip(self, other)])
#減法
def __sub__(self, other):
return self+(-other)
def __rsub__(self, other):
return (-self)+other
#乗法
def __mul__(self,other):
pass
def __rmul__(self,other):
return self.__scale__(other)
#スカラー倍
def __scale__(self, r):
return Modulo_Vector([r * vi for vi in self])
#内積
def inner(self,other):
assert self.size == other.size, f"2つのベクトルのサイズが異なります. ({self.size}, {other.size})"
return sum(vi * wi % Mod for vi, wi in zip(self, other)) % Mod
#累乗
def __pow__(self,n):
pass
#等号
def __eq__(self, other):
return self.vec == other.vec
def __len__(self):
return self.size
#不等号
def __neq__(self, other):
return not (self == other)
def __getitem__(self,index):
assert isinstance(index,int)
return self.vec[index]
def __setitem__(self,index,val):
assert isinstance(index,int)
self.vec[index]=val
class Modulo_Vector_Space:
def __init__(self, dim):
""" 次元が dim のベクトル空間の部分空間を生成する.
"""
self.dim=dim
self.basis=[]
self.__ind=[]
def __contains__(self, v):
for i,u in zip(self.__ind, self.basis):
v-=v[i]*u
return not bool(v)
def add_vectors(self, *S):
for v in S:
assert len(v)==self.dim
for i,u in zip(self.__ind, self.basis):
v-=v[i]*u
if bool(v):
for j in range(self.dim):
if v[j]:
self.__ind.append(j)
break
v=pow(v[j], Mod-2, Mod) * v
self.basis.append(v)
for k in range(len(self.basis)-1):
self.basis[k]-=self.basis[k][j]*v
def dimension(self):
return len(self.basis)
def __le__(self, other):
for u in self.basis:
if u not in other:
return False
return True
def __ge__(self, other):
return other<=self
def __eq__(self, other):
return (self<=other) and (other<=self)
def refresh(self):
I=sorted(range(len(self.__ind)), key=lambda i:self.__ind[i])
self.basis=[self.basis[i] for i in I]
self.__ind=[self.__ind[i] for i in I]
def projection(self, v):
for i,u in zip(self.__ind, self.basis):
v-=v[i]*u
return v
def Kernel_Space(A):
""" 行列 A の核空間 Ker A (Ax=0 となる x の空間) を求める.
"""
row,col=A.size
T=deepcopy(A.ele)
p=[]; q=[]
rnk=0
for j in range(col):
for i in range(rnk,row):
if T[i][j]!=0:
break
else:
q.append(j)
continue
if j==col:
return Modulo_Vector_Space(col)
p.append(j)
T[rnk],T[i]=T[i],T[rnk]
inv=pow(T[rnk][j], Mod-2, Mod)
for k in range(col):
T[rnk][k]=(inv*T[rnk][k])%Mod
for s in range(row):
if s==rnk:
continue
c=-T[s][j]
for t in range(col):
T[s][t]=(T[s][t]+c*T[rnk][t])%Mod
rnk+=1
x=[0]*col
for i in range(rnk):
x[p[i]]=T[i][-1]
ker_dim=col-rnk
ker=[[0]*col for _ in range(ker_dim)]
for i in range(ker_dim):
ker[i][q[i]]=1
for i in range(ker_dim):
for j in range(rnk):
ker[i][p[j]]=-T[j][q[i]]%Mod
Ker=Modulo_Vector_Space(col)
Ker.add_vectors(*[Modulo_Vector(v) for v in ker])
return Ker
#==================================================
def solve():
L, M, N = map(int, input().split())
A = [None] * L
for i in range(L):
A[i] = list(map(int, input().split()))
B = [None] * M
for i in range(M):
B[i] = list(map(int, input().split()))
def calc(mod):
global Mod; Mod = mod
X = Modulo_Matrix(A)
Y = Modulo_Matrix(B)
return Y.rank() == N and X.rank() == L and L == M - N
return all(calc(mod) for mod in [998244353, 10**9 + 7, 10**9 + 9])
#==================================================
print("Yes" if solve() else "No")