【洛谷P3390】矩阵快速幂

矩阵快速幂

题目描述

矩阵乘法：

A[n*m]*B[m*k]=C[n*k];

C[i][j]=sum(A[i][1~n]+B[1~n][j])

为了便于赋值和定义，我们定义一个结构体储存矩阵：

struct Matrix{
    long long m[][];
};

X*Y：

 Matrix cheng(Matrix X,Matrix Y)
 {
     Matrix C;
     for(long long i=;i<=n;i++)
      for(long long j=;j<=n;j++)
      {
          C.m[i][j]=;
          for(long long l=;l<=n;l++)
              C.m[i][j]=(C.m[i][j]+X.m[i][l]*Y.m[l][j])%MOD;
      }
      return C;
 }

快速幂：

把k转化为二进制，

　　如k=10(10)=1010(2);

　　a^10=a^(2^3) * a^(2^1)=(a^8)*(a^2)

代码：

Matrix qsort(Matrix X,long long p)
{
    Matrix S=E;
    while(p)
    {
        if(p&) S=cheng(S,X);
        X=cheng(X,X);
        p>>=;
    }
    return S;
}

其中E是一个矩阵，相当于数字1，任何一个矩阵A*E=A。

当n=10时，E等于

1,0,0,0,0,0,0,0,0,0

0,1,0,0,0,0,0,0,0,0

0,0,1,0,0,0,0,0,0,0

0,0,0,1,0,0,0,0,0,0

0,0,0,0,1,0,0,0,0,0

0,0,0,0,0,1,0,0,0,0

0,0,0,0,0,0,1,0,0,0

0,0,0,0,0,0,0,1,0,0

0,0,0,0,0,0,0,0,1,0

0,0,0,0,0,0,0,0,0,1

生成矩阵E：

for(long long i=;i<=n;i++)
     E.m[i][i]=;

#include<iostream>
#include<cstdio>
#include<cstring>
#include<algorithm>
using namespace std;
long long n,k;
const long long MOD=;
struct Matrix{
    long long m[][];
};
Matrix A,E,ANS;
Matrix cheng(Matrix X,Matrix Y)
{
    Matrix C;
    for(long long i=;i<=n;i++)
     for(long long j=;j<=n;j++)
     {
         C.m[i][j]=;
         for(long long l=;l<=n;l++)
             C.m[i][j]=(C.m[i][j]+(X.m[i][l]*Y.m[l][j]))%MOD;
     }
     return C;
}
Matrix qsort(Matrix X,long long p)
{
    Matrix S=E;
    while(p)
    {
        if(p&) S=cheng(S,X);
        X=cheng(X,X);
        p>>=;
    }
    return S;
}
int main()
{
    scanf("%lld%lld",&n,&k);
    for(long long i=;i<=n;i++)
     E.m[i][i]=;
    for(long long i=;i<=n;i++)
     for(long long j=;j<=n;j++)
      scanf("%lld",&A.m[i][j]);
    ANS=qsort(A,k);
    for(long long i=;i<=n;i++)
    {
        for(long long j=;j<=n;j++)
         printf("%lld ",ANS.m[i][j]);
        puts("");
    }
    return ;
}