poj 3735 稀疏矩阵矩阵快速幂

Posted 2021-01-07 shandongs1

设人数为 $n$，构造 $(n + 1) imes (n + 1)$ 的矩阵

得花生：
将改行的最后一列元素 $+ 1$

egin{gather}
egin{bmatrix}
1 & 0 & 0 & 1 \
0 & 1 & 0 & 0 \
0 & 0 & 1 & 0 \
0 & 0 & 0 & 1
end{bmatrix}
imes
egin{bmatrix}
x \
y \
z \
1 \
end{bmatrix}
=
egin{bmatrix}
x + 1 \
y \
z \
1\
end{bmatrix}
end{gather}

吃花生：
将一行的元素全部清零

egin{gather}
egin{bmatrix}
1 & 0 & 0 & 0\
0 & 0 & 0 & 0 \
0 & 0 & 1 & 0 \
0 & 0 & 0 & 1
end{bmatrix}
imes
egin{bmatrix}
x \
y \
z \
1 \
end{bmatrix}
=
egin{bmatrix}
x \
0 \
z \
1\
end{bmatrix}
end{gather}

交换两行

egin{gather}
egin{bmatrix}
0 & 1 & 0 & 0 \
1 & 0 & 0 & 0 \
0 & 0 & 1 & 0 \
0 & 0 & 0 & 1
end{bmatrix}
imes
egin{bmatrix}
x \
y \
z \
1 \
end{bmatrix}
=
egin{bmatrix}
y \
x \
z \
1\
end{bmatrix}
end{gather}

普通矩阵快速幂
TLE
稀疏矩阵矩阵快速幂

 

#include <iostream>
#include <cstdio>
#include <algorithm> 
#include <cstring>

using namespace std;
const int N = 110;

#define LL long long

LL n, m, k;
struct Matrix {LL M[N][N];} A;

Matrix operator * (Matrix &a, Matrix &b) {
    Matrix ret;
    memset(ret.M, 0, sizeof ret.M);
    for(int i = 1; i <= n; i ++)
        for(int j = 1; j <= n; j ++)
            if(a.M[i][j]) 
                for(int K = 1; K <= n; K ++)
                    ret.M[i][K] = (ret.M[i][K] + a.M[i][j] * b.M[j][K]);
    return ret;
}



Matrix Ksm(int p) {
    Matrix Ans;
    memset(Ans.M, 0, sizeof Ans.M);
    for(int i = 1; i <= n; i ++) Ans.M[i][i] = 1;
    while(p) {
        if(p & 1) Ans = Ans * A;
        A = A * A;
        p >>= 1;
    }
    return Ans;
}

int main() {
    while(1) {
        cin >> n >> k >> m;
        if(n == 0 && m == 0 && k == 0) return 0;
        n ++;
        char s[10];
        memset(A.M, 0, sizeof A.M);
        for(int i = 1; i <= n; i ++) A.M[i][i] = 1;
        for(int i = 1; i <= m; i ++) {
            scanf("%s", s);
            if(s[0] == ‘g‘) {
                int x; std:: cin >> x;
                A.M[x][n] ++;
            } else if(s[0] == ‘e‘) {
                int x; std:: cin >> x;
                for(int j = 1; j <= n; j ++) A.M[x][j] = 0;
            } else {
                int x, y; std:: cin >> x >> y;
                swap(A.M[x], A.M[y]);
            }
        }
        Matrix Answer = Ksm(k);
        for(int i = 1; i < n; i ++) cout << Answer.M[i][n] << " ";
        printf("
");
    }
    return 0;
}