首页 > 代码库 > [HDU 1757] A Simple Math Problem (矩阵快速幂)
[HDU 1757] A Simple Math Problem (矩阵快速幂)
题目大意:
If x < 10 f(x) = x.
If x >= 10 f(x) = a0 * f(x-1) + a1 * f(x-2) + a2 * f(x-3) + …… + a9 * f(x-10);
And ai(0<=i<=9) can only be 0 or 1 .
现在给你k和m,求f(k) % m。
解题思路:
f(x) = a0 * f(x-1) + a1 * f(x-2) + a2 * f(x-3) + …… + a9 * f(x-10)
其中k为10^9数量级,必然不能用递推的方式做。这类题目可以通过构造矩阵,用矩阵快速幂来做。
构造的矩阵是:
|0
|0
|................1|
|a9
代码:
/* ID: wuqi9395@126.com PROG: LANG: C++ */ #include<map> #include<set> #include<queue> #include<stack> #include<cmath> #include<cstdio> #include<vector> #include<string> #include<fstream> #include<cstring> #include<ctype.h> #include<iostream> #include<algorithm> #define INF (1<<30) #define PI acos(-1.0) #define mem(a, b) memset(a, b, sizeof(a)) #define For(i, n) for (int i = 0; i < n; i++) typedef long long ll; using namespace std; const int maxn = 20; const int maxm = 20; ll mod, k; struct Matrix { int n, m; int a[maxn][maxm]; void clear() { n = m = 0; memset(a, 0, sizeof(a)); } Matrix operator * (const Matrix &b) const { //实现矩阵乘法 Matrix tmp; tmp.clear(); tmp.n = n; tmp.m = b.m; for (int i = 0; i < n; i++) for (int j = 0; j < b.m; j++) for (int k = 0; k < m; k++) { tmp.a[i][j] += a[i][k] * b.a[k][j]; tmp.a[i][j] %= mod; } return tmp; } }; Matrix A, B; void init() { A.clear(); //矩阵A是构造的矩阵 A.n = A.m = 10; for (int i = 0; i < 9; i++) A.a[i][i + 1] = 1; B.clear(); B.n = 10; B.m = 1; //矩阵B是f(x)的前10个数 for (int i = 0; i < 10; i++) B.a[i][0] = i; } Matrix Matrix_pow(Matrix A, ll k, ll mod) { //矩阵快速幂 Matrix res; res.clear(); res.n = res.m = 10; for (int i = 0; i < 10; i++) res.a[i][i] = 1; while(k) { if (k & 1) res = A * res; k >>= 1; A = A * A; } return res; } int main () { init(); while(cin>>k>>mod) { int x; for (int i = 0; i < 10; i++) { scanf("%d", &x); A.a[9][9 - i] = x; } if (k < 10) { printf("%lld\n", k % mod); } else { Matrix res = Matrix_pow(A, k - 9, mod); res = res * B; cout<<res.a[9][0]<<endl; } } return 0; }
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