[BZOJ1009]GT考试[矩阵快速幂+kmp]

\(f_{i,j}\) 表示X串前 \(i\) 位 匹配到A串第 \(j\) 位的方案数，因为不能匹配到A串第 \(m\) 位，所以答案是 \(\sum_{i=0}^{m-1}f_{n,i}\)

\(f[i][j] = \sum_{k=0}^{m-1} f[i-1][k] * g[k][j]\)

g[i][j] 是 从第 i 位匹配到第 j 位的方案数

\(f_{i-1} \times G=f_i\)

其实初始的 \(f_0\) 应该是[1,0,0,0,0,0....] 即 \(f_{0,0}=1\)
\(f_0\times G^n = G^n\) 的第一行 所以就没有再写一遍乘法了

namespace QvvQ {
int n = in, m = in, mod = in, nxt[25];
char s[24];
struct Mod {
  int a;
  Mod(int _a) : a(_a) {}
  Mod(ll _a) : a(_a % mod) {}
  Mod() {}
  int inv() {return Pow(a, mod - 2, mod);}
  inline Mod &operator += (const int b) {a += b; if (a < 0) a += mod; else if (a >= mod) a -= mod; return *this;}
  inline Mod &operator -= (const int b) {a -= b; if (a < 0) a += mod; else if (a >= mod) a -= mod; return *this;}
  inline Mod &operator *= (const int b) {a = a * 1ll * b % mod; if (a < 0) a += mod; return *this;}
  inline Mod &operator /= (const int b) {a = a * 1ll * Pow(b, mod - 2, mod) % mod; if (a < 0) a += mod; return *this;}
  inline Mod &operator += (const Mod rhs) {return *this += rhs.a;}
  inline Mod &operator -= (const Mod rhs) {return *this -= rhs.a;}
  inline Mod &operator *= (const Mod rhs) {return *this *= rhs.a;}
  inline Mod &operator /= (const Mod rhs) {return *this /= rhs.a;}
  friend Mod inline operator + (Mod c, const Mod rhs) {return c += rhs.a;}
  friend Mod inline operator - (Mod c, const Mod rhs) {return c -= rhs.a;}
  friend Mod inline operator * (Mod c, const Mod rhs) {return c *= rhs.a;}
  friend Mod inline operator / (Mod c, const Mod rhs) {return c /= rhs.a;}
  inline Mod &operator = (const int x) {a = x; return *this;}
  inline Mod &operator = (const ll x) {a = x % mod; return *this;}
  inline Mod &operator = (const Mod rhs) {a = rhs.a; return *this;}
};
// f[i][j] = \sum_{k=0}^m-1 f[i-1][k] * g[k][j]
struct Matrix {
  int n;
  Mod a[23][21];
  Matrix(int _n = 0) {n = _n; lo0(i, n) lo0(j, n) a[i][j] = 0;}
  Matrix operator * (const Matrix &rhs) const {
    Matrix res(n);
    lo0(k, n)
    lo0(i, n) {
      if (!a[i][k].a) continue;
      lo(j, 0, rhs.n) res.a[i][j] += a[i][k] * 1ll * rhs.a[k][j];
    }
    return res;
  }
  Matrix operator ^ (int x) const {
    Matrix res(n), A; A.n = n;
    lo0(i, n) res.a[i][i] = 1;
    lo0(i, n) lo0(j, n) A.a[i][j] = a[i][j];
    while (x) {
      if (x & 1) res = res * A;
      A = A * A;
      x >>= 1;
    }
    return res;
  }
} G(m);
void init() {
  in, s + 1;
  nxt[1] = 0;
  for (int i = 2, j = 0; i <= m; ++i) {
    while (j && s[j + 1] != s[i]) j = nxt[j];
    j += s[j + 1] == s[i], nxt[i] = j;
  }
  lo0(i, m)
  lo(cc, '0', '9') {
    int j = i;
    while (j && s[j + 1] != cc) j = nxt[j];
    j += s[j + 1] == cc, G.a[i][j] += 1;
  }
}

void solve() {
  //其实初始的f_0应该是[1,0,0,0,0,0....] 即f_{0,0}=1
  //f_n*G_n = G_n的第一行 所以就没有再写乘法了
  G = G ^ n;
  Mod ans = 0;
  
  lo0(i, m) ans += G.a[0][i];
  out, ans.a;
}

}