20240306 lengli's submission for CF621E (#124)

20240306 lengli's submission for CF621E

daily_problems/2024/03/0306/personal_submission/cf621e_lengli_star.cpp

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+//首先不难想到dp,但是数据范围很令人头疼,但是如果可以矩阵快速幂那就很好！
+//定义dp(i,j)表示选了i个数字,模x余j的方案数
+//那么枚举当前选择的数字k,则有转移dp(i+1,(j+k)%x)+=dp(i,j)
+//既然使用矩阵快速幂,那么我们只需要考虑dp(j)即可
+//tip:  '#'为手动转置QAQ
+//那么我们要构造一个转移矩阵A，使其乘上[dp(0),dp(1)...dp(x-1)]#
+//那么新的dp要根据不同数字贡献组成的行向量进行线性变换得到，那么利用递推式构造A即可
+//初始化仅有一个列向量
+//左乘A^b后，f[k][0]即为答案。
+
+
+/*
+lengli_QAQ
+Hope there are no bugs!!!
+*/
+#include <bits/stdc++.h>
+#define fastio ios::sync_with_stdio(0); cin.tie(0); cout.tie(0)
+#define endl '\n'
+#define all(x) x.begin(),x.end()
+#define pb push_back
+
+using namespace std;
+
+#ifdef LOCAL
+#include "algo/debug.h"
+#else
+#define debug(...) "lengli"
+#endif
+
+const int N=100010;
+
+template <unsigned M_> struct ModInt {
+  static constexpr unsigned M = M_;
+  unsigned x;
+  constexpr ModInt() : x(0U) {}
+  constexpr ModInt(unsigned x_) : x(x_ % M) {}
+  constexpr ModInt(unsigned long long x_) : x(x_ % M) {}
+  constexpr ModInt(int x_) : x(((x_ %= static_cast<int>(M)) < 0) ? (x_ + static_cast<int>(M)) : x_) {}
+  constexpr ModInt(long long x_) : x(((x_ %= static_cast<long long>(M)) < 0) ? (x_ + static_cast<long long>(M)) : x_) {}
+  ModInt &operator+=(const ModInt &a) { x = ((x += a.x) >= M) ? (x - M) : x; return *this; }
+  ModInt &operator-=(const ModInt &a) { x = ((x -= a.x) >= M) ? (x + M) : x; return *this; }
+  ModInt &operator*=(const ModInt &a) { x = (static_cast<unsigned long long>(x) * a.x) % M; return *this; }
+  ModInt &operator/=(const ModInt &a) { return (*this *= a.inv()); }
+  ModInt pow(long long e) const {
+    if (e < 0) return inv().pow(-e);
+    ModInt a = *this, b = 1U; for (; e; e >>= 1) { if (e & 1) b *= a; a *= a; } return b;
+  }
+  ModInt inv() const {
+    unsigned a = M, b = x; int y = 0, z = 1;
+    for (; b; ) { const unsigned q = a / b; const unsigned c = a - q * b; a = b; b = c; const int w = y - static_cast<int>(q) * z; y = z; z = w; }
+    assert(a == 1U); return ModInt(y);
+  }
+  ModInt operator+() const { return *this; }
+  ModInt operator-() const { ModInt a; a.x = x ? (M - x) : 0U; return a; }
+  ModInt operator+(const ModInt &a) const { return (ModInt(*this) += a); }
+  ModInt operator-(const ModInt &a) const { return (ModInt(*this) -= a); }
+  ModInt operator*(const ModInt &a) const { return (ModInt(*this) *= a); }
+  ModInt operator/(const ModInt &a) const { return (ModInt(*this) /= a); }
+  template <class T> friend ModInt operator+(T a, const ModInt &b) { return (ModInt(a) += b); }
+  template <class T> friend ModInt operator-(T a, const ModInt &b) { return (ModInt(a) -= b); }
+  template <class T> friend ModInt operator*(T a, const ModInt &b) { return (ModInt(a) *= b); }
+  template <class T> friend ModInt operator/(T a, const ModInt &b) { return (ModInt(a) /= b); }
+  explicit operator bool() const { return x; }
+  bool operator==(const ModInt &a) const { return (x == a.x); }
+  bool operator!=(const ModInt &a) const { return (x != a.x); }
+  bool operator<(const ModInt &a) const { return (x < a.x); }
+  bool operator>(const ModInt &a) const { return (x > a.x); }
+  bool operator<=(const ModInt &a) const { return (x <= a.x); }
+  bool operator>=(const ModInt &a) const { return (x >= a.x); }
+  friend std::ostream &operator<<(std::ostream &os, const ModInt &a) { return os << a.x; }
+  friend std::istream &operator>>(std::istream &is, ModInt &a) {int v;is >> v;a = ModInt(v);return is;}
+};
+constexpr unsigned MO = 1000000007;
+using Mint = ModInt<MO>;
+
+template<typename T> struct matrix {
+	int n, m;
+	std::vector<T> data;
+	matrix () : matrix(0, 0) {}
+	matrix (int n) : matrix(n, n) {}
+	matrix (int n, int m) : n(n), m(m), data(n * m) {}
+	matrix (const std::vector<std::vector<T> > &a) : n(a.size()), m(a[0].size()) {
+		data.resize(n*m);
+		for (int i = 0; i < n; i++) {
+			copy(a[i].begin(), a[i].end(), data.begin()+i*m);
+		}
+	}
+	auto operator [] (int i) { return data.begin() + i * m; }
+	auto operator [] (int i) const { return data.cbegin() + i * m; }
+	static matrix mul_ident(int n) {
+		matrix res(n, n);
+		for (int i = 0; i < n; i++) res[i][i] = 1;
+		return res;
+	}
+	matrix operator + (const matrix &rhs) {
+		assert(n == rhs.n && m == rhs.m);
+		matrix res(n,m);
+		for (int i = 0; i < n ; i++) {
+			for (int j = 0; j < m; j++)
+				res[i][j] = (*this)[i][j] + rhs[i][j];
+		}
+		return res;
+	}
+	matrix operator - (const matrix &rhs) {
+		assert(n == rhs.n && m == rhs.m);
+		matrix res(n,m);
+		for (int i = 0; i < n ; i++) {
+			for (int j = 0; j < m; j++)
+				res[i][j] = (*this)[i][j] - rhs[i][j];
+		}
+		return res;
+	}
+	matrix operator * (const matrix &rhs) {
+		assert(m == rhs.n);
+		matrix res(n, rhs.m);
+		for (int i = 0; i < n; i++) for (int j = 0; j < m; j++) for (int k = 0; k < rhs.m; k++)
+			res[i][k] += (*this)[i][j] * rhs[j][k];
+		return res;
+	}
+	template<typename int_t> matrix operator ^ (int_t x) {
+		assert(n == m);
+		matrix res = mul_ident(n);
+		while (x) {
+			if (x & 1) res = matrix(*this)*res;
+			*this = matrix(*this)*matrix(*this);
+			x>>=1;
+		}
+		return res;
+	}
+	matrix &operator += (const matrix &rhs) { return *this = *this + rhs;}
+	matrix &operator -= (const matrix &rhs) { return *this = *this - rhs;}
+	matrix &operator *= (const matrix &rhs) { return *this = *this * rhs;}
+	template<typename int_t> matrix &operator ^= (int_t x) { return *this = *this ^ x;}
+	bool operator == (const matrix &rhs) const { return m == rhs.m && data == rhs.data; }
+	bool operator != (const matrix &rhs) const { return m != rhs.m || data != rhs.data; }
+	std::pair<bool, matrix> inv() {
+		assert(n == m);
+		matrix a = *this;
+		matrix r = mul_ident(n);
+		for (int i = 0; i < n; i++) {
+			int id = -1;
+			for (int j = i; j < n; j++) if (a[j][i] != T(0)) { id = j; break; }
+			if (id == -1) return {false, matrix()};
+			for (int j = i; j < n; j++) std::swap(a[i][j], a[id][j]);
+			for (int j = 0; j < n; j++) std::swap(r[i][j], r[id][j]);
+			auto t = T(1) / a[i][i];
+			for (int j = i; j < n; j++) a[i][j] *= t;
+			for (int j = 0; j < n; j++) r[i][j] *= t;
+			for (int j = 0; j < n; j++) if (i != j) {
+				auto s = a[j][i];
+				for (int k = i; k < n; k++) a[j][k] -= a[i][k] * s;
+				for (int k = 0; k < n; k++) r[j][k] -= r[i][k] * s;
+			}
+		}
+		return {true, r};
+	}
+	std::pair<bool, matrix> inv2() {
+		assert(n == m);
+		matrix a = *this;
+		std::vector<std::pair<int, int> > swaps;
+		for (int i = 0; i < n; i++) {
+			int id = -1;
+			for (int j = i; j < n; j++) if (a[j][i] != T(0)) { id = j; break; }
+			if (id == -1) return {false, matrix()};
+			if (id != i) {
+				swaps.push_back({id, i});
+				for (int j = 0; j < n; j++) std::swap(a[i][j], a[id][j]);
+			}
+			a[i][i] =  T(1) / a[i][i];
+			for (int j = 0; j < n; j++) if (j != i) a[i][j] *= a[i][i];
+			for (int j = 0; j < n; j++) if (j != i) {
+				for (int k = 0; k < n; k++) if (k != i) a[j][k] -= a[j][i] * a[i][k];
+				a[j][i] *= -a[i][i];
+			}
+		}
+		for (int i = swaps.size(); i--; ) 
+			for (int j = 0; j < n; j++) std::swap(a[j][swaps[i].first], a[j][swaps[i].second]);
+		return {true, a};
+	}
+	T det() const {
+		assert(n == m);
+		matrix a = *this;
+		T res = 1;
+		for (int i = 0; i < n; i++) {
+			int id = -1;
+			for (int j = i; j < n; j++) if (a[j][i] != T(0)) { id = j; break; }
+			if (id == -1) return 0;
+			if (id != i) {
+				res = -res;
+				for (int j = i; j < n; j++) std::swap(a[id][j], a[i][j]);
+			}
+			res *= a[i][i];
+
+			T t = T(1) / a[i][i];
+			for (int j = i; j < n; j++) a[i][j] *= t;
+			for (int j = i + 1; j < n; j++) {
+				auto s = a[j][i];
+				for (int k = i; k < n; k++) a[j][k] -= a[i][k] * s;
+			}
+		}
+		return res;
+	}
+};
+
+int n,b,k,x;
+
+void solve(){
+    cin>>n>>b>>k>>x;
+    vector<int> cnt(10,0);
+    for(int i=1;i<=n;i++){
+        int x;
+        cin>>x;
+        cnt[x]++;
+    }
+    matrix<Mint> a(100,100);
+    for(int i=0;i<x;i++){
+        for(int j=0;j<max(x,10);j++){
+            for(int k=0;k<10;k++){
+                if((j*10%x+k)%x==i) {
+                    a[i][j]+=cnt[k];
+                }
+            }
+        }
+    }
+    matrix<Mint> res(100,100);
+    for(int i=0;i<=9;i++){
+        if(cnt[i]) res[i][0]=cnt[i];
+    }
+    a=a^(b-1);
+    res=a*res;
+    cout<<res[k][0]<<endl;
+}
+
+signed main(){
+    fastio;
+    int T;
+    T=1;
+    while(T--) solve();
+
+    return 0;
+}