luoguP1357 花园

Posted 2021-03-25 lylyl

首先对于(n<10^5)的范围,我们可以直接状压(dp).

设(dp[i][s])表示(dp)到了第(i)位,最后(m)个状态的二进制压缩为(s)的方案数.

( herefore dp[i][s]=dp[i-1][frac{s}{2}]+dp[i-1][(frac{s}{2}+2^{m-1}]).

但是这个题显然线性做法会超时.

于是使用套路:矩阵快速幂!

认真观察这个(dp)方程,我们可以发现对于任意一个合法状态的转移总是单一的,并且总是那个合法的(bool)矩阵.

于是我们可以对于中间状态(T)自乘(n)次,在对于每一个合法的初始状态,算出它的贡献,求和即可.

#pragma GCC optimize(3)
#include<bits/stdc++.h>
#define il inline
#define rg register
using namespace std;
typedef long long ll;
const int mod = 1e9 + 7, O = 55;
ll n;
int m, k, res, ans, sta[O];
struct Matrix {
    int a[O][O];
    Matrix() { memset(a, 0, sizeof a); }
    il int * operator [] (int x) { return a[x]; }
    il Matrix operator * (Matrix rhs) const {
        Matrix c;
        for (int k = 0; k <= res; ++k)
            for (int i = 0; i <= res; ++i)
                for (int j = 0; j <= res; ++j)
                    (c[i][j] += 1ll * a[i][k] * rhs[k][j] % mod) %= mod;
        return c;
    }
}S, T, e;
il Matrix qpow(Matrix b, ll x) {
    Matrix Res = e;
    while (x) {
        if (x & 1) Res = Res * b;
        b = b * b; x >>= 1;
    }
    return Res;
}
int main() {
    scanf("%lld%d%d", &n, &m, &k);
    for (int i = 0; i < O; ++i) e[i][i] = 1;
    res = (1 << m) - 1;
    for (int s = 0; s <= res; ++s) {
        int tot = 0, num = s;
        while (num) tot += num & 1, num >>= 1;
        if (tot > k) continue;
        sta[s] = 1;
        T[s >> 1][s] = 1;
        if (tot != k || s & 1) T[(s >> 1) + (1 << m - 1)][s] = 1;
    }
    T = qpow(T, n);
    for (int s = 0; s <= res; ++s)
        if (sta[s]) {
            S[0][s] = 1;
            S = S * T;
            (ans += S[0][s]) %= mod;
            memset(S.a, 0, sizeof S.a);
        }
    printf("%d
", ans);
    return 0;
}