BZOJ3329: Xorequ(二进制数位dp 矩阵快速幂)

题意

题目链接

sol

挺套路的一道题

首先把式子移一下项

\(x \oplus 2x = 3x\)

有一件显然的事情：\(a \oplus b \leqslant c\)

又因为\(a \oplus b + 2(a \& b) = c\)

那么\(x \& 2x = 0\)

也就是说，\(x\)的二进制表示下不能有相邻位

第一问直接数位dp即可

第二问比较interesting，设\(f[i]\)表示二进制为\(i\)的方案数，转移时考虑上一位选不选

如果能选，方案数为\(f[i - 2]\)

不选的方案数为\(f[i - 1]\)

#include<bits/stdc++.h>
#define ll long long 
//#define int long long 
#define file {freopen("a.in", "r", stdin); freopen("a.out", "w", stdout);}
using namespace std;
const int maxn = 233, mod = 1e9 + 7;
inline ll read() {
    char c = getchar(); ll x = 0, f = 1;
    while(c < '0' || c > '9') {if(c == '-') f = -1; c = getchar();}
    while(c >= '0' && c <= '9') x = x * 10 + c - '0', c = getchar();
    return x * f;
}
ll n;
struct matrix {
    int m[3][3];
    matrix() {
        memset(m, 0, sizeof(m));
    }
    matrix operator * (const matrix &rhs) const {
        matrix ans;
        for(int k = 1; k <= 2; k++)
            for(int i = 1; i <= 2; i++)
                for(int j = 1; j <= 2; j++)
                    (ans.m[i][j] += 1ll * m[i][k] * rhs.m[k][j] % mod) %= mod;
        return ans;
    }
};
matrix matrixpow(matrix a, ll p) {
    matrix base;
    for(int i = 1; i <= 2; i++) base.m[i][i] = 1;
    while(p) {
        if(p & 1) base = base * a;
        a = a * a; p >>= 1;
    }
    return base;
}
ll num[maxn], tot; ll f[maxn][2];
ll dfs(int x, bool lim, bool pre) {
    if(!lim && (~f[x][pre])) return f[x][pre];
    if(x == 0) return 1;
    ll ans = 0;
    if(!pre && (num[x] == 1 || (!lim))) ans += dfs(x - 1, lim, 1);
    ans += dfs(x - 1, lim && num[x] == 0, 0);

    if(!lim) f[x][pre] = ans;
    return ans;
}
ll dp(ll x) {
    tot = 0;
    while(x) num[++tot] = x & 1, x >>= 1;
    return dfs(tot, 1, 0);
}   
main() {
//  file;
    memset(f, -1, sizeof(f));
    int t = read();
    while(t--) {
        n = read();
        printf("%lld\n", dp(n) - 1);
        matrix a;
        a.m[1][1] = 1; a.m[1][2] = 1;
        a.m[2][1] = 1; a.m[2][2] = 0;
        a = matrixpow(a, n);
        printf("%d\n", (a.m[1][1] + a.m[1][2]) % mod);      
    }

    return 0;
}
/*
1
5
*/