CCF 201312-4 有趣的数 (数位DP, 状压DP, 组合数学+暴力枚举, 推公式, 矩阵快速幂)

时间：2024-07-02 19:06:02
问题描述
　　我们把一个数称为有趣的，当且仅当：
　　1. 它的数字只包含0, 1, 2, 3，且这四个数字都出现过至少一次。
　　2. 所有的0都出现在所有的1之前，而所有的2都出现在所有的3之前。
　　3. 最高位数字不为0。
　　因此，符合我们定义的最小的有趣的数是2013。除此以外，4位的有趣的数还有两个：2031和2301。
　　请计算恰好有n位的有趣的数的个数。由于答案可能非常大，只需要输出答案除以1000000007的余数。
输入格式
　　输入只有一行，包括恰好一个正整数n (4 ≤ n ≤ 1000)。
输出格式
　　输出只有一行，包括恰好n 位的整数中有趣的数的个数除以1000000007的余数。
样例输入
4
样例输出
3
析：先说第一种方法，数位DP：
首先只有0123，并且有约束条件，那么就只有6种状态，{2},{20},{201},{23},{203},{2031}，然后依次对每一位进行计算即可。
第二种，状压DP：
因为只有4种数字，所以可能用状压DP，1表示0， 2表示1， 4表示2，8表示3.剩下的就简单了。
第三种，组合数学+暴力枚举：
先枚举01的数量，那么最少2个，最多n-2个，由于0是不能放在首位，所以就有c(n-1, i)，i 表示有多少个01，然后再考虑有多种方法，
应该有 n-i 种方法，因为0的数量是从1个到 i-1 个，同理，23就有 n-i-1 种，然后乘起来就好。
第四种，推公式：
这个方法和第三种一样的，只不过是把第三种简化了，然后成一个递推公式，an=（n*n-5n+4）*2^（n-3）+n-1。
第五种，矩阵快速幂：
对每一个组成数字进行分析，字符串必定以2开头，把字符串中字符放入一个集合，我们可以统计长度为N，字符集合为S的合法字符串个数，
然后把它们的关系表示成一个矩阵，就OK了。
代码如下：
数位DP：
#pragma comment(linker, "/STACK:1024000000,1024000000")

#include <cstdio>

#include <string>

#include <cstdlib>

#include <cmath>

#include <iostream>

#include <cstring>

#include <set>

#include <queue>

#include <algorithm>

#include <vector>

#include <map>

#include <cctype>

#include <stack>

#define freopenr freopen("in.txt", "r", stdin)

#define freopenw freopen("out.txt", "w", stdout)

using namespace std;

typedef long long LL;

typedef pair<int, int> P;

const int INF = 0x3f3f3f3f;

const double inf = 0x3f3f3f3f3f3f;

const LL LNF = 100000000000000000;

const double PI = acos(-1.0);

const double eps = 1e-8;

const int maxn = 1e3 + 5;

const int mod = 1e9 + 7;

const char *mark = "+-*";

const int dr[] = {-1, 0, 1, 0};

const int dc[] = {0, 1, 0, -1};

int n, m;

inline bool is_in(int r, int c){

    return r >= 0 && r < n && c >= 0 && c < m;

}

inline LL Max(LL a, LL b){  return a < b ? b : a; }

inline LL Min(LL a, LL b){  return a > b ? b : a; }

inline int Max(int a, int b){  return a < b ? b : a; }

inline int Min(int a, int b){  return a > b ? b : a; }

LL dp[maxn][5];

int main(){

    while(cin >> n){

        dp[4][0] = 7;  dp[4][1] = 5; dp[4][2] = 3;  dp[4][3] = 9;  dp[4][4] = 3;

        for(int i = 5; i <= n; ++i){

            dp[i][0] = (1 + 2 * dp[i-1][0]) % mod;

            dp[i][1] = (dp[i-1][0] + 2 * dp[i-1][1]) % mod;

            dp[i][2] = (1 + dp[i-1][2]) % mod;

            dp[i][3] = (dp[i-1][0] + dp[i-1][2] + 2 * dp[i-1][3]) % mod;

            dp[i][4] = (dp[i-1][1] + dp[i-1][3] + 2 * dp[i-1][4]) % mod;

        }

        cout << dp[n][4] << endl;

    }

}

状压DP：

#pragma comment(linker, "/STACK:1024000000,1024000000")

#include <cstdio>

#include <string>

#include <cstdlib>

#include <cmath>

#include <iostream>

#include <cstring>

#include <set>

#include <queue>

#include <algorithm>

#include <vector>

#include <map>

#include <cctype>

#include <cmath>

#include <stack>

#define frer freopen("in.txt", "r", stdin)

#define frew freopen("out.txt", "w", stdout)

using namespace std;

typedef long long LL;

typedef pair<int, int> P;

const int INF = 0x3f3f3f3f;

const double inf = 0x3f3f3f3f3f3f;

const double PI = acos(-1.0);

const double eps = 1e-8;

const int maxn = 1e3 + 5;

const int mod = 1e9 + 7;

const int dr[] = {-1, 1, 0, 0};

const int dc[] = {0, 0, 1, -1};

const char *de[] = {"0000", "0001", "0010", "0011", "0100", "0101", "0110", "0111", "1000", "1001", "1010", "1011", "1100", "1101", "1110", "1111"};

int n, m;

const int mon[] = {0, 31, 28, 31, 30, 31, 30, 31, 31, 30, 31, 30, 31};

const int monn[] = {0, 31, 29, 31, 30, 31, 30, 31, 31, 30, 31, 30, 31};

inline int Min(int a, int b){ return a < b ? a : b; }

inline int Max(int a, int b){ return a > b ? a : b; }

inline LL Min(LL a, LL b){ return a < b ? a : b; }

inline LL Max(LL a, LL b){ return a > b ? a : b; }

inline bool is_in(int r, int c){

    return r >= 0 && r < n && c >= 0 && c < m;

}

LL dp[maxn][(1<<4)+1];

int main(){

    while(cin >> n){

        memset(dp, 0, sizeof dp);

        dp[0][0] = 1;

        for(int i = 0; i < n; ++i){

            for(int j = 0; j < 16; ++j){

                if(!dp[i][j])  continue;

                if(!(j & 2) && i)  dp[i+1][j|1] = (dp[i+1][j|1] + dp[i][j]) % mod;

                dp[i+1][j|2] = (dp[i+1][j|2] + dp[i][j]) % mod;

                if(!(j & 8))  dp[i+1][j|4] = (dp[i+1][j|4] + dp[i][j]) % mod;

                dp[i+1][j|8] = (dp[i+1][j|8] + dp[i][j]) % mod;

            }

        }

        cout << dp[n][15] << endl;

    }

    return 0;

}

组合数学+暴力枚举：

#pragma comment(linker, "/STACK:1024000000,1024000000")

#include <cstdio>

#include <string>

#include <cstdlib>

#include <cmath>

#include <iostream>

#include <cstring>

#include <set>

#include <queue>

#include <algorithm>

#include <vector>

#include <map>

#include <cctype>

#include <cmath>

#include <stack>

#define frer freopen("in.txt", "r", stdin)

#define frew freopen("out.txt", "w", stdout)

using namespace std;

typedef long long LL;

typedef pair<int, int> P;

const int INF = 0x3f3f3f3f;

const double inf = 0x3f3f3f3f3f3f;

const double PI = acos(-1.0);

const double eps = 1e-8;

const int maxn = 1e3 + 5;

const int mod = 1e9 + 7;

const int dr[] = {-1, 1, 0, 0};

const int dc[] = {0, 0, 1, -1};

const char *de[] = {"0000", "0001", "0010", "0011", "0100", "0101", "0110", "0111", "1000", "1001", "1010", "1011", "1100", "1101", "1110", "1111"};

int n, m;

const int mon[] = {0, 31, 28, 31, 30, 31, 30, 31, 31, 30, 31, 30, 31};

const int monn[] = {0, 31, 29, 31, 30, 31, 30, 31, 31, 30, 31, 30, 31};

inline int Min(int a, int b){ return a < b ? a : b; }

inline int Max(int a, int b){ return a > b ? a : b; }

inline LL Min(LL a, LL b){ return a < b ? a : b; }

inline LL Max(LL a, LL b){ return a > b ? a : b; }

inline bool is_in(int r, int c){

    return r >= 0 && r < n && c >= 0 && c < m;

}

LL c[maxn][maxn];

void init(){

    for(int i = 0; i < n; ++i) c[i][0] = c[i][i] = 1;

    for(int i = 2; i < n; ++i)

        for(int j = 1; j < i; ++j)

            c[i][j] = (c[i-1][j] + c[i-1][j-1]) % mod;

}

int main(){

    while(cin >> n){

        init();

        LL ans = 0;

        for(int i = 2; i < n-1; ++i)

            ans = (ans + (c[n-1][i] * (i-1) * (n-i-1))  % mod) % mod;

        cout << ans << endl;

    }

    return 0;

}

推公式：

#pragma comment(linker, "/STACK:1024000000,1024000000")

#include <cstdio>

#include <string>

#include <cstdlib>

#include <cmath>

#include <iostream>

#include <cstring>

#include <set>

#include <queue>

#include <algorithm>

#include <vector>

#include <map>

#include <cctype>

#include <cmath>

#include <stack>

#define frer freopen("in.txt", "r", stdin)

#define frew freopen("out.txt", "w", stdout)

using namespace std;

typedef long long LL;

typedef pair<int, int> P;

const int INF = 0x3f3f3f3f;

const double inf = 0x3f3f3f3f3f3f;

const double PI = acos(-1.0);

const double eps = 1e-8;

const int maxn = 1e3 + 5;

const int mod = 1e9 + 7;

const int dr[] = {-1, 1, 0, 0};

const int dc[] = {0, 0, 1, -1};

const char *de[] = {"0000", "0001", "0010", "0011", "0100", "0101", "0110", "0111", "1000", "1001", "1010", "1011", "1100", "1101", "1110", "1111"};

int n, m;

const int mon[] = {0, 31, 28, 31, 30, 31, 30, 31, 31, 30, 31, 30, 31};

const int monn[] = {0, 31, 29, 31, 30, 31, 30, 31, 31, 30, 31, 30, 31};

inline int Min(int a, int b){ return a < b ? a : b; }

inline int Max(int a, int b){ return a > b ? a : b; }

inline LL Min(LL a, LL b){ return a < b ? a : b; }

inline LL Max(LL a, LL b){ return a > b ? a : b; }

inline bool is_in(int r, int c){

    return r >= 0 && r < n && c >= 0 && c < m;

}

LL quick_pow(LL a, LL b){

    LL ans = 1;

    while(b){

        if(b & 1)  ans = (ans * a) % mod;

        b >>= 1;

        a = (a * a) % mod;

    }

    return ans;

}

int main(){

    while(cin >> n){

        LL ans = (n * n - 5 * n + 4) % mod;

        ans = (ans * quick_pow(2LL, n-3) + n - 1) % mod;

        cout << ans << endl;

    }

    return 0;

}

矩阵快速幂：

#pragma comment(linker, "/STACK:1024000000,1024000000")

#include <cstdio>

#include <string>

#include <cstdlib>

#include <cmath>

#include <iostream>

#include <cstring>

#include <set>

#include <queue>

#include <algorithm>

#include <vector>

#include <map>

#include <cctype>

#include <cmath>

#include <stack>

#define frer freopen("in.txt", "r", stdin)

#define frew freopen("out.txt", "w", stdout)

using namespace std;

typedef long long LL;

typedef pair<int, int> P;

const int INF = 0x3f3f3f3f;

const double inf = 0x3f3f3f3f3f3f;

const double PI = acos(-1.0);

const double eps = 1e-8;

const int maxn = 1e3 + 5;

const int mod = 1e9 + 7;

const int dr[] = {-1, 1, 0, 0};

const int dc[] = {0, 0, 1, -1};

const char *de[] = {"0000", "0001", "0010", "0011", "0100", "0101", "0110", "0111", "1000", "1001", "1010", "1011", "1100", "1101", "1110", "1111"};

int n, m;

const int mon[] = {0, 31, 28, 31, 30, 31, 30, 31, 31, 30, 31, 30, 31};

const int monn[] = {0, 31, 29, 31, 30, 31, 30, 31, 31, 30, 31, 30, 31};

inline int Min(int a, int b){ return a < b ? a : b; }

inline int Max(int a, int b){ return a > b ? a : b; }

inline LL Min(LL a, LL b){ return a < b ? a : b; }

inline LL Max(LL a, LL b){ return a > b ? a : b; }

inline bool is_in(int r, int c){

    return r >= 0 && r < n && c >= 0 && c < m;

}

struct Matrix{

    LL a[6][6];

};

Matrix mul(Matrix x, Matrix y){

    Matrix ans;

    memset(ans.a, 0, sizeof ans.a);

    for(int i = 0; i < 6; ++i){

        for(int j = 0; j < 6; ++j){

            for(int k = 0; k < 6; ++k){

                ans.a[i][j] += (x.a[i][k] * y.a[k][j]) % mod;

            }

            ans.a[i][j] %= mod;

        }

    }

    return ans;

}

Matrix quick_pow(Matrix x, int b){

    Matrix ans;

    memset(ans.a, 0, sizeof ans.a);

    for(int i = 0; i < 6; ++i)

            ans.a[i][i] = 1;

    while(b){

        if(b & 1)  ans = mul(ans, x);

        b >>= 1;

        x = mul(x, x);

    }

    return ans;

}

int main(){

    Matrix x, y;

    memset(x.a, 0, sizeof x.a);

    memset(y.a, 0, sizeof y.a);

    x.a[3][3] = x.a[0][0] = x.a[1][0] = x.a[2][1] = x.a[3][0] = x.a[4][1] = x.a[4][3] = x.a[5][2] = x.a[5][4] = x.a[5][5] = 1;

    x.a[1][1] = x.a[4][4] = x.a[5][5] = x.a[2][2] = 2;

    y.a[0][0] = 1;

    while(cin >> n){

        Matrix ans = mul(quick_pow(x, n-1), y);

        cout << ans.a[5][0] << endl;

    }

    return 0;

}

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