【小结】SG生成函数(Grundy函数)

$SG生成函数(Grundy函数)小结$

• 转移到子游戏$x$&$y$，则 $$sg[now]=sg[x] \land sg[y]$$
• 模板

#include <cstdio>
#include <cstring>
#include <algorithm>
using namespace std;

const int MAX = 100007;
const int MAX_S = 128;
const int MAX_X = MAX;
int sg[MAX], stone[MAX_S];
int vis[MAX];
int X[MAX_S];

void get_sg()
{
    sg[0] = 0;
    for (int i = 1; i < MAX_X; ++i)
    {
        for (int j = 1; j <= MAX_S; ++j)
        {
            if (i >= stone[j])
            {
                vis[sg[i - stone[j]]] = i;
            }
        }

        int g = -1;
        while (vis[++g] == i);
        sg[i] = g;
    }
}

void gao()
{
    get_sg();

    int res = 0;
    for (int i = 1; i <= MAX_S; ++i)
    {
        res ^= sg[X[i]];
    }

    puts(res ? "player1 wins." : "player2 wins.");
}

int main()
{
    return 0;
}

$POJ2960$

#include <cstdio>
#include <cstring>
#include <set>
#include <algorithm>
using namespace std;

const int MAX = 100007;
const int MAX_S = 107;
int vis[MAX];
int X[MAX_S][MAX_S];
int max_xx[MAX_S];
int I[MAX_S];
int sg[MAX];
int s[MAX_S];

int main()
{
    int k, m;
    while (~scanf(" %d", &k) && k)
    {
        for (int i = 1; i <= k; ++i)
        {
            scanf(" %d", s + i);
        }
        scanf(" %d", &m);
        for (int i = 1; i <= m; ++i)
        {
            scanf(" %d", I + i);
            max_xx[i] = 0;
            for (int j = 1; j <= I[i]; ++j)
            {
                scanf(" %d", &X[i][j]);
                if (X[i][j] > max_xx[i])
                {
                    max_xx[i] = X[i][j];
                }
            }
        }

        sg[0] = 0;

        int max_x = 0;
        for (int i = 1; i <= m; ++i)
        {
            if (max_x < max_xx[i])
            {
                max_x = max_xx[i];
            }
        }

        memset(vis, 0, sizeof(vis));
        for (int i = 1; i <= max_x; ++i)
        {
            for (int j = 1; j <= k; ++j)
            {
                if (s[j] <= i)
                {
                    vis[sg[i - s[j]]] = i;
                }
            }

            int g = 0;
            while (vis[g] == i)
            {
                ++g;
            }
            sg[i] = g;
        }

        for (int i = 1; i <= m; ++i)
        {
            int res = 0;
            for (int j = 1; j <= I[i]; ++j)
            {
                res ^= sg[X[i][j]];
            }
            putchar(res ? 'W' : 'L');
        }
        putchar('\n');
    }
    return 0;
}

$poj2425$

• 在所给图中随便搞一下，任何状态都可以转移到拓扑序更小的状态，后者即为子状态
• 当前状态的$sg$值需要后更新，因此后续遍历即可

#include <cstdio>
#include <cstring>
#include <set>
#include <vector>
#include <algorithm>
using namespace std;

const int MAX = 1024;
vector<int> G[MAX];
int sg[MAX];
bool used[MAX];
bool vis[MAX][MAX];
//set<int> vis[MAX];

void dfs(int v)
{
    used[v] = true;
    for (int i = 0; i < G[v].size(); ++i)
    {
        if (!used[G[v][i]])
        {
            dfs(G[v][i]);
        }
        vis[v][sg[G[v][i]]] = true;
        //vis[v].insert(sg[G[v][i]]);
    }
    int g = -1;
    while (vis[v][++g]);
    sg[v] = g;
}

inline void read(int& x)
{
    char c;
    while ((c = getchar()) < '0' || c > '9');
    x = c - '0';
    while ((c = getchar()) >= '0' && c <= '9')
    {
        x = (x << 3) + (x << 1) + c - '0';
    }
}

int main()
{
    int n;
    while (~scanf(" %d", &n))
    {
        for (int i = 0; i < n; ++i)
        {
            G[i].clear();
        }
        memset(vis, false, sizeof(vis));
        int m, v;
        for (int i = 0; i < n; ++i)
        {
            read(m);
            //scanf(" %d", &m);
            while (m--)
            {
                read(v);
                //scanf(" %d", &v);
                G[i].push_back(v);
            }
        }
        memset(used, false, sizeof(used));
        for (int i = 0; i < n; ++i)
        {
            if (!used[i])
            {
                dfs(i);
            }
        }

        /*
            for (int i = 0; i < n; ++i)
            {
                printf("sg[%d] = %d\n", i, sg[i]);
            }
        */
        while (true)
        {
            read(m);
            if (m == 0)
            {
                break;
            }
            int res = 0;
            while (m--)
            {
                read(v);
                //scanf(" %d", &v);
                res ^= sg[v];
                //printf("res = %d\n", res);
            }
            puts(res ? "WIN" : "LOSE");
        }
    }
    return 0;
}

$hdu3980$

• 当前$\left(n, m\right)$状态可以拿掉$m$个弹珠，枚举拿的位置$x$，于是分成$x,\ n-m-x$两个状态。

#include <cstdio>
#include <cstring>
#include <algorithm>
using namespace std;

const int MAX = 1024;
int sg[MAX];

int get_sg(int n, int m)
{
    if (n < m)
    {
        return sg[n] = 0;
    }
    else if (sg[n] != -1)
    {
        return sg[n];
    }

    bool vis[MAX] = {0};
    for (int i = 0; i <= n - m; ++i)
    {
        vis[get_sg(i, m) ^ get_sg(n - m - i, m)] = true;
    }
    int g = -1;
    while (vis[++g]);
    return sg[n] = g;
}

int main()
{
    int T, n, m;
    scanf(" %d", &T);
    while (T--)
    {
        scanf(" %d %d", &n, &m);
        memset(sg, -1, sizeof(sg));

        static int __ = 1;
        printf("Case #%d: %s\n", __++, (n >= m && !get_sg(n - m, m)) ? "aekdycoin" : "abcdxyzk");
    }
    return 0;
}