算法笔记--极大极小搜索及alpha-beta剪枝

参考1：https://www.zhihu.com/question/27221568

参考2：https://blog.csdn.net/hzk_cpp/article/details/79275772

参考3：https://blog.csdn.net/BIT1120172185/article/details/80963609

极小极大搜索算法即minimax搜索算法

主要应用于零和博弈（非胜即负，如围棋，象棋，井子棋等），完全信息（玩家知道之前所有的步骤。象棋就是完全信息，因为玩家是交替着落子，且之前的步骤都能在棋盘上体现）

这个算法采用搜索算法递归实现，一层为先手，记为a, 一层为后手，记为b, 交替出现

对于最终局面，有一个分数（比如：先手胜分数为1， 平局分数为0，先手输分数为-1）

先手a想要让这个分数越大越好，后手b想要让这个分数越小越好，于是搜索到先手这一层，取最大返回，搜索到后手这一层，取最小返回

如下图：

于是伪代码为：

int MaxMin(position,int d)
{
    int bestvalue,value;
    if(game over)   //检查游戏是否结束
        return evaluation(p);// 游戏结束，返回估值
    if(depth<=)    //检查是否是叶子节点
        return evaluation(p);//叶子节点，返回估值
    if(max)         //极大值点
        bestvalue=-INFINTY;
    else            //极小值点
        bestvalue=INFINTY;
    for(each possibly move m)
    {
        MakeMove(m);    //走棋
        value=MaxMin(p,d-);
        UnMakeMove(m);  //恢复当前局面
        if(max)
            bestvalue=max(value,bestvalue);//取最大值
        else
            bestvalue=min(value,bestvalue);//取最小值
    }
    return bestvalue;
}

关于alpha-beta剪枝：

如果当前层为取最小，如果取最小后比上一层当前最大值还小，则不需要往下搜索，因为上一层根本不会选择当前节点往下搜，还有更好的选择

同理，如果当前层为取最大，如果取最大后比上一层当前最小值还大，则不需要往下搜索，因为上一层根本不会选择当前节点往下搜

具体推荐看最上面的知乎链接点赞最多的回答。

alpha-beta剪枝后的伪代码：

int AlphaBeta(nPlay,nAlpha,nBeta)
{
    if(game over)
        return Eveluation;   //胜负已分，返回估值
    if(nPly==)
        return  Eveluation;  //叶子节点返回估值
    if(Is Min Node)          //判断 节点类型
    {        // 极小值节点
        for(each possible move m)
        {
            make move m;      //生成新节点
            score=AlphaBeta(nPly-,nAlpha,nBeta)//递归搜索子节点
            unmake move m;//撤销搜索过的节点
            if(score<nBeta)
            {
                nBeta=score;//取极小值
                if(nAlpha>=nBeta)
                    return nAlpha;//alpha剪枝，抛弃后继节点
             }
        }
        return nBeta;//返回最小值
    }
    else
    {//取极大值的节点
        for(each possible move m)
        {
            make move m;      //生成新节点
            score=AlphaBeta(nPly-,nAlpha,nBeta)//递归搜索子节点
            unmake move m;//撤销搜索过的节点
            if(score>nAlpha)
            {
                nAlpha=score;//取极小值
                if(nAlpha>=nBeta)
                    return nBeta;//nBeta剪枝，抛弃后继节点
             }
        }
        return nAlpha;//返回最小值
    }
} 

例题1：POJ - 1568

思路：井子棋下的步数小于等于4必平局

代码：

#pragma GCC optimize(2)
#pragma GCC optimize(3)
#pragma GCC optimize(4)
#include<cstdio>
#include<iostream>
using namespace std;
#define fi first
#define se second
#define pi acos(-(long double)1.0)
#define LL long long
//#define mp make_pair
#define pb push_back
#define ls rt<<1, l, m
#define rs rt<<1|1, m+1, r
#define ULL unsigned LL
#define pll pair<LL, LL>
#define pli pair<LL, int>
#define pii pair<int, int>
#define piii pair<pii, int>
#define pdd pair<long double, long double>
#define mem(a, b) memset(a, b, sizeof(a))
#define fio ios::sync_with_stdio(false);cin.tie(0);cout.tie(0);
#define fopen freopen("in.txt", "r", stdin);freopen("out.txt", "w", stout);
//head

char s[], mp[][];
int cnt = , ansx, ansy;
int check(char c) {
    for (int i = ; i < ; i++) {
        int a = ;
        for (int j = ; j < ; j++) {
            if(mp[i][j] == c) a++;
        }
        if(a == ) return ;
        a = ;
        for (int j = ; j < ; j++) {
            if(mp[j][i] == c) a++;
        }
        if(a == ) return ;
    }
    int a = ;
    for (int i = ; i < ; i++) if(mp[i][i] == c) a++;
    if(a == ) return ;
    a = ;
    for (int i = ; i < ; i++) if(mp[i][-i] == c) a++;
    if(a == ) return ;
    return ;
}
int dfs(int step, int a, int b) {
    if((cnt-step)% == ) {
        int tmp = check('o');
        if(tmp || step == ) return -tmp;
    }
    else {
        int tmp = check('x');
        if(tmp || step == ) return tmp;
    }
    if((cnt-step)% == ) {
        for (int i = ; i < ; i++) {
            for (int j = ; j < ; j++) {
                if(mp[i][j] == '.') {
                    mp[i][j] = 'x';
                    int tmp = dfs(step-, a, b);
                    mp[i][j] = '.';
                    if(tmp >= a) {
                        if(step == cnt) {
                            ansx = i;
                            ansy = j;
                        }
                        a = tmp;
                        if(b <= a) return b;
                    }
                }
            }
        }
        return a;
    }
    else {
        for (int i = ; i < ; i++) {
            for (int j = ; j < ; j++) {
                if(mp[i][j] == '.') {
                    mp[i][j] = 'o';
                    int tmp = dfs(step-, a, b);
                    mp[i][j] = '.';
                    if(tmp <= b) {
                        b = tmp;
                        if(a >= b) return a;
                    }
                }
            }
        }
        return b;
    }
}
int main() {
    while(~scanf("%s", s)) {
        if(s[] == '$') return ;
        for (int i = ; i < ; i++) scanf("%s", mp[i]);
        cnt = ;
        for (int i = ; i < ; i++) for (int j = ; j < ; j++) if(mp[i][j] == '.') cnt++;
        if(cnt >= ) {
            printf("#####\n");
            continue;
        }
        int ans = dfs(cnt, -, );
        if(ans == ) printf("(%d,%d)\n", ansx, ansy);
        else printf("#####\n");
    }
    return ;
}

例题2：POJ - 1085

思路1：记忆化搜索dp, dp[s]表示从状态s出发先手所能获得的最大利益。

特点：空间大，时间短

代码：

#pragma GCC optimize(2)
#pragma GCC optimize(3)
#pragma GCC optimize(4)
#include<cstdio>
#include<iostream>
#include<cstring>
#include<algorithm>
#include<cmath>
#include<climits>
using namespace std;
#define fi first
#define se second
#define pi acos(-1.0)
#define LL long long
//#define mp make_pair
#define pb push_back
#define ls rt<<1, l, m
#define rs rt<<1|1, m+1, r
#define ULL unsigned LL
#define pll pair<LL, LL>
#define pli pair<LL, int>
#define pii pair<int, int>
#define piii pair<pii, int>
#define pdd pair<long double, long double>
#define mem(a, b) memset(a, b, sizeof(a))
#define fio ios::sync_with_stdio(false);cin.tie(0);cout.tie(0);
#define fopen freopen("in.txt", "r", stdin);freopen("out.txt", "w", stout);
//head

int line[][] = {{, }, {, }, {, }, {, }, {, }, {, }, {, }, {, }, {, }, {, }, {, }, {, }, {, }, {, }, {, }, {, }, {, }, {, }};
int tri[][] = {{, , }, {, , }, {, , }, {, , }, {, , }, {, , }, {, , }, {, , }, {, , }};
int num[][];
int dp[<<];
int count(int s) {
    int cnt = ;
    for (int i = ; i < ; i++) if((s&(<<tri[i][])) && (s&(<<tri[i][])) && (s&(<<tri[i][]))) cnt++;
    return cnt;
}
int dfs(int s) {
    if(~dp[s]) return dp[s];
    if(__builtin_popcount(s) == ) return ;
    int prenum = count(s);
    int ans = INT_MIN;
    for(int i = ; i < ; i++) {
        if(!(s&(<<i))) {
            int nownum = count(s|(<<i));
            if(nownum == prenum) {
                ans = max(ans, -dfs(s|<<i));
            }
            else {
                ans = max(ans, nownum - prenum + dfs(s|<<i));
            }
        }
    }
    return dp[s] = ans;
}
int main() {
    int T, m, u, v;
    for (int i = ; i < ; i++) num[line[i][]][line[i][]] = num[line[i][]][line[i][]] = i;
    mem(dp, -);
    scanf("%d", &T);
    for(int cs = ; cs <= T; cs++) {
        int st = ;
        scanf("%d", &m);
        int prenum = , nownum = ;
        int cnt = , step = ;
        for (int i = ; i <= m; i++) {
            scanf("%d %d", &u, &v);
            st |= <<num[u][v];
            nownum = count(st);
            if(nownum > prenum) {
                if(step% == ) cnt += nownum - prenum;
                else cnt -= nownum - prenum;
            }
            else step++;
            prenum = nownum;
        }
        if(step% == ) cnt += dfs(st);
        else cnt -= dfs(st);
        if(cnt > ) printf("Game %d: A wins.\n", cs);
        else printf("Game %d: B wins.\n", cs);
    }
    return ;
}

思路2：极大极小搜索+alpha-beta剪枝

特点：空间小，时间长

代码：

#pragma GCC optimize(2)
#pragma GCC optimize(3)
#pragma GCC optimize(4)
#include<cstdio>
#include<iostream>
#include<cstring>
#include<algorithm>
#include<cmath>
#include<climits>
using namespace std;
#define fi first
#define se second
#define pi acos(-1.0)
#define LL long long
//#define mp make_pair
#define pb push_back
#define ls rt<<1, l, m
#define rs rt<<1|1, m+1, r
#define ULL unsigned LL
#define pll pair<LL, LL>
#define pli pair<LL, int>
#define pii pair<int, int>
#define piii pair<pii, int>
#define pdd pair<long double, long double>
#define mem(a, b) memset(a, b, sizeof(a))
#define fio ios::sync_with_stdio(false);cin.tie(0);cout.tie(0);
#define fopen freopen("in.txt", "r", stdin);freopen("out.txt", "w", stout);
//head

int line[][] = {{, }, {, }, {, }, {, }, {, }, {, }, {, }, {, }, {, }, {, }, {, }, {, }, {, }, {, }, {, }, {, }, {, }, {, }};
int tri[][] = {{, , }, {, , }, {, , }, {, , }, {, , }, {, , }, {, , }, {, , }, {, , }};
int num[][];
int count(int s) {
    int cnt = ;
    for (int i = ; i < ; i++) if((s&(<<tri[i][])) && (s&(<<tri[i][])) && (s&(<<tri[i][]))) cnt++;
    return cnt;
}
int dfs(int s, int a, int b, int alpha, int beta, int step) {
    if(__builtin_popcount(s) == ) {
        if(a > b) return ;
        else return -;
    }
    int prenum = count(s);
    for(int i = ; i < ; i++) {
        if(!(s&(<<i))) {
            int nownum = count(s|(<<i));
            if(nownum == prenum) {
                if(step% == ) {
                    int tmp = dfs(s|<<i, a, b, alpha, beta, step+);
                    if(tmp >= alpha) {
                        alpha = tmp;
                        if(alpha >= beta) return alpha;
                    }
                }
                else {
                    int tmp = dfs(s|<<i, a, b, alpha, beta, step+);
                    if(tmp <= beta) {
                        beta = tmp;
                        if(alpha >= beta) return beta;
                    }
                }
            }
            else {
                if(step% == ) {
                    int tmp = dfs(s|<<i, a+nownum-prenum, b, alpha, beta, step);
                    if(tmp >= alpha) {
                        alpha = tmp;
                        if(alpha >= beta) return alpha;
                    }
                }
                else {
                    int tmp = dfs(s|<<i, a, b+nownum-prenum, alpha, beta, step);
                    if(tmp <= beta) {
                        beta = tmp;
                        if(alpha >= beta) return beta;
                    }
                }
            }
        }
    }
    if(step% == ) return alpha;
    else return beta;
}
int main() {
    int T, m, u, v;
    for (int i = ; i < ; i++) num[line[i][]][line[i][]] = num[line[i][]][line[i][]] = i;
    scanf("%d", &T);
    for(int cs = ; cs <= T; cs++) {
        int st = ;
        scanf("%d", &m);
        int prenum = , nownum = ;
        int step = , a = , b = ;
        for (int i = ; i <= m; i++) {
            scanf("%d %d", &u, &v);
            st |= <<num[u][v];
            nownum = count(st);
            if(nownum > prenum) {
                if(step% == ) a += nownum - prenum;
                else b += nownum - prenum;
            }
            else step++;
            prenum = nownum;
        }
        int cnt = dfs(st, a, b, -, , step);
        if(cnt > ) printf("Game %d: A wins.\n", cs);
        else printf("Game %d: B wins.\n", cs);
    }
    return ;
}