骑士共存问题【网络流24题】

题目链接：https://www.luogu.org/problem/P3355

首先，攻击关系不会出现奇环，所以可以对棋盘进行二分染色。

然后就和方格取数那个题一样了。

S连每个黑，边权为1.

每个黑连其能攻击的点，边权为inf。

每个白连T，边权为1。

#include <bits/stdc++.h>
#define rep(i, a, b) for(int i = (a); i <= (b); i++)
using namespace std;
const int N = 4e4+100;
int n,m,S,T,tot;
int head[N];
struct node{
    int v,cap,nxt;
}edge[int(3e6+100)];
int cur[N],deep[N];
void ae(int u,int v,int cap){  //前向星加边
    edge[++tot] = node{v,cap,head[u]};
    head[u] = tot;
    edge[++tot] = node{u,0,head[v]};
    head[v] = tot;
}
bool bfs(){  //构建分层图
    memset(deep,-1,sizeof(deep));  //-1表示未访问
    queue<int>q;
    q.push(S);
    deep[S] = 0;
    while(!q.empty()){
        int u = q.front();
        for(int i = head[u]; ~i;i = edge[i].nxt){
            int v = edge[i].v;
            if(deep[v]==-1&&edge[i].cap>0){
                q.push(v);
                deep[v] = deep[u]+1;
                if(v==T) return 1;   //提前退出
            }
        }
        q.pop();
    }
    return 0;
}
int dfs(int u,int f){  //寻找增广网
    int flow = 0,d;
    if(u==T||f==0) return f;
    for(int &i = cur[u]; ~i;i = edge[i].nxt){ //当前弧优化
        int v = edge[i].v;
        if(deep[v]>deep[u]&&edge[i].cap>0&&(d=dfs(v,min(f,edge[i].cap)))){
            edge[i].cap -= d;
            edge[i^1].cap += d;
            f -= d;
            flow += d;
            if(!f) break;
        }
    }
    if(flow==0) deep[u] = -1;  //删除无法增广的点
	return flow;
}
int dinic(){
    int ans = 0;
    while(bfs()){
        memcpy(cur, head, sizeof(head));  //重置cur数组
        ans += dfs(S,1e9);
    }
    return ans;
}

int dx[8]={-2,-1,1,2,2,1,-1,-2};
int dy[8]={1,2,2,1,-1,-2,-2,-1};
bool flag[220][220];
int col[220][220];
bool ok(int x,int y) {
    if(x<0||x>=n||y<0||y>=n) return 0;
    if(col[x][y]) return 0;
    return 1;
}
void Dfs(int x,int y,int now) {
    col[x][y] = now;
    rep(i, 0, 7) {
        int xx = x+dx[i];
        int yy = y+dy[i];
        if(ok(xx,yy)) Dfs(xx,yy,-now);
    }
}
int id(int x,int y) {
    return x*n+y;
}
int sum;
int main() {
    //freopen("a.txt","r",stdin);
    ios::sync_with_stdio(0);
    cin>>n>>m;
    sum = n*n-m;
    memset(head,-1,sizeof(head));  //head初始化-1
    tot = -1;

    rep(i, 1, m) {
        int x,y;
        cin>>x>>y;
        x--;y--;
        flag[x][y] = 1;
    }

    rep(i, 0, n-1)
        rep(j, 0, n-1) {
            if(col[i][j]) continue;
            Dfs(i,j,1);
        }
    S = n*n;
    T = n*n+1;
    rep(i, 0, n*n-1) {
        int x = i/n;
        int y = i%n;
        if(flag[x][y]) continue;
        if(col[x][y]==1) {
            ae(S,i,1);
            rep(j, 0, 7) {
                int xx = x+dx[j];
                int yy = y+dy[j];
                if((xx>=0&&xx<n&&yy>=0&&yy<n) && (!flag[xx][yy]))
                    ae(i,id(xx,yy),1e9);
            }
        }
        else ae(i,T,1);
    }

    cout<<sum-dinic()<<endl;
    return 0;
}