luogu P3355 骑士共存问题

本题和方格取数一样，也可以分成黑白点，本题加上特判一个点是否有障碍即可，其余和那题没什么区别，挂一下大佬的证明(二分图最大独立集)

 

#include<bits/stdc++.h>
using namespace std;
#define lowbit(x) ((x)&(-x))
typedef long long LL;

const int maxm = 1e6+5;
const int INF = 0x3f3f3f3f;
const int dx[] = {-2,-2,-1,-1,1,1,2,2};
const int dy[] = {-1,1,-2,2,2,-2,1,-1};

struct edge{
    int u, v, cap, flow, nex;
} edges[maxm];

int head[maxm], cur[maxm], cnt, level[40005], num[205][205], ID;
bool block[205][205];

void init() {
    memset(head, -1, sizeof(head));
}

void add(int u, int v, int cap) {
    edges[cnt] = edge{u, v, cap, 0, head[u]};
    head[u] = cnt++;
}

void addedge(int u, int v, int cap) {
    add(u, v, cap), add(v, u, 0);
}

void bfs(int s) {
    memset(level, -1, sizeof(level));
    queue<int> q;
    level[s] = 0;
    q.push(s);
    while(!q.empty()) {
        int u = q.front();
        q.pop();
        for(int i = head[u]; i != -1; i = edges[i].nex) {
            edge& now = edges[i];
            if(now.cap > now.flow && level[now.v] < 0) {
                level[now.v] = level[u] + 1;
                q.push(now.v);
            }
        }
    }
}

int dfs(int u, int t, int f) {
    if(u == t) return f;
    for(int& i = cur[u]; i != -1; i = edges[i].nex) {
        edge& now = edges[i];
        if(now.cap > now.flow && level[u] < level[now.v]) {
            int d = dfs(now.v, t, min(f, now.cap - now.flow));
            if(d > 0) {
                now.flow += d;
                edges[i^1].flow -= d;
                return d;
            }

        }
    }
    return 0;
}

int dinic(int s, int t) {
    int maxflow = 0;
    for(;;) {
        bfs(s);
        if(level[t] < 0) break;
        memcpy(cur, head, sizeof(head));
        int f;
        while((f = dfs(s, t, INF)) > 0)
            maxflow += f;
    }
    return maxflow;
}

void run_case() {
    int m, n;
    LL sum = 0;
    init();
    cin >> n >> m;
    int s = 0, t = n*n+1;
    sum += n*n;
    for(int i = 0; i < m; ++i) {
        int x, y; cin >> x >> y;
        block[x][y] = true;
    }
    for(int i = 1; i <= n; ++i) {
        for(int j = 1; j <= n; ++j) {
            num[i][j] = ++ID;
            if((i+j)%2==1) {
                if(!block[i][j]) addedge(s, ID, 1);
            } else { 
                if(!block[i][j])addedge(ID, t, 1); 
            }
        }
    }
    for(int i = 1; i <= n; ++i)
        for(int j = 1; j <= n; ++j) {
            if((i+j)%2==0) continue;
            for(int k = 0; k < 8; ++k) {
                int nx = i+dx[k], ny = j+dy[k];
                if(nx > n || nx < 1 || ny > n || ny < 1) continue;
                addedge(num[i][j], num[nx][ny], INF);
            }
        }
    sum -= dinic(s, t);
    cout << sum-m;
}

int main() {
    ios::sync_with_stdio(false), cin.tie(0);
    run_case();
    cout.flush();
    return 0;
}