Luogu P3355 骑士共存问题

题目链接 \(Click\) \(Here\)

二分图最大独立集。对任意两个可以相互攻击的点，我们可以选其中一个。对于不会互相攻击的，可以全部选中。所以我们只需要求出最大匹配，根据定理，二分图最大独立集等于点数减去最大匹配，就得到了答案。

#include <bits/stdc++.h>
using namespace std;

const int N = 80010;
const int M = 800010;
const int INF = 0x3f3f3f3f;

int n, m, ban[210][210];
int cnt = -1, head[N];
int mv[8][2] = {{2, 1},{2, -1},{1, 2},{1, -2},{-2, 1},{-2, -1},{-1, 2},{-1, -2}};

bool in_map (int x, int y) {return 1 <= x && x <= n && 1 <= y && y <= n;}

struct edge {
	int nxt, to, f;
}e[M];

void add_edge (int from, int to, int flw) {
	e[++cnt].nxt = head[from];
	e[cnt].to = to;
	e[cnt].f = flw;
	head[from] = cnt;
}

void add_len (int u, int v, int f) {
	add_edge (u, v, f);
	add_edge (v, u, 0);
}

int nd1 (int x, int y) {return n * n * 0 + (x - 1) * n + y;}
int nd2 (int x, int y) {return n * n * 1 + (x - 1) * n + y;}

queue <int> q;
int cur[N], deep[N];

bool bfs (int s, int t) {
	memcpy (cur, head, sizeof (head));
	memset (deep, 0x3f, sizeof (deep));
	q.push (s); deep[s] = 0;
	while (!q.empty ()) {
		int u = q.front (); q.pop ();
		for (int i = head[u]; ~i; i = e[i].nxt) {
			int v = e[i].to;
			if (deep[v] == INF && e[i].f) {
				deep[v] = deep[u] + 1;
				q.push (v);
			}
		}
	}
	return deep[t] != INF;
}

int dfs (int u, int t, int lim) {
	if (u == t || !lim) {
		return lim;
	}
	int tmp = 0, flow = 0;
	for (int &i = cur[u]; ~i; i = e[i].nxt) {
		int v = e[i].to;
		if (deep[v] == deep[u] + 1) {
			tmp = dfs (v, t, min (lim, e[i].f));
			lim -= tmp;
			flow += tmp;
			e[i ^ 0].f -= tmp;
			e[i ^ 1].f += tmp;
			if (!lim) break;
		}
	}
	return flow;
}

int main () {
	memset (head, -1, sizeof (head));
	cin >> n >> m;
	for (int i = 1; i <= m; ++i) {
		int x, y;
		cin >> x >> y;
		ban[x][y] = true;
	}
	int s = n * n * 2 + 1, t = n * n * 2 + 2;
	for (int i = 1; i <= n; ++i) {
		for (int j = 1; j <= n; ++j) {
			if (ban[i][j]) continue;
			add_len (s, nd1 (i, j), 1);
			add_len (nd2 (i, j), t, 1);
			for (int k = 0; k < 8; ++k) {
				int tx = i + mv[k][0];
				int ty = j + mv[k][1];
				if (in_map (tx, ty) && !ban[tx][ty]) {
					add_len (nd1 (i, j), nd2 (tx, ty), 1);
				}
			}
		}
	}
	int max_flow = 0;
	while (bfs (s, t)) {
		getchar ();
		max_flow += dfs (s, t, INF);
	}
	// printf ("max_flow = %d\n", max_flow);
	cout << n * n - m - max_flow / 2 << endl;
}