题解 P3355 【骑士共存问题】
Ireliaღ
2019-04-10 17:36:14
**最大流=最小割**
ISAP+当前弧优化巨快
### 解题思路
不难看出,马能攻击到的点和它本身所在的点黑白颜色不同,所以我们考虑建立二分图进行匹配
### 建图
* 设$0$为原点,$n ^ 2 + 1$为汇点,把格子上的每一个点映射到$[1, n ^ 2]$
* 对于$\forall i \in black$,如果**$i$点没有障碍**,从$0$向$i$建立容量为$1$的边
* 对于$\forall j \in white$,如果**$j$点没有障碍**,从$j$向$n ^ 2 + 1$建立容量为$1$的边
* 对于$\forall i \in black$,从$i$向能攻击到的八个白点建立容量为$\infty$的边
然后跑最大流,用总格数-障碍数-最大流即可
### 代码
看有人发匈牙利和dinic的题解,说奇数建图和偶数建图有一种过不了,但是用ISAP两种都能过。
```cpp
// luogu-judger-enable-o2
// luogu-judger-enable-o2
#include <iostream>
#include <cstring>
#include <algorithm>
#include <queue>
using namespace std;
const int MAXN = 205;
const int INF = 0x3f3f3f3f;
const int DIRX[8] = {-2, -1, 1, 2, 2, 1, -1, -2};
const int DIRY[8] = {-1, -2, -2, -1, 1, 2, 2, 1};
int n, m;
bool trap[MAXN][MAXN];
int ID(int x, int y) {
return (x - 1) * n + y;
}
struct Edge{
int to, val;
Edge *next, *ops;
Edge(int to, int val, Edge *next): to(to), val(val), next(next){}
};
Edge *head[MAXN * MAXN], *cur[MAXN * MAXN];
void AddEdge(int u, int v, int w) {
head[u] = new Edge(v, w, head[u]);
head[v] = new Edge(u, 0, head[v]);
head[u]->ops = head[v]; head[v]->ops = head[u];
}
int dep[MAXN * MAXN], gap[MAXN * MAXN], s, t, res;
void Bfs() {
memset(dep, -1, sizeof(dep));
memset(gap, 0, sizeof(gap));
dep[t] = 0; gap[0]++;
queue<int> q; q.push(t);
while (!q.empty()) {
int u = q.front(); q.pop();
for (Edge *e = head[u]; e; e = e->next) {
int v = e->to;
if (dep[v] != -1) continue;
q.push(v);
dep[v] = dep[u] + 1;
gap[dep[v]]++;
}
}
}
int Dfs(int u, int flow) {
if (u == t) {
res += flow;
return flow;
}
int used = 0;
for (Edge *&e = cur[u]; e; e = e->next) {
int v = e->to;
if (e->val && dep[v] == dep[u] - 1) {
int mi = Dfs(v, min(e->val, flow - used));
if (mi) {
used += mi;
e->val -= mi;
e->ops->val += mi;
}
if (used == flow) return used;
}
}
gap[dep[u]]--;
cur[u] = head[u];
if (gap[dep[u]] == 0) dep[s] = n * n + 3;
dep[u]++;
gap[dep[u]]++;
return used;
}
void Work() {
memcpy(cur, head, sizeof(head));
res = 0;
Bfs();
while (dep[s] <= n * n + 2) Dfs(s, INF);
}
int main() {
ios :: sync_with_stdio(false); cin.tie(NULL); cout.tie(NULL);
cin >> n >> m;
s = 0; t = n * n + 1;
for (int i = 1; i <= m; i++) {
int x, y;
cin >> x >> y;
trap[x][y] = true;
}
for (int i = 1; i <= n; i++) {
for (int j = 1; j <= n; j++) {
if ((i + j) % 2 == 0) {
if (!trap[i][j]) AddEdge(0, ID(i, j), 1);
for (int k = 0; k < 8; k++) {
int x = i + DIRX[k], y = j + DIRY[k];
if (x < 1 || x > n || y < 1 || y > n) continue;
AddEdge(ID(i, j), ID(x, y), INF);
}
} else {
if (!trap[i][j]) AddEdge(ID(i, j), n * n + 1, 1);
}
}
}
Work();
cout << n * n - m - res << endl;
return 0;
}
```