Luogu P2774 方格取数问题

题目描述

https://www.luogu.com.cn/problem/P2774

Solution

二分图染色之后,答案即为最大点权独立集,即总点权减掉最小点权覆盖

建图:

$s$ 连 黑点 $(i,j)$,容量为 $a[i][j]$

黑点 连 周围的白点,容量为 $\infty$

白点 $(i,j)$ 连 $t$,容量为 $a[i][j]$

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#include <iostream>
#include <cstdio>
#include <queue>
#define maxn 100010
#define INF 1000000000
using namespace std;

int n, m;

inline int id(int i, int j) { return (i - 1) * m + j; }

int dx[] = { 0, 0, 1, -1 };
int dy[] = { 1, -1, 0, 0 };

struct Edge {
int to, next, w;
} e[1000000]; int c1, head[maxn];
inline void add_edge(int u, int v, int w) {
e[c1].to = v; e[c1].w = w;
e[c1].next = head[u]; head[u] = c1++;
}

inline void Add_edge(int u, int v, int w) {
add_edge(u, v, w); add_edge(v, u, 0);
}

int s, t, dep[maxn];
bool bfs() {
fill(dep, dep + maxn, 0); dep[s] = 1;
queue<int> Q; Q.push(s);
while (!Q.empty()) {
int u = Q.front(); Q.pop();
for (int i = head[u]; ~i; i = e[i].next) {
int v = e[i].to, w = e[i].w;
if (w > 0 && !dep[v]) {
dep[v] = dep[u] + 1;
Q.push(v); if (v == t) return 1;
}
}
}
return 0;
}

int dfs(int u, int _w) {
if (u == t || !_w) return _w;
int s = 0;
for (int i = head[u]; ~i; i = e[i].next) {
int v = e[i].to, w = e[i].w;
if (dep[v] == dep[u] + 1 && w > 0) {
int di = dfs(v, min(_w - s, w));
e[i].w -= di; e[i ^ 1].w += di;
s += di; if (s == _w) break;
}
}
if (s < _w) dep[u] = 0;
return s;
}

int mf;
int dinic() {
while (bfs()) mf += dfs(s, INF);
return mf;
}

int ans;
int main() { fill(head, head + maxn, -1);
ios::sync_with_stdio(false);
cin.tie(nullptr); cout.tie(nullptr);

cin >> n >> m; s = 0; t = n * m + 1;
for (int i = 1; i <= n; ++i)
for (int j = 1; j <= m; ++j) {
int x; cin >> x; ans += x;
if (i + j & 1) Add_edge(s, id(i, j), x);
else Add_edge(id(i, j), t, x);
}
for (int i = 1; i <= n; ++i)
for (int j = 1; j <= m; ++j)
if (i + j & 1)
for (int k = 0; k < 4; ++k) {
int x = i + dx[k], y = j + dy[k];
if (x < 1 || x > n || y < 1 || y > m) continue;
Add_edge(id(i, j), id(x, y), INF);
}
cout << ans - dinic() << "\n";
return 0;
}