题解 P4016 【负载平衡问题】
Ireliaღ
2019-04-13 18:46:43
**Zkw费用流指针版**
### 解题思路
这道题是想让所有节点最终货物值相等，也就是最终所有都会变成它们的平均值，所以可以使用费用流。
### 建图
* 设$0$为超级源点，$n + 1$为超级汇点，平均值为$ave$
* 对于每一个需要“送”的点，从$0$向它连边，容量$val_i - ave$，费用$0$
* 对于每一个需要“拿”的点，从它向$n + 1$连边，容量$ave - val_i$，费用$0$
* 从每个点向相邻点连容量$\infty$，费用$1$的边
### 代码
Zkw费用流，指针存图，有当前弧
```cpp
// luogu-judger-enable-o2
// luogu-judger-enable-o2
#include <iostream>
#include <algorithm>
#include <cstring>
#include <deque>
#include <cstdio>
using namespace std;
const int MAXN = 105;
const int INF = 0x3f3f3f3f;
int n;
struct Edge{
int to, val, cost;
Edge *next, *ops;
Edge(int to, int val, int cost, Edge *next): to(to), val(val), cost(cost), next(next){}
};
Edge *head[MAXN], *cur[MAXN];
void AddEdge(int u, int v, int w, int c) {
head[u] = new Edge(v, w, c, head[u]);
head[v] = new Edge(u, 0, -c, head[v]);
head[u]->ops = head[v]; head[v]->ops = head[u];
}
int s, t, res, ans;
int dis[MAXN];
bool vis[MAXN];
bool Spfa() {
memset(dis, INF, sizeof(dis));
memset(vis, false, sizeof(vis));
deque<int> q; q.push_back(s); vis[s] = true; dis[s] = 0;
while (!q.empty()) {
int u = q.front(); q.pop_front();
for (Edge *e = head[u]; e; e = e->next) {
int v = e->to;
if (e->val && dis[v] > dis[u] + e->cost) {
dis[v] = dis[u] + e->cost;
if (!vis[v]) {
if (!q.empty() && dis[v] < dis[q.front()]) q.push_front(v);
else q.push_back(v);
}
}
}
}
return dis[t] < INF;
}
int Dfs(int u, int flow) {
vis[u] = true;
if (u == t) {
res += flow;
return flow;
}
int used = 0;
for (Edge *&e = cur[u]; e; e = e->next) {
int v = e->to;
if ((v == t || !vis[v]) && e->val && dis[v] == dis[u] + e->cost) {
int mi = Dfs(v, min(e->val, flow - used));
if (mi) {
used += mi;
e->val -= mi;
e->ops->val += mi;
ans += e->cost * mi;
}
if (used == flow) break;
}
}
return used;
}
void Dinic() {
res = ans = 0;
while (Spfa()) {
vis[t] = true;
while (vis[t]) {
memset(vis, false, sizeof(vis));
memcpy(cur, head, sizeof(head));
Dfs(s, INF);
}
}
}
int num[MAXN];
int main() {
#ifndef ONLINE_JUDGE
freopen("in.txt", "r", stdin);
freopen("out.txt", "w", stdout);
#endif
ios :: sync_with_stdio(false); cin.tie(NULL); cout.tie(NULL);
int sum = 0;
cin >> n;
s = 0; t = n + 1;
for (int i = 1; i <= n; i++) {
cin >> num[i];
sum += num[i];
}
int ave = sum / n;
for (int i = 1; i <= n; i++) {
if (num[i] > ave) AddEdge(s, i, num[i] - ave, 0);
else AddEdge(i, t, ave - num[i], 0);
}
for (int i = 2; i <= n; i++) {
AddEdge(i - 1, i, INF, 1);
AddEdge(i, i - 1, INF, 1);
}
AddEdge(1, n, INF, 1);
AddEdge(n, 1, INF, 1);
Dinic();
cout << ans << endl;
return 0;
}
```