最小费用最大流

2020-03-07

数据结构与算法

字数统计: 701字 | 阅读时长≈ 4分钟

最小费用最大流的总结

P4016 负载平衡问题

题目链接

https://www.luogu.org/problem/P4016

解题思路

分源点、汇点，源点所有超过均值的点流量为1，费用为0、所有未超过均值的点连汇点流量为1，费用为0、相邻两点连边，流量为INF，费用为1，跑最小费用最大流即可。

AC代码

复制#include <bits/stdc++.h>
#define go(i,a,b) for(int (i)=(a);(i)<=(b);(i)++)
#define dip(i,a,b) for(int (i)=(a);(i)<=(b);(i)++)
#define maxn 2010
#define maxm 1200010
#define inf 0x3f3f3f3f
using namespace std;
struct Edge{
    int to, nex, cap, flow, cost;
}edge[maxm];
int head[maxn], tol, pre[maxn], dis[maxn];
bool vis[maxn];
int N;
void init(int n){
    N = n;
    tol = 0;
    memset(head, -1, sizeof(head));
}
void addedge(int u, int v, int cap, int cost){
    edge[tol].to = v;
    edge[tol].cap = cap;
    edge[tol].cost = cost;
    edge[tol].flow = 0;
    edge[tol].nex = head[u];
    head[u] = tol++;
    edge[tol].to = u;
    edge[tol].cap = 0;
    edge[tol].cost = -cost;
    edge[tol].flow = 0;
    edge[tol].nex = head[v];
    head[v] = tol++;
}
bool spfa(int s, int t){
    queue<int>q;
    for(int i = 0; i < N; i++){
        dis[i] = inf;
        vis[i] = false;
        pre[i] = -1;
    }
    dis[s] = 0;
    vis[s] = true;
    q.push(s);
    while(!q.empty()){

        int u = q.front();
        q.pop();
        vis[u] = false;
        for(int i = head[u]; i != -1; i = edge[i].nex){
            int v = edge[i].to;

            if(edge[i].cap > edge[i].flow && dis[v] > dis[u] + edge[i].cost){
                dis[v] = dis[u] + edge[i].cost;
                pre[v] = i;
                if(!vis[v]){
                    vis[v] = true;
                    q.push(v);
                }
            }
        }
    }
    if(pre[t] == -1)return false;
    else return true;
}
int minCostMaxflow(int s, int t, int &cost){
    int flow = 0;
    cost = 0;
    while(spfa(s, t)){
        int Min = inf;
        for(int i = pre[t]; i != -1; i = pre[edge[i^1].to]){
            if(Min > edge[i].cap - edge[i].flow)
                Min = edge[i].cap - edge[i].flow;
        }
        for(int i = pre[t]; i != -1; i = pre[edge[i^1].to]){
            edge[i].flow += Min;
            edge[i^1].flow -= Min;
            cost += edge[i].cost * Min;
        }
        flow += Min;
    }
    return flow;
}
int a[105];
int main()
{
    int n, sum = 0;
    scanf("%d", &n);
    init(n + 2);
    go(i,1,n)scanf("%d", &a[i]), sum += a[i];
    sum /= n;
    go(i,1,n){
        if(a[i] > sum)addedge(0, i, a[i] - sum, 0);
        else addedge(i, n + 1, sum - a[i], 0);
    }
    go(i,1,n-1)addedge(i,i+1,inf,1),addedge(i+1,i,inf,1);
    addedge(1,n,inf,1),addedge(n,1,inf,1);
    int ans;
    minCostMaxflow(0,n+1,ans);
    printf("%d\n", ans);
    return 0;
}

1	复制#include <bits/stdc++.h>
2	#define go(i,a,b) for(int (i)=(a);(i)<=(b);(i)++)
3	#define dip(i,a,b) for(int (i)=(a);(i)<=(b);(i)++)
4	#define maxn 2010
5	#define maxm 1200010
6	#define inf 0x3f3f3f3f
7	using namespace std;
8	struct Edge{
9	int to, nex, cap, flow, cost;
10	}edge[maxm];
11	int head[maxn], tol, pre[maxn], dis[maxn];
12	bool vis[maxn];
13	int N;
14	void init(int n){
15	N = n;
16	tol = 0;
17	memset(head, -1, sizeof(head));
18	}
19	void addedge(int u, int v, int cap, int cost){
20	edge[tol].to = v;
21	edge[tol].cap = cap;
22	edge[tol].cost = cost;
23	edge[tol].flow = 0;
24	edge[tol].nex = head[u];
25	head[u] = tol++;
26	edge[tol].to = u;
27	edge[tol].cap = 0;
28	edge[tol].cost = -cost;
29	edge[tol].flow = 0;
30	edge[tol].nex = head[v];
31	head[v] = tol++;
32	}
33	bool spfa(int s, int t){
34	queue<int>q;
35	for(int i = 0; i < N; i++){
36	dis[i] = inf;
37	vis[i] = false;
38	pre[i] = -1;
39	}
40	dis[s] = 0;
41	vis[s] = true;
42	q.push(s);
43	while(!q.empty()){
44
45	int u = q.front();
46	q.pop();
47	vis[u] = false;
48	for(int i = head[u]; i != -1; i = edge[i].nex){
49	int v = edge[i].to;
50
51	if(edge[i].cap > edge[i].flow && dis[v] > dis[u] + edge[i].cost){
52	dis[v] = dis[u] + edge[i].cost;
53	pre[v] = i;
54	if(!vis[v]){
55	vis[v] = true;
56	q.push(v);
57	}
58	}
59	}
60	}
61	if(pre[t] == -1)return false;
62	else return true;
63	}
64	int minCostMaxflow(int s, int t, int &cost){
65	int flow = 0;
66	cost = 0;
67	while(spfa(s, t)){
68	int Min = inf;
69	for(int i = pre[t]; i != -1; i = pre[edge[i^1].to]){
70	if(Min > edge[i].cap - edge[i].flow)
71	Min = edge[i].cap - edge[i].flow;
72	}
73	for(int i = pre[t]; i != -1; i = pre[edge[i^1].to]){
74	edge[i].flow += Min;
75	edge[i^1].flow -= Min;
76	cost += edge[i].cost * Min;
77	}
78	flow += Min;
79	}
80	return flow;
81	}
82	int a[105];
83	int main()
84	{
85	int n, sum = 0;
86	scanf("%d", &n);
87	init(n + 2);
88	go(i,1,n)scanf("%d", &a[i]), sum += a[i];
89	sum /= n;
90	go(i,1,n){
91	if(a[i] > sum)addedge(0, i, a[i] - sum, 0);
92	else addedge(i, n + 1, sum - a[i], 0);
93	}
94	go(i,1,n-1)addedge(i,i+1,inf,1),addedge(i+1,i,inf,1);
95	addedge(1,n,inf,1),addedge(n,1,inf,1);
96	int ans;
97	minCostMaxflow(0,n+1,ans);
98	printf("%d\n", ans);
99	return 0;
100	}