[网络流24题] 最小路径覆盖问题(匈牙利 最大流)
定理: 最小路径覆盖数=|G|-二分图最大匹配数
所以匈牙利算法就来了,内置匹配,简单易上手
#include <bits/stdc++.h>
const int maxn = 310;
using namespace std;
typedef long long ll;
vector<int> maps[maxn];
int pipei[maxn], n, m;
bool mark[maxn];
int in[maxn], son[maxn];
bool dfs(int u) //找增广路径,匹配
{
for (int i = 0, len = maps[u].size(); i < len; i++){
int v = maps[u][i];
if (!mark[v]){
mark[v] = true;
if (!pipei[v] || dfs(pipei[v])){
pipei[v] = u;
return true;
}
}
}
return false;
}
int main()
{
scanf("%d%d", &n, &m);
for (int i = 0; i < m; i++){
int a, b;
scanf("%d%d", &a, &b);
maps[a].push_back(n + b); //建图
}
int ans = 0;
for (int i = 1; i <= n; i++){
memset(mark, false, sizeof(mark));
if (dfs(i))
ans++;
}
for (int i = n + 1; i <= 2 * n; i++){
if (pipei[i]){
int u = pipei[i], v = i - n;
son[u] = v; //u --> v
in[v]++; //入度
}
}
for (int i = 1; i <= n; i++){ //输出答案
if (in[i] == 0){
int u = i;
printf("%d", u);
while ((u = son[u]))
printf(" %d", u);
printf("\n");
}
}
printf("%d\n", n - ans); //最小路径覆盖数=|G|-二分图最大匹配数
getchar();
getchar();
return 0;
}
好吧,从二分图,入手用匈牙利算法,好像就是个模板题了
最大流:用最大流的话,主要还是建图,还有反向弧为满流,就是匹配边
#include <bits/stdc++.h>
const int maxn = 160;
const int emax = 6000 * 3;
const int inf = 0x3f3f3f3f;
using namespace std;
typedef long long ll;
struct note{
int u, v, w;
int next;
} e[emax];
int head[maxn * 2], cnt;
int level[maxn * 2], son[maxn], in[maxn];
int n, m, s, t;
void add(int u, int v, int w){
e[cnt].u = u, e[cnt].v = v, e[cnt].w = w;
e[cnt].next = head[u], head[u] = cnt++; //正向弧
e[cnt].u = v, e[cnt].v = u, e[cnt].w = 0;
e[cnt].next = head[v], head[v] = cnt++; //反向弧
}
bool bfs(){
memset(level, -1, sizeof(level));
level[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].v;
if (e[i].w > 0 && level[v] == -1){
level[v] = level[u] + 1;
q.push(v);
}
}
}
return level[t] != -1;
}
int dfs(int u, int delta){
if (u == t)
return delta;
int flow = 0;
for (int i = head[u]; ~i; i = e[i].next){
int v = e[i].v;
if (e[i].w > 0 && level[u] + 1 == level[v]){
int tmp = dfs(v, min(delta - flow, e[i].w));
e[i].w -= tmp;
e[i ^ 1].w += tmp;
flow += tmp;
}
}
if (flow == inf)
level[u] = -1;
return flow;
}
int Dinic()
{
int maxflow = 0, tmp;
while (bfs()){
while ((tmp = dfs(s, inf)))
maxflow += tmp;
}
return maxflow;
}
int main()
{
scanf("%d%d", &n, &m);
s = 0, t = 2 * n + 1;
memset(head, -1, sizeof(head));
for (int i = 0; i < m; i++){
int a, b;
scanf("%d%d", &a, &b);
add(a, b + n, 1); //建边
}
for (int i = 1; i <= n; i++)
add(s, i, 1); //建边
for (int i = n + 1; i <= 2 * n; i++)
add(i, t, 1); //建边
int ans = Dinic();
for (int i = 0; i < cnt; i += 2)
{
if (e[i].u == s || e[i ^ 1].u == t)
continue;
if (e[i ^ 1].w){ //反向弧
son[e[i].u] = e[i].v - n;
in[e[i].v - n]++;
}
}
for (int i = 1; i <= n; i++){
if (in[i] == 0){
int u = i;
printf("%d", u);
while ((u = son[u]))
printf(" %d", u);
printf("\n");
}
}
printf("%d\n", n - ans);
return 0;
}
好吧,其实如果,做过类似,最大流去解决二分图匹配的话,又变成模板题了