解题报告：hdu 3572 Task Schedule（当前弧优化Dinic算法）

Problem Description

Our geometry princess XMM has stoped her study in computational geometry to concentrate on her newly opened factory. Her factory has introduced M new machines in order to process the coming N tasks. For the i-th task, the factory has to start processing it at or after day Si, process it for Pi days, and finish the task before or at day Ei. A machine can only work on one task at a time, and each task can be processed by at most one machine at a time. However, a task can be interrupted and processed on different machines on different days.
Now she wonders whether he has a feasible schedule to finish all the tasks in time. She turns to you for help.

Input

On the first line comes an integer T(T<=20), indicating the number of test cases.
You are given two integer N(N<=500) and M(M<=200) on the first line of each test case. Then on each of next N lines are three integers Pi, Si and Ei (1<=Pi, Si, Ei<=500), which have the meaning described in the description. It is guaranteed that in a feasible schedule every task that can be finished will be done before or at its end day.

Output

For each test case, print “Case x: ” first, where x is the case number. If there exists a feasible schedule to finish all the tasks, print “Yes”, otherwise print “No”.
Print a blank line after each test case.

Sample Input

4 3

1 3 5

1 1 4

2 3 7

3 5 9

2 2

2 1 3

1 2 2

Sample Output

Case 1: Yes

Case 2: Yes

解题思路：此题关键在于建图，跑最大流来判断是否已达到满流。题意：有n个任务，m台机器。每个任务有最早才能开始做的时间s_i，截止时间e_i，和完成该任务所需要的时间p_i。每个任务可以分段进行，但在同一天一台机器最多只能执行一个任务，问是否有可行的工作时间来完成所有任务。做法：建立一个超级源点s=0和一个超级汇点t=1001。源点s和每个任务i建边，边权为p_i，表示完成该任务需要的天数。对于每一个任务i，将编号i与编号n+[s_i～e_i]中每个编号建边，边权为1，表示将任务i分在第n+s_i天～第n+e_i天来完成，再将编号n+[s_i～e_i]与汇点t建边，边权为m，表示每一天(编号为n+j)最多同时运行m台机器来完成8天的任务单位量。但由于建的边数太多，用裸的Dinic跑最大流会TLE，怎么优化呢？采用当前弧优化：因为每次dfs找增广路的过程中都是从每个顶点指向的第1(编号为0)条边开始遍历的，而如果第1条边已达到满流，则会继续遍历第2条边....直至找到汇点，事实上这个递归的过程就造成了很多不必要浪费的时间，所以在dfs的过程中应标记一下每个顶点v当前能到达的第curfir[v]条边，说明顶点v的第0条边～第curfir[v]-1条边都已达满流或者流不到汇点，这样时间复杂度就大大降低了。注意：每次bfs给图重新分层次时，需要清空当前弧数组cirfir[]，表示初始时从每个顶点v的第1(编号为0)条边开始遍历。

AC代码(124ms)：

 #include<bits/stdc++.h>

 using namespace std;

 const int INF=0x3f3f3f3f;

 const int maxn=;

 struct edge{ int to,cap;size_t rev;

     edge(int _to, int _cap, size_t _rev):to(_to),cap(_cap),rev(_rev){}

 };

 int T,n,m,p,s,e,tot,level[maxn];queue<int> que;vector<edge> G[maxn];size_t curfir[maxn];//当前弧数组

 void add_edge(int from,int to,int cap){

     G[from].push_back(edge(to,cap,G[to].size()));

     G[to].push_back(edge(from,,G[from].size()-));

 }

 bool bfs(int s,int t){

     memset(level,-,sizeof(level));

     while(!que.empty())que.pop();

     level[s]=;

     que.push(s);

     while(!que.empty()){

         int v=que.front();que.pop();

         for(size_t i=;i<G[v].size();++i){

             edge &e=G[v][i];

             if(e.cap>&&level[e.to]<){

                 level[e.to]=level[v]+;

                 que.push(e.to);

             }

         }

     }

     return level[t]<?false:true;

 }

 int dfs(int v,int t,int f){

     if(v==t)return f;

     for(size_t &i=curfir[v];i<G[v].size();++i){//从v的第curfir[v]条边开始，采用引用的方法，同时改变本身的值

         //因为节点v的第0~curfir[v]-1条边已达到满流了，所以无需重新遍历--->核心优化

         edge &e=G[v][i];

         if(e.cap>&&(level[v]+==level[e.to])){

             int d=dfs(e.to,t,min(f,e.cap));

             if(d>){

                 e.cap-=d;

                 G[e.to][e.rev].cap+=d;

                 return d;

             }

         }

     }

     return ;

 }

 int max_flow(int s,int t){

     int f,flow=;

     while(bfs(s,t)){

         memset(curfir,,sizeof(curfir));//重新将图分层之后就清空数组，从第0条边开始遍历

         while((f=dfs(s,t,INF))>)flow+=f;

     }

     return flow;

 }

 int main(){

     while(~scanf("%d",&T)){

         for(int cas=;cas<=T;++cas){

             scanf("%d%d",&n,&m);tot=;

             for(int i=;i<maxn;++i)G[i].clear();

             for(int i=;i<=;++i)add_edge(+i,,m);

             for(int i=;i<=n;++i){

                 scanf("%d%d%d",&p,&s,&e);

                 add_edge(,i,p);tot+=p;//tot为总时间

                 for(int j=s;j<=e;++j)add_edge(i,+j,);

             }

             printf("Case %d: %s\n\n",cas,max_flow(,)==tot?"Yes":"No");

         }

     }

     return ;

 }

巴特西