王道考研复试机试题解 2020版--第十一章图论(3)

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本文深入探讨了图算法中的三种核心应用：拓扑排序确定赛事名次、任务安排的临界路径寻找，以及合法性检查。通过具体代码实例，详细解析了如何使用邻接表表示图，如何计算顶点的入度，以及如何运用队列实现拓扑排序，最终确定图中各节点的正确顺序或找出关键路径。

1.legal or not

#include<iostream>
#include<cstring>
#include<queue>
#include<vector>

using namespace std;
const int maxn=500;

vector<int> graph[maxn];
int inDegree[maxn];

bool TopologicalSort(int n){
	queue<int> node;
	for(int i=0;i<n;i++){
		if(inDegree[i]==0){
			node.push(i);
		}
	}
	int number=0;
	while(!node.empty()){
		int u=node.front();
		node.pop();
		number++;
		for(int i=0;i<graph[u].size();i++){
			int v=graph[u][i];
			inDegree[v]--;
			if(inDegree[v]==0)
			node.push(v);
		}
	}
	return n==number;
}

int main(){
	int n,m;
	while(cin>>n>>m){
		if(n==0&&m==0)break;
		memset(graph,0,sizeof(graph));
		memset(inDegree,0,sizeof(inDegree));
		while(m--){
			int from,to;
			cin>>from>>to;
			graph[from].push_back(to);
			inDegree[to]++;
		}
		if(TopologicalSort(n))
		cout<<"YES"<<endl;
		else
		cout<<"NO"<<endl;
	} 
	return 0;
}

2.确定比赛名次

#include<iostream>
#include<cstring>
#include<vector>
#include<queue>

using namespace std;

const int maxn=501;

vector<int> graph[maxn];
int inDegree[maxn];

vector<int> TopologicalSort(int n){
	vector<int> topology; //拓扑序列
	priority_queue<int,vector<int>,greater<int> > node;
	
	for(int i=1;i<=n;i++){
		if(inDegree[i]==0)
		node.push(i);
	}
	
	while(!node.empty()){
		int u=node.top();
		node.pop();
		topology.push_back(u);
		for(int i=0;i<graph[u].size();i++){
			int v=graph[u][i];
			inDegree[v]--;
			if(inDegree[v]==0)
			node.push(v);
		}
	} 
	return topology;
}

int main(){
	int n,m;
	while(cin>>n>>m){
		memset(graph,0,sizeof(graph));
		memset(inDegree,0,sizeof(inDegree));
		while(m--){
			int from,to;
			cin>>from>>to;
			graph[from].push_back(to);
			inDegree[to]++;
		}
		vector<int>answer =TopologicalSort(n);
		for(int i=0;i<answer.size();i++){
			if(i==0){
				cout<<answer[i];
			}
			else cout<<" "<<answer[i];
		}
		cout<<endl;
	}
	return 0;
}

3.Instructions Arrangement

#include<iostream>
#include<cstring>
#include<vector>
#include<queue>
#include<algorithm>
#include<climits>

using namespace std;

const int maxn=1001;
const int INF=INT_MAX;

struct Edge{
	int to;//终点
	int length; //距离
	Edge(int t,int l):to(t),length(l){
	} 
};

vector<Edge> graph[maxn];
int earliest[maxn];  //最早开始时间
int lastest[maxn];   //最晚开始时间
int inDegree[maxn];  //入度

void CriticalPath(int n){
	vector<int>topology;
	queue<int> node;
	for(int i=0;i<n;i++){
		if(inDegree[i]==0){
			node.push(i);
			earliest[i]=1;
		}
	}
	while(!node.empty()){
		int u=node.front();
		topology.push_back(u);
		node.pop();
		for(int i=0;i<graph[u].size();++i){
			int v=graph[u][i].to;
			int l=graph[u][i].length;
			earliest[v]=max(earliest[v],earliest[u]+l);
			inDegree[v]--;
			if(inDegree[v]==0){
				node.push(v);
			}
		}
	}
	for(int i=topology.size()-1;i>=0;i--){
		int u=topology[i];
		if(graph[u].size()==0){
			lastest[u]=earliest[u];  //汇点的最晚开始时间初始化
			 
		}else {
			lastest[u]=INF;
		} 
		for(int j=0;j<graph[u].size();++j){
			int v=graph[u][j].to;
			int l=graph[u][j].length;
			lastest[u]=min(lastest[u],lastest[v]-l);
		}
	}
} 

int main(){
	int n,m;
	while(cin>>n>>m){
		memset(graph,0,sizeof(graph));
		memset(earliest,0,sizeof(earliest));
		memset(lastest,0,sizeof(lastest));
		memset(inDegree,0,sizeof(inDegree));
		while(m--){
			int from,to,length;
			cin>>from>>to>>length;
			graph[from].push_back(Edge(to,length));
			inDegree[to]++;
		}
		
		CriticalPath(n);
		int answer=0;
		for(int i=0;i<n;i++){
			answer=max(answer,earliest[i]);
		}
		cout<<answer<<endl;
	}
	return 0;
}