C++ 图的实现

图可以说是算法与数据结构中十分重要的一个部分，然而对于图的实现，还是有一点点繁琐，今天参考清华大学出版社《数据结构》一书进行了一些回顾，记录于此。

本文并不对基本概念进行过多探讨，而着眼于实现。基于对途中边集的存储有邻接矩阵以及邻接表两种主要形式。本文将着重实现三个类：Graph基类，包含大量的virtual函数以待在派生类中实现；Graph的派生类Graphmtx(邻接矩阵实现图的存储)、Graphlnk(邻接表实现图的存储)。并通过简单的test对以上实现加以测试。废话不多说，直接贴代码，代码中加了比较详细的注释说明。

• 基类Graph定义：

//FileName : Graph.h
#pragma once
#include<iostream>
using namespace std;


#define INF 100000 //表示正无穷
const int DefaultVertices = 30;

template<class T,class E>
class Graph
{
public:
	static const E maxWeight =  INF;
	Graph(int size = DefaultVertices){};
	~Graph(){};
	bool GraphEmpty()const //检查为空
	{
		if (numEdges ==0 )return true;
		else return false;
	}
	bool GraphFull()const //检查为满
	{
		if(numVertices==maxVertices ||numEdges==maxVertices*(maxVertices-1)/2)
			return true;
		else
			return false;
	}
	int NumberOfVertices(){return numVertices;}	//返回当前顶点数
	int NumberOfEdges(){return numEdges;}		//返回当前边数
	virtual T getValue(int i)=0;					//取顶点i的值，i不合理返回0
	virtual E getWeight(int v1,int v2)=0;			//取边（v1，v2）的权值
	virtual int getFirstNeighbor(int v)=0;		//取顶点v的第一个邻接顶点
	virtual int getNextNeighbor(int v,int w)=0;	//取邻接顶点w的下一个邻接顶点
	virtual bool insertVertex(const T& vertex)=0;	//插入一个顶点vertex
	virtual bool insertEdge(int v1, int v2,E cost)=0;//插入边（v1,v2）,权值cost
	virtual bool removeVertex(int v)=0;			//删除顶点v和所有与之关联的边
	virtual bool removeEdge(int v1,int v2)=0;		//删除边（v1,v2）
	virtual int getVertexPos(T vertex)=0;

protected:
	int maxVertices;
	int numEdges;
	int numVertices;
	
};

• 基于邻接表实现边集存储的派生子类Graphlnk定义及实现：

//Filename : Grapglnk.h
#include "Graph.h"

template<class T ,class E>
struct Edge		//边界点的定义
{
	int dest;	//边的另一顶点位置
	E  cost;	//权值
	Edge<T ,E> *link;//下一条边链指针
	Edge(){}		//构造函数
	Edge(int num , E weight):dest(num),cost(weight),link(NULL){} //构造函数
	bool operator != (Edge<T,E> &R)const{	//判边不等否
		return (dest!=R.dest)? true:false;
	}
};

template<class T ,class E >
struct Vertex{	//顶点的定义
	T data;		//顶点名字
	Edge<T ,E> *adj;	//边链表的头指针
};

template <class T ,class E>
class Graphlnk: public Graph<T,E>
{
public:
	Graphlnk(int sz=DefaultVertices);
	~Graphlnk();
	T getValue(int i)
	{
		return (i>=0 && i<numVertices)? NodeTable[i].data : 0;
	}
	E getWeight(int v1,int v2);
	int getFirstNeighbor(int v);
	int getNextNeighbor(int v,int w);
	bool insertVertex(const T& vertex);
	bool insertEdge(int v1, int v2,E cost);
	bool removeVertex(int v);
	bool removeEdge(int v1,int v2);
	void inputGraph();
	void outputGraph();
	int getVertexPos( T vertex){
		for (int i = 0;i<numVertices;i++)
		{
			if(NodeTable[i].data == vertex)
				return i;
		}
		return -1; //找不到就返回-1
	}

private:
	Vertex<T,E> * NodeTable; //顶点表
	
};

template <class T ,class E>
Graphlnk<T,E>::Graphlnk(int sz)  //构造函数
{
	maxVertices = sz;
	numVertices = 0;
	numEdges = 0;
	NodeTable = new Vertex<T,E>[maxVertices];
	if(NodeTable ==NULL){ cerr<<"存储分配错！"<<endl;exit(1);}
	for (int i = 0;i<maxVertices;i++)
	{
		NodeTable[i].adj = NULL;
	}
}

template<class T ,class E>
Graphlnk<T,E>::~Graphlnk() //析构函数
{
	for(int i = 0; i<numVertices;i++)
	{
		Edge<T,E> *p = NodeTable[i].adj;
		while( p!= NULL)
		{
			NodeTable[i].adj = p->link;
			delete p ;
			p = NodeTable[i].adj;
		}
		
	}
	delete []NodeTable;
}

template <class T ,class E>
E Graphlnk<T,E>::getWeight(int v1,int v2) //返回边（v1，v2）的权重，边不存在则返回0
{
	if(v1 != -1 && v2 !=-1)
	{
		Edge<T,E> *p = NodeTable[v1].adj;
		while(p!=NULL && p->dest!=v2)
			p = p->link;
		if(p!=NULL)
			return p->cost;
	}
	return 0;
}

template<class T,class E>
int Graphlnk<T,E>::getFirstNeighbor(int v) //获得v的第一个邻接顶点，找不到则返回-1
{
	if(v!=-1)
	{
		Edge<T,E> *p = NodeTable[v].adj;
		if(p!=NULL)
			return p->dest;
	}
	return -1;
}
template<class T ,class E>
int Graphlnk<T,E>::getNextNeighbor(int v,int w) //获得v的邻接顶点w的下一个邻接顶点
{
	if(v!=-1 )
	{
		Edge<T,E> *p = NodeTable[v].adj;
		while(p!=NULL && p->dest != w)		//寻找邻接顶点w
			p = p->link;
		if(p!=NULL && p->link!=NULL)	//找到w且存在下一个邻接顶带你
			return p->link->dest;
	}
	return -1;
}


template<class T ,class E>
bool Graphlnk<T,E>::insertVertex(const T& vertex)	//插入点
{
	if(numVertices == maxVertices)	return false;	//图已满，插入失败
	NodeTable[numVertices].data = vertex;	
	numVertices++;
	return true;
}

template<class T ,class E>
bool Graphlnk<T,E>::removeVertex(int v)			//删除点
{
	if(numVertices ==1 || v<0 ||v>=numVertices)		return false;
	Edge<T,E> *p ,*s ,*t;
	int  k;
	while(NodeTable[v].adj != NULL)		//删除该顶点，以及与之邻接的顶点中的记录
	{
		p = NodeTable[v].adj;
		k = p->dest;
		s = NodeTable[k].adj;	//以找对称存放的边节点
		t = NULL;
		while (s!=NULL && s->dest!= v) //在对称点的邻接表里面找v，删除掉
		{
			t = s;
			s = s->link;
		}
		if(s!=NULL)
		{
			if(t==NULL) //第一个邻接顶点就是v
				NodeTable[k].adj = s->link;
			else
				t->link = s->link;
			delete s;
		}
		NodeTable[v].adj = p->link;
		delete p;
		numEdges --;
	}
	numVertices--;
	NodeTable[v].data = NodeTable[numVertices].data;
	p = NodeTable[v].adj = NodeTable[numVertices].adj;
	while(p!=NULL)
	{
		s = NodeTable[p->dest].adj;
		while(s!=NULL)
		{
			if(s->dest == numVertices)
			{
				s->dest = v;
				break;
			}
			else 
				s = s->link;
		}
		p = p->link;
	}
	return true;
}

template<class T ,class E>
bool Graphlnk<T,E>::insertEdge(int v1, int v2,E weight)	//插入一条边,若边已存在，或参数不合理，返回false
{
	if(v1>=0 && v1< numVertices && v2>=0 && v2< numVertices)
	{
		Edge<T,E> *q ,*p = NodeTable[v1].adj;
		//先检查该边是否已经存在
		while(p!=NULL && p->dest!= v2)
			p = p->link;
		if(p!=NULL)//找到该边
		{
			cout<<"该边已经存在，插入失败！"<<endl;
			return false;
		}
		p = new Edge<T,E>;
		q = new Edge<T,E>;
		p->dest = v2;
		p->cost = weight;
		p->link = NodeTable[v1].adj; 
		NodeTable[v1].adj = p;	//插入到邻接表表头
		q->dest = v1;
		q->cost = weight;
		q->link = NodeTable[v2].adj;
		NodeTable[v2].adj = q;
		numEdges ++;
		return true;
	}
	return false;
}

template<class T, class E>
bool Graphlnk<T,E>::removeEdge(int v1,int v2)
{
	if(v1 >=0 && v1< numVertices && v2>=0 && v2< numVertices)
	{
		Edge<T,E> *p = NodeTable[v1].adj, *q = NULL ,*s = p;
		while(p!=NULL && p->dest!= v2) //先找该边
		{
			q = p;
			p = p->link;
		}
		if(p!=NULL) //找到该边
		{
			if(p==s)//第一个节点就找到
				NodeTable[v1].adj = p->link;
			else 
				q->link = p->link;
			delete p;
		}
		else return false; //找不到边
		p = NodeTable[v2].adj; q= NULL; s = p;
		while(p!=NULL && p->dest!=v1)
		{
			q = p;
			p = p->link;
		}
		if(p==s)
			NodeTable[v2].adj = p->link;
		else
			q->link = p->link;
		delete p;
		return true;
	}
	return false;
}

template<class T ,class E>
void Graphlnk<T,E>::inputGraph()
{
	//通过从输入流对象in输入n的顶点和e条五项边的信息建立邻接矩阵表示的图G。邻接矩阵初始化工作在构造函数完成
	int i,j,k,m,n;
	T e1,e2;
	E weight;
	cout<<"请输入顶点数和边数(空格隔开):"<<endl;
	cin >> n >> m;	//输入点数n的边数m
	cout<<"请依次输入顶点:"<<endl;
	for(i=0;i<n;i++)//输入顶点，建立顶点表
	{
		cin>>e1;
		this->insertVertex(e1);
		//G.insertVertex(e1);
	}
	cout<<"请依次输入边，形如 v1 v2 weight ："<<endl;
	i=0;
	while(i<m)
	{
		cin >> e1>>e2>>weight;
		j = this->getVertexPos(e1);//查顶点号
		k = this->getVertexPos(e2);
		if(j==-1 || k==-1)
		{
			cout<<"边两端点信息有误，重新输入！"<<endl;
		}
		else
		{
			if(this->insertEdge(j,k,weight))
				i++;
		}
	}

}
template<class T ,class E>
void Graphlnk<T,E>::outputGraph()
{
	//输出图的所有顶点和边信息
	int i,j,n,m;
	T e1,e2;
	E weight;
	n = this->NumberOfVertices();	 //点数
	m = this->NumberOfEdges();		//边数
	cout<<"顶点数的边数为：";
	cout<<n<<","<<m<<endl;		//输出点数和边数
	cout<<"各边依次为："<<endl;
	for(i=0;i<n;i++)
	{
		for(j=i+1;j<n;j++)
		{
			weight = this->getWeight(i,j);
			if(weight>0 && weight< maxWeight)
			{
				e1 = this->getValue(i);
				e2 = this->getValue(j);
				cout<<"("<<e1<<","<<e2<<","<<weight<<")"<<endl;
			}
		}
	}
}

• 基于邻接矩阵实现边集存储的派生子类Graphmtx定义及实现：

//Filename : Graphmtc.h

#include "Graph.h"
#include<iostream>
using namespace std;

template<class T,class E>
class Graphmtx: public Graph<T,E>
{

public:
	Graphmtx(int sz=DefaultVertices);							//构造
	~Graphmtx()													//析构
	{	
		delete []VerticesList;
		delete []Edge;
	}
	T getValue(int i)											//取顶点i的值，若i不合理返回NULL
	{
		if(i>=0 && i<numVertices) return VerticesList[i];
		else return NULL;
	}
	E getWeight(int v1,int v2)									//取边（v1,v2）的权值，不合理返回0
	{
		if(v1!=-1 && v2!=-1)
			return Edge[v1][v2];
		else
			return 0;
	}
	int getFirstNeighbor(int v);
	int getNextNeighbor(int v,int w);
	bool insertVertex(const T& vertex);
	bool insertEdge(int v1, int v2,E cost);
	bool removeVertex(int v);
	bool removeEdge(int v1,int v2);
	void inputGraph();
	void outputGraph();
	int getVertexPos(T vertex)									//给出顶点在图中的位置
	{
		for(int i=0;i<numVertices;i++)
			if(VerticesList[i]==vertex)return i;
		return -1;												//找不到返回-1
	}
private:
	T *VerticesList;											//顶点表
	E * *Edge;													//邻接矩阵
	
};

template<class T ,class E>
Graphmtx<T,E>::Graphmtx(int sz)	//构造函数
{
	maxVertices = sz;
	numVertices = 0;
	numEdges = 0;
	int i,j;
	VerticesList = new T[maxVertices];
	Edge = new E *[maxVertices];
	for (i=0;i<maxVertices;i++)
		Edge[i]= new E[maxVertices];
	for(i=0;i<maxVertices;i++)
		for(j=0;j<maxVertices;j++)
			Edge[i][j] = (i==j) ? 0 :maxWeight;
}

template<class T,class E>
int Graphmtx<T,E>::getFirstNeighbor(int v)//返回v的第一个邻接顶点的位置
{
	if(v!=-1)
	{
		for(int col =0;col<maxVertices;col++)
			if(Edge[v][col]>0 && Edge[v][col]<maxWeight)
				return col;
	}
	return -1;
}

template<class T,class E>
int Graphmtx<T,E>::getNextNeighbor(int v,int w)//返回v的邻接顶点w的下一个邻接顶点
{
	if (v!=-1 && w!=-1)
	{
		for (int col =w+1;col<maxVertices;col++)
		{
			if(Edge[v][col]>0 &&Edge[v][col]<maxWeight)
				return col;
		}
	}
	return -1;
}

template<class T ,class E>
bool Graphmtx<T,E>::insertVertex(const T& vertex) //插入一个顶点
{
	if(numVertices == maxVertices)return false; //顶点表已满，返回false
	VerticesList[numVertices++]=vertex;
	return true;
}

template<class T ,class E>
bool Graphmtx<T,E>::insertEdge(int v1, int v2,E cost)//插入一条边
{
	if(v1>-1 &&  v1<numVertices && v2>-1  && v2<numVertices) //检查条件
	{
		if( Edge[v1][v2]==maxWeight)
		{
			Edge[v1][v2]=Edge[v2][v1] = cost;
			numEdges++;
			return true;
		}
		else
		{
			cout<<"该边已存在，添加失败"<<endl;
			return false;
		}	
	}
	else return false;
}

template<class T,class E>
bool Graphmtx<T,E>::removeVertex(int v)				//删除一个顶点
{
	if(v<0 ||v>numVertices)	return false;			//v不在图中
	if(numVertices==1)	return false;				//只剩一个顶点，不删除
	int i,j;
	VerticesList[v]=VerticesList[numVertices-1];	//顶点表中删除
	for( i=0;i<numVertices;i++)						//边数调整
		if(Edge[i][v]>0 && Edge[i][v]<maxWeight)
			numEdges--;
	for(i=0;i<numVertices;i++)
		Edge[i][v]=Edge[i][numVertices-1];
	numVertices--;									//顶点数调整
	for(j=0;j<numVertices;j--)
		Edge[v][j]=Edge[numVertices][j];
	return true;
}

template<class T,class E>
bool Graphmtx<T,E>::removeEdge(int v1,int v2)		//删除边
{
	if(v1>-1 && v1<numVertices && v2>-1 &&v2<numVertices && Edge[v1][v2]>0 && Edge[v1][v2]<maxWeight)
	{
		Edge[v1][v2] = Edge[v1][v2] = maxWeight;
		numEdges--;
		return true;
	}
	else return false;
};

template<class T ,class E>
void Graphmtx<T,E>::inputGraph()
{
	//通过从输入流对象in输入n的顶点和e条五项边的信息建立邻接矩阵表示的图G。邻接矩阵初始化工作在构造函数完成
	int i,j,k,m,n;
	T e1,e2;
	E weight;
	cout<<"请输入顶点数和边数(空格隔开):"<<endl;
	cin >> n >> m;	//输入点数n的边数m
	cout<<"请依次输入顶点:"<<endl;
	for(i=0;i<n;i++)//输入顶点，建立顶点表
	{
		cin>>e1;
		this->insertVertex(e1);
		//G.insertVertex(e1);
	}
	cout<<"请依次输入边，形如 v1 v2 weight ："<<endl;
	i=0;
	while(i<m)
	{
		cin >> e1>>e2>>weight;
		j = this->getVertexPos(e1);//查顶点号
		k = this->getVertexPos(e2);
		if(j==-1 || k==-1)
		{
			cout<<"边两端点信息有误，重新输入！"<<endl;
		}
		else
		{
			if(this->insertEdge(j,k,weight))
				i++;
		}
	}

}
template<class T ,class E>
void Graphmtx<T,E>::outputGraph()
{
	//输出图的所有顶点和边信息
	int i,j,n,m;
	T e1,e2;
	E weight;
	n = this->NumberOfVertices();	 //点数
	m = this->NumberOfEdges();		//边数
	cout<<"顶点数的边数为：";
	cout<<n<<","<<m<<endl;		//输出点数和边数
	cout<<"各边依次为："<<endl;
	for(i=0;i<n;i++)
	{
		for(j=i+1;j<n;j++)
		{
			weight = this->getWeight(i,j);
			if(weight>0 && weight< maxWeight)
			{
				e1 = this->getValue(i);
				e2 = this->getValue(j);
				cout<<"("<<e1<<","<<e2<<","<<weight<<")"<<endl;
			}
		}
	}
}

• 编写测试程序加以简单测试：

//Filename： test.cpp
#include "Graphmtx.h"
#include "Graphlnk.h"

void test_Graphmtx()
{
	char ch1,ch2;
	int weight;
	Graphmtx<char,int> g(30);
	g.inputGraph();		//构造图
	g.outputGraph();	//显示图

	cout<<"顶点数和边数："<<g.NumberOfVertices()<<" "<<g.NumberOfEdges()<<endl;
	cout<<"查看第一个邻接顶点:";
	cin>>ch1;
	cout<<g.getValue(g.getFirstNeighbor(g.getVertexPos(ch1)))<<endl;
	cout<<"查看后一个邻接顶点:";
	cin>>ch1>>ch2;
	cout<<g.getValue(g.getNextNeighbor(g.getVertexPos(ch1),g.getVertexPos(ch2)))<<endl;

	cout<<"插入顶点:";
	cin >>ch1;
	g.insertVertex(ch1);  //插入点
	cout<<"插入边："<<endl;
	cin >>ch1>>ch2>>weight;
	g.insertEdge(g.getVertexPos(ch1),g.getVertexPos(ch2),weight);//插入边
	g.outputGraph();

	cout<<"删除边：";
	cin >>ch1>>ch2;
	g.removeEdge(g.getVertexPos(ch1),g.getVertexPos(ch2)); //删除边

	cout<<"删除点：";
	cin>>ch1;
	g.removeVertex(g.getVertexPos(ch1));
	g.outputGraph();
	
}

void test_Graphlnk()
{
	cout<<"-----------Grapglnk Test-----------"<<endl;
	char ch1,ch2;
	int weight;
	Graphmtx<char,int> g(30);
	g.inputGraph();		//构造图
	g.outputGraph();	//显示图

	cout<<"顶点数和边数："<<g.NumberOfVertices()<<" "<<g.NumberOfEdges()<<endl;
	cout<<"查看第一个邻接顶点:";
	cin>>ch1;
	cout<<g.getValue(g.getFirstNeighbor(g.getVertexPos(ch1)))<<endl;
	cout<<"查看后一个邻接顶点:";
	cin>>ch1>>ch2;
	cout<<g.getValue(g.getNextNeighbor(g.getVertexPos(ch1),g.getVertexPos(ch2)))<<endl;

	cout<<"插入顶点:";
	cin >>ch1;
	g.insertVertex(ch1);  //插入点
	cout<<"插入边："<<endl;
	cin >>ch1>>ch2>>weight;
	g.insertEdge(g.getVertexPos(ch1),g.getVertexPos(ch2),weight);//插入边
	g.outputGraph();

	cout<<"删除边：";
	cin >>ch1>>ch2;
	g.removeEdge(g.getVertexPos(ch1),g.getVertexPos(ch2)); //删除边

	cout<<"删除点：";
	cin>>ch1;
	g.removeVertex(g.getVertexPos(ch1));
	g.outputGraph();
}

//Filename : main.cpp
#include "Graphmtx.h"
extern void test_Graphmtx();
extern void test_Graphlnk();

int main()
{
	
	test_Graphmtx();
	test_Graphlnk();
	system("pause");
	return 0;
}

经测试，以上代码可以正确运行。