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[数据结构]图

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数据结构——图

定义：

图是由顶点的又穷非空集合和顶点之间边的集合，通常表示为：G(V,E)，G是一个图，V是图G中顶点的集合，E是图G中边的集合。

有向图：两顶点之间的边没有方向

无向图：两顶点之间的边有方向

带权的图（网）：两顶点之间的连线具有权重

图的存储结构：

1、邻接矩阵

图的邻接矩阵存储是用两个数组表示图，其中一个一维数组表示顶点，一个二维数组表示图中边的关系（为0表示两个顶点不相连，不为0表示两顶点之间边的权重）

struct Graph
{
    //顶点
    int vertexes[MAX];
    //边
    int arc[MAX][MAX];
    //顶点总数、边的总数
    int sum_vertexes,sum_edges;
};

2、邻接表

邻接表是数组与链表相结合来存储图：

一个一维数组表示顶点，然后每个顶点的所有相连的顶点构成一个线性表。

struct graphNode
{
    int name;
    int weight;
    struct graphNode *next;
};

struct vertexesNode
{
    int name;
    graphNode *firstNode;
};

struct Graph
{
    vertexesNode vertexes[MAX];
    int sum_vertexes,sum_edges;
};

图的建立：

1、邻接矩阵

<span style="font-size:12px;">//无向表
void createGraph_1(Graph &graph)
{
    cout<<"输入顶点的个数"<<endl;
    cin>>graph.sum_vertexes;
    cout<<"输入边的个数"<<endl;
    cin>>graph.sum_edges;
    for(int i=0; i<graph.sum_vertexes; i++)
    {
        cout<<"输入第"<<i+1<<"个顶点的标识"<<endl;
        cin>>graph.vertexes[i];
    }
    //矩阵初始化
    for(int i=0; i<graph.sum_vertexes; i++)
    {
        for(int j=0; j<graph.sum_vertexes; j++)
        {
            graph.arc[i][j]=0;
        }
    }
    for(int i=0; i<graph.sum_edges; i++)
    {
        int x;
        int y;
        int weight;
        cout<<"输入第"<<i+1<<"条边的两个定点和该边的权重"<<endl;
        cin>>x>>y>>weight;
        graph.arc[x][y]=graph.arc[y][x]=weight;
    }

}
//有向表
void createGraph_2(Graph &graph)
{
    cout<<"输入顶点的个数"<<endl;
    cin>>graph.sum_vertexes;
    cout<<"输入边的个数"<<endl;
    cin>>graph.sum_edges;
    for(int i=0; i<graph.sum_vertexes; i++)
    {
        cout<<"输入第"<<i+1<<"个顶点的标识"<<endl;
        cin>>graph.vertexes[i];
    }
    //矩阵初始化
    for(int i=0; i<graph.sum_vertexes; i++)
    {
        for(int j=0; j<graph.sum_vertexes; j++)
        {
            graph.arc[i][j]=0;
        }
    }
    for(int i=0; i<graph.sum_edges; i++)
    {
        int x;
        int y;
        int weight;
        cout<<"输入第"<<i+1<<"条弧的弧尾 弧头 权重"<<endl;
        cin>>x>>y>>weight;
        graph.arc[x][y]=weight;
    }

}</span>

2、邻接表

<span style="font-size:12px;">//无向图
void createGraph_1(Graph &graph)
{
    cout<<"输入图的顶点的个数"<<endl;
    cin>>graph.sum_vertexes;
    cout<<"输入图的边的个数"<<endl;
    cin>>graph.sum_edges;
    for(int i=0; i<graph.sum_vertexes; i++)
    {
        cout<<"输入第"<<i+1<<"个顶点的标识"<<endl;
        cin>>graph.vertexes[i].name;
        graph.vertexes[i].firstNode=NULL;
    }
    for(int i=0; i<graph.sum_edges; i++)
    {
        int x,y,weight;
        int x_pos,y_pos;
        cout<<"输入第"<<i+1<<"条边的两个顶点和权重"<<endl;
        cin>>x>>y>>weight;
        for(int j=0; j<graph.sum_vertexes; j++)
        {
            if(graph.vertexes[j].name==x)
            {
                x_pos=j;
                break;
            }
        }
        for(int j=0; j<graph.sum_vertexes; j++)
        {
            if(graph.vertexes[j].name==y)
            {
                y_pos=j;
                break;
            }
        }
        graphNode *temp_x=new graphNode();
        temp_x->weight=weight;
        temp_x->name=y;
        temp_x->next=graph.vertexes[x_pos].firstNode;
        graph.vertexes[x_pos].firstNode=temp_x;
        graphNode *temp_y=new graphNode();
        temp_y->weight=weight;
        temp_y->name=x;
        temp_y->next=graph.vertexes[y_pos].firstNode;
        graph.vertexes[y_pos].firstNode=temp_y;

    }
}

//有向图
void createGraph_2(Graph &graph)
{
    cout<<"输入图的顶点的个数"<<endl;
    cin>>graph.sum_vertexes;
    cout<<"输入图的边的个数"<<endl;
    cin>>graph.sum_edges;
    for(int i=0; i<graph.sum_vertexes; i++)
    {
        cout<<"输入第"<<i+1<<"个顶点的标识"<<endl;
        cin>>graph.vertexes[i].name;
        graph.vertexes[i].firstNode=NULL;
    }
    for(int i=0; i<graph.sum_edges; i++)
    {
        int x,y,weight;
        int x_pos,y_pos;
        cout<<"输入第"<<i+1<<"条弧的弧尾 弧首 权重"<<endl;
        cin>>x>>y>>weight;
        for(int j=0; j<graph.sum_vertexes; j++)
        {
            if(graph.vertexes[j].name==x)
            {
                x_pos=j;
                break;
            }
        }
        graphNode *temp_x=new graphNode();
        temp_x->weight=weight;
        temp_x->name=y;
        temp_x->next=graph.vertexes[x_pos].firstNode;
        graph.vertexes[x_pos].firstNode=temp_x;

    }
}</span>

图的遍历：

1、深度优先遍历

a)邻接矩阵

<span style="font-size:12px;">void DFS(Graph graph,int i)
{
    visited[i]=1;
    //do something
    cout<<graph.vertexes[i]<<endl;
    for(int j=0; j<graph.sum_vertexes; j++)
    {
        if(graph.arc[i][j]!=0&&visited[j]==0)
        {
            DFS(graph,j);

        }
    }
}

void DFSTraverse(Graph graph)
{
    for(int i=0; i<graph.sum_vertexes; i++)
        visited[i]=0;
    for(int i=0; i<graph.sum_vertexes; i++)
        if(visited[i]==0)
            DFS(graph,i);
}</span>

b)邻接表

<span style="font-size:12px;">void DFS(Graph graph,int i)
{
    visited[i]=1;
    cout<<graph.vertexes[i].name<<endl;
    graphNode *node=graph.vertexes[i].firstNode;
    while(node)
    {
        if(visited[node->name]==0)
            DFS(graph,node->name);
        node=node->next;
    }
}

void DFSTraverse(Graph graph)
{
    for(int i=0; i<graph.sum_vertexes; i++)
        visited[i]=0;
    for(int i=0; i<graph.sum_vertexes; i++)
        if(visited[i]==0)
            DFS(graph,i);
}
</span>

2、宽度优先遍历

a)邻接矩阵

<span style="font-size:12px;">oid BFSTraverse(Graph graph)
{
    queue<int> q;
    for(int i=0; i<graph.sum_vertexes; i++)
        visited[i]=0;
    for(int i=0; i<graph.sum_vertexes; i++)
    {
        if(visited[i]==0)
        {
            visited[i]=1;
            //do something
            cout<<graph.vertexes[i]<<endl;

            q.push(graph.vertexes[i]);
            while(!q.empty())
            {
                int temp=q.front();
                q.pop();
                for(int j=0; j<graph.sum_vertexes; j++)
                {
                    if(graph.arc[temp][j]!=0&&visited[j]==0)
                    {
                        visited[j]=1;
                        //do something
                        cout<<graph.vertexes[j]<<endl;
                        q.push(graph.vertexes[j]);
                    }
                }
            }
        }
    }

}</span>

b)邻接表

<span style="font-size:12px;">void BFSTraverse(Graph graph)
{
    queue<int> q;
    for(int i=0; i<graph.sum_vertexes; i++)
        visited[i]=0;
    for(int i=0; i<graph.sum_vertexes; i++)
    {
        if(visited[i]==0)
        {
            visited[i]=1;
            //do something
            cout<<graph.vertexes[i].name<<endl;
            q.push(graph.vertexes[i].name);
            while(!q.empty())
            {
                int temp=q.front();
                q.pop();
                graphNode *node=graph.vertexes[temp].firstNode;
                while(node)
                {
                    if(visited[node->name]==0)
                    {
                        q.push(node->name);
                        //do something
                        cout<<node->name<<endl;
                        visited[node->name]=1;
                    }
                    node=node->next;
                }

            }
        }
    }
}</span>

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