数据结构基础(20) --图的存储结构

图的结构定义

图是由一个顶点集 V 和一个弧集 E构成的数据结构。

Graph = (V , E )

其中，E ＝ {<v,w>| v,w∈V 且 P(v,w)} <v,w>表示从 v 到 w 的一条弧，并称 v 为弧尾，w 为弧头。谓词 P(v,w) 定义了弧 <v,w>的意义或信息。

由顶点集和边集构成的图称作无向图。

如果”弧”是有方向的，则称由顶点集和弧集构成的图为有向图。

邻接矩阵

定义:矩阵的元素为

有向图的邻接矩阵为非对称矩阵, 而无向图的邻接矩阵为对称矩阵;

//无向图的邻接矩阵
const int MAX_VERTS = 20;
//顶点
template <typename Type>
class Vertex
{
public:
    Vertex(const Type &_node = Type())
        : node(_node) {}

private:
    Type node;
};
//图
template <typename Type>
class Graph
{
public:
    Graph();
    ~Graph();

    void addVertex(const Type &vertex);
    void addEdge(int start, int end);
    void printMatrix();

private:
    Vertex<Type>* vertexList[MAX_VERTS];
    int nVerts;
    int adjMatrix[MAX_VERTS][MAX_VERTS];
};

template <typename Type>
Graph<Type>::Graph():nVerts(0)
{
    for (int i = 0; i < MAX_VERTS; ++i)
        for (int j = 0; j < MAX_VERTS; ++j)
            adjMatrix[i][j] = 0;
}
template <typename Type>
Graph<Type>::~Graph()
{
    for (int i = 0; i < nVerts; ++i)
        delete vertexList[i];
}

template <typename Type>
void Graph<Type>::addVertex(const Type &vertex)
{
    vertexList[nVerts ++] = new Vertex<Type>(vertex);
}
template <typename Type>
void Graph<Type>::addEdge(int start, int end)
{
    //无向图
    adjMatrix[start][end] = 1;
    adjMatrix[end][start] = 1;
}

template <typename Type>
void Graph<Type>::printMatrix()
{
    for (int i = 0; i < nVerts; ++i)
    {
        for (int j = 0; j < nVerts; ++j)
            cout << adjMatrix[i][j] << ‘ ‘;
        cout << endl;
    }
}

//测试代码
int main()
{
    Graph<char> g;
    g.addVertex(‘A‘);   //0
    g.addVertex(‘B‘);   //1
    g.addVertex(‘C‘);   //2
    g.addVertex(‘D‘);   //3
    g.addVertex(‘E‘);   //4

    g.addEdge(0, 1);    //A-B
    g.addEdge(0, 3);    //A-D
    g.addEdge(1, 0);    //B-A
    g.addEdge(1, 4);    //B-E
    g.addEdge(2, 4);    //C-E
    g.addEdge(3, 0);    //D-A
    g.addEdge(3, 4);    //D-E
    g.addEdge(4, 1);    //E-B
    g.addEdge(4, 2);    //E-C
    g.addEdge(4, 3);    //E-D

    g.printMatrix();

    return 0;
}

邻接表

注意:在有向图的邻接表中不易找到指向该顶点的弧。

//无向图的邻接表
template <typename Type>
class Graph
{
public:
    Graph(int _size = 10);
    ~Graph();

    void addVertex(const Type &vertex);
    void addEdge(int start, int end);
    void printVertex();
    void printAdjList();

private:
    Type *vertexList;
    list<int> *headNode;
    int size;
    int nVertex;
};

template <typename Type>
Graph<Type>::Graph(int _size):size(_size), nVertex(0)
{
    vertexList = new Type[size];
    headNode = new list<int>[size];
}
template <typename Type>
Graph<Type>::~Graph()
{
    delete []vertexList;
    delete []headNode;
}

template <typename Type>
void Graph<Type>::addVertex(const Type &vertex)
{
    vertexList[nVertex ++] = vertex;
}
template <typename Type>
void Graph<Type>::addEdge(int start, int end)
{
    headNode[start].push_back(end);
}

template <typename Type>
void Graph<Type>::printVertex()
{
    cout << vertexList[0];
    for (int i = 1; i < nVertex; ++i)
        cout << ‘ ‘ << vertexList[i];
    cout << endl;
}
template <typename Type>
void Graph<Type>::printAdjList()
{
    for (int i = 0; i < nVertex; ++i)
    {
        cout << i;
        for (list<int>::iterator iter = headNode[i].begin();
                iter != headNode[i].end();
                ++iter)
            cout << " -> " << *iter;
        cout << endl;
    }
}

//测试代码
int main()
{
    Graph<char> g;
    g.addVertex(‘A‘);   //0
    g.addVertex(‘B‘);   //1
    g.addVertex(‘C‘);   //2
    g.addVertex(‘D‘);   //3
    g.addVertex(‘E‘);   //4
    g.printVertex();

    g.addEdge(0, 1);    //A-B
    g.addEdge(0, 3);    //A-D
    g.addEdge(1, 0);    //B-A
    g.addEdge(1, 4);    //B-E
    g.addEdge(2, 4);    //C-E
    g.addEdge(3, 0);    //D-A
    g.addEdge(3, 4);    //D-E
    g.addEdge(4, 1);    //E-B
    g.addEdge(4, 2);    //E-C
    g.addEdge(4, 3);    //E-D

    g.printAdjList();

    return 0;
}