有向图的可达性

        有向图中，我们可以使用深度优先遍历方式来测试有向图的可达性。这样给出一个节点，我们就可以知道是否存在一个路径从起点(就是从该点进行深度优先遍历)到达该节点。它解决了单点连通性的问题，使用用例可以判定其他顶点和给定的起点是否连通。

        在存储图的数据结构中，我们使用邻接表来存储有向图。下图就是一个存储的示例：

源代码如下：

#include <iostream>#include <vector>#include <algorithm>using namespace std;class Digraph{public:Digraph(size_t nvertax) : arr(nvertax), vertaxs(nvertax), edges(0) {}void addEdge(size_t a, size_t b);void removeEdge(size_t a, size_t b);void addVertax(size_t v);void removeVertax(size_t v);Digraph reverse() const;vector<size_t> adj(size_t v) const;void showAdj(size_t v) const;void showDigraph() const;size_t edge() const { return edges; }size_t vertax() const { return vertaxs; }size_t arrSize() const { return arr.size(); }private:vector<vector<size_t>> arr; // 邻接表size_t vertaxs; // 顶点个数size_t edges;// 边的个数};/// 使用深度优先遍历测试有向图的可达性class DirectedDFS{public:DirectedDFS(const Digraph &graph, size_t _start) : marked(graph.arrSize()), start(_start){for (size_t i = 0; i < graph.arrSize(); i++)marked[i] = false;dfs(graph, start);}/// 节点v是可达的吗？起点startbool isDirected(size_t v) const { return (v < marked.size()) ? marked[v] : false; }private:void dfs(const Digraph &graph, size_t v){marked[v] = true;vector<size_t> vec = graph.adj(v);for (size_t i = 0; i < vec.size(); i++){if (!marked[vec[i]])dfs(graph, vec[i]);}}vector<bool> marked;size_t start; // 起点};int main(void){Digraph graph(5);graph.addEdge(0, 1);graph.addEdge(0, 2);graph.addEdge(1, 3);graph.addEdge(2, 3);//graph.addEdge(3, 4);graph.addEdge(4, 2);cout << graph.vertax() << endl;cout << graph.edge() << endl;graph.showDigraph();cout << endl;DirectedDFS dir(graph, 0);cout << "0 -> 3: ";if (dir.isDirected(3))cout << "true" << endl;elsecout << "false" << endl;cout << "0 -> 4: ";if (dir.isDirected(4))cout << "true" << endl;elsecout << "false" << endl;system("pause");return 0;}void Digraph::addEdge(size_t a, size_t b){if (!(a < vertax() && b < vertax()))return;arr[a].push_back(b);edges++;}void Digraph::removeEdge(size_t a, size_t b){if (!(a < vertax() && b < vertax()))return;vector<size_t>::iterator iter = find(arr[a].begin(), arr[a].end(), b);if (iter != arr[a].end()){arr[a].erase(iter);edges--;}}void Digraph::addVertax(size_t v){if (v != arrSize())return;arr.push_back(vector<size_t>());vertaxs++;}void Digraph::removeVertax(size_t v){if (v >= arrSize())return;vector<size_t>::iterator iter;for (size_t i = 0; i < arrSize(); i++){if (i == v){edges -= arr[i].size(); // 减去头部是v的边的个数arr[i].clear();vertaxs--;}else{iter = find(arr[i].begin(), arr[i].end(), v);if (iter != arr[i].end()){arr[i].erase(iter);edges--;}}}}Digraph Digraph::reverse() const{Digraph graph(vertax());vector<size_t> vec;for (size_t i = 0; i < arrSize(); i++){vec = adj(i);for (size_t j = 0; j < vec.size(); j++){graph.addEdge(vec[j], i); // 取得该图的反向图}}return graph;}vector<size_t> Digraph::adj(size_t v) const{if (v < arrSize())return arr[v];elsevector<size_t>();}void Digraph::showAdj(size_t v) const{if (v >= arrSize())return;vector<size_t> vec = adj(v);for (size_t i = 0; i < vec.size(); i++)cout << vec[i] << " ";cout << endl;}void Digraph::showDigraph() const{for (size_t i = 0; i < arr.size(); i++){cout << i << ": ";showAdj(i);}}

参考资料：

        1、https://github.com/luoxn28/algorithm_data_structure （graph文件夹下有常用的关于图的各种算法）

        2、《算法 第4版》中关于图的章节