有向图是否有环

        在和有向图相关的实际应用中，有向环特别的重要。其中就可以解决一组任务的执行顺序问题，因为有的任务必须在另一个任务完成之后才能开始进行，此时就需要判断图中是否有环，有环的话是不能对这些任务的排序的。

 左图就是有环图

        下图没有形成环，就可以进行排序操作。

源代码示例：

#include <iostream>#include <vector>#include <stack>#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};/// 使用深度优先遍历在有向图中寻找有向环class DirectedCycle{public:DirectedCycle(const Digraph &graph) : marked(graph.arrSize()), edgeto(graph.arrSize()), on_stack(graph.arrSize()){for (size_t i = 0; i < graph.arrSize(); i++)marked[i] = on_stack[i] = false;for (size_t i = 0; i < graph.arrSize(); i++){if (!marked[i])dfs(graph, i);}}bool hasCycle() const { return !sta.empty(); }stack<size_t> cycle() const { return sta; }private:void dfs(const Digraph &graph, size_t v){on_stack[v] = true;marked[v] = true;vector<size_t> vec = graph.adj(v);for (size_t i = 0; i < vec.size(); i++){if (this->hasCycle())return;else if (!marked[vec[i]]){edgeto[vec[i]] = v;dfs(graph, vec[i]);}else if (on_stack[vec[i]]){for (size_t x = v; x != vec[i]; x = edgeto[x])sta.push(x);sta.push(vec[i]);sta.push(v);}}on_stack[v] = false;}vector<bool> marked;vector<size_t> edgeto;vector<bool> on_stack;stack<size_t> sta;};int main(void){Digraph graph(4);graph.addEdge(0, 1);graph.addEdge(1, 2);graph.addEdge(2, 3);graph.addEdge(3, 1);cout << graph.vertax() << endl;cout << graph.edge() << endl;graph.showDigraph();cout << endl;DirectedCycle cycle(graph);stack<size_t> sta;if (cycle.hasCycle()){cout << "graph has cycle" << endl;sta = cycle.cycle();while (!sta.empty()){cout << sta.top() << " ";sta.pop();}cout << endl;}elsecout << "graph hasn't cycle" << 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 （里面有常用的数据结构源代码，图相关算法在graph文件夹中）

       2、《算法 第4版》 图有关章节