Topological Sorting拓扑排序

定义：

Topological Sorting is a method of arranging the vertices in a directed acyclic graph (DAG有向无环图) as a sequence, such that for every directed edge(u,v), 即u-> v, vertex u comes before v in the ordering

注意：

• A topological ordering is possible if and only if the graph has no directed cycles
• Topological orderings are NOT unique

应用场景：

• build systems:  a program cannot be built unless its dependencies are first built
• pre-requisite problems: a course cannot be selected unless its pre-requisite courses are taken
• dependency problems
• task schedule

时间复杂度:

O(V+E)

Topological Sort(){
    1. Call DFS to compute finish time f[v] for each vertex
    2. At each vertex is finished, insert it onto the front of a linked list
    3. Return the linked list of vertices
}

内部原理：

比如从任意一个vertex出发（比如，随便选个NodeH）， 来DFS搜索

从NodeH出发，DFS可以走 NodeH-> NodeJ

从NodeJ出发， DFS可以 NodeJ -> NodeM

发现NodeM后面没有元素了， 我们把NodeM放到Stack里面（基于Stack的FILO特性）

从NodeM往回走，到NodeJ， 从NodeJ出发，DFS可以NodeJ-> NodeL

发现NodeL后面没有元素了， 我们把NodeL放到Stack里面

以此类推，DFS走完整个有向无环图，并用Stack保证了u-> v, vertex u comes before v in the ordering

综上，

1. 从有向无环图的任意点出发

2. 进行DFS搜索， 直到搜到当前元素已经"nowhere left to go"了

3. 将该元素放到Stack里，并将该元素mark为visited

4. 从该元素backtrack，继续DFS