（译）割点

注：本文翻译自http://www.geeksforgeeks.org/articulation-points-or-cut-vertices-in-a-graph/。如有翻译错误请指正。

 

一个无向联通图（undirected connected graph）中的顶点，当且仅当去掉它会使图不再联通，就是割点（articulation point/cut vertex）。割点表示一个连通网络的脆弱性——一个点出问题会将整个网络分成2个或更多个部分。在设计可靠的网络中，它们很有用。

 

对于一个无向联通图，一个割点是一个去掉之后会增加连通分量（connected component）数量的点。

 

下面是一些例子，割点用红色圆圈圈出：

 

 

怎么找到一个图里的全部割点？

 

一个简单方法就是一个接一个地去掉所有顶点，然后看去掉该点是否导致图不再连通。下面是这种简单方法的步骤：

 

对于每个顶点v

　　a) 将v从图中去掉

　　b) 看图是否仍是连通的（既可以使用BFS，也可以使用DFS）

　　c) 将v重新加入图

 

上述方法的时间复杂度为O(V*(V+E))，如果使用邻接表存储图。可以更好吗？

 

寻找所有割点的O(V+E)算法

 

方法是使用DFS（depth first search，深度优先搜索）。在DFS中，我们用树的形式跟踪顶点，叫做DFS树。在DFS树中，一个顶点u是另一个顶点v的父节点（parent），如果v已经被u发现（很明显v在图上与u相邻）。在DFS树中，如果以下两个条件之一成立，则u是割点：

 

1) u是DFS树的根节点，并且有两个以上的子节点。

 

2) u不是DFS树的根节点，它有一个子节点v，并且以v为根节点的子树中，没有一个顶点有通向任意一个u的祖先的回边。

 

下面的图片显示了与上面一样的概念，还有一点补充，就是DFS树中的叶节点不可能是割点。

 

 

对于给定的图，我们用DFS遍历和附加代码，寻找割点（articulation points，APs）。在DFS遍历中，我们维护一个parent[]数组，parent[u]存储u的父节点。在上面提及的两种情况中，第一种情况很好检测。对于每个顶点，数出其子节点数量。如果目前访问过的顶点u是根节点（parent[u]为NIL），并且有两个以上的子节点，输出。

 

第二种情况怎么处理？第二种情况更复杂。我们维护一个disc[]数组，存储一个顶点被发现的时刻（译者注：即从根节点开始，该点是第几个被访问的）。对于每个节点u，我们需要找到最早访问过的顶点（访问时刻最低的顶点），且该顶点可以从以u为根节点的子树（的某个顶点）到达。因此我们维护一个额外的数组low[]，定义如下。

 

low[u] = min(disc[u], disc[w])

 

w为u的祖先，并且存在一条回边，可以从u的后代到达w。

 

下面是用来寻找割点的Tarjan算法的C++、Java和Python实现。

 

C++

 

// A C++ program to find articulation points in an undirected graph
#include<iostream>
#include <list>
#define NIL -1
using namespace std;
 
// A class that represents an undirected graph
class Graph
{
    int V;    // No. of vertices
    list<int> *adj;    // A dynamic array of adjacency lists
    void APUtil(int v, bool visited[], int disc[], int low[], 
                int parent[], bool ap[]);
public:
    Graph(int V);   // Constructor
    void addEdge(int v, int w);   // function to add an edge to graph
    void AP();    // prints articulation points
};
 
Graph::Graph(int V)
{
    this->V = V;
    adj = new list<int>[V];
}
 
void Graph::addEdge(int v, int w)
{
    adj[v].push_back(w);
    adj[w].push_back(v);  // Note: the graph is undirected
}
 
// A recursive function that find articulation points using DFS traversal
// u --> The vertex to be visited next
// visited[] --> keeps tract of visited vertices
// disc[] --> Stores discovery times of visited vertices
// parent[] --> Stores parent vertices in DFS tree
// ap[] --> Store articulation points
void Graph::APUtil(int u, bool visited[], int disc[], 
                                      int low[], int parent[], bool ap[])
{
    // A static variable is used for simplicity, we can avoid use of static
    // variable by passing a pointer.
    static int time = 0;
 
    // Count of children in DFS Tree
    int children = 0;
 
    // Mark the current node as visited
    visited[u] = true;
 
    // Initialize discovery time and low value
    disc[u] = low[u] = ++time;
 
    // Go through all vertices aadjacent to this
    list<int>::iterator i;
    for (i = adj[u].begin(); i != adj[u].end(); ++i)
    {
        int v = *i;  // v is current adjacent of u
 
        // If v is not visited yet, then make it a child of u
        // in DFS tree and recur for it
        if (!visited[v])
        {
            children++;
            parent[v] = u;
            APUtil(v, visited, disc, low, parent, ap);
 
            // Check if the subtree rooted with v has a connection to
            // one of the ancestors of u
            low[u]  = min(low[u], low[v]);
 
            // u is an articulation point in following cases
 
            // (1) u is root of DFS tree and has two or more chilren.
            if (parent[u] == NIL && children > 1)
               ap[u] = true;
 
            // (2) If u is not root and low value of one of its child is more
            // than discovery value of u.
            if (parent[u] != NIL && low[v] >= disc[u])
               ap[u] = true;
        }
 
        // Update low value of u for parent function calls.
        else if (v != parent[u])
            low[u]  = min(low[u], disc[v]);
    }
}
 
// The function to do DFS traversal. It uses recursive function APUtil()
void Graph::AP()
{
    // Mark all the vertices as not visited
    bool *visited = new bool[V];
    int *disc = new int[V];
    int *low = new int[V];
    int *parent = new int[V];
    bool *ap = new bool[V]; // To store articulation points
 
    // Initialize parent and visited, and ap(articulation point) arrays
    for (int i = 0; i < V; i++)
    {
        parent[i] = NIL;
        visited[i] = false;
        ap[i] = false;
    }
 
    // Call the recursive helper function to find articulation points
    // in DFS tree rooted with vertex 'i'
    for (int i = 0; i < V; i++)
        if (visited[i] == false)
            APUtil(i, visited, disc, low, parent, ap);
 
    // Now ap[] contains articulation points, print them
    for (int i = 0; i < V; i++)
        if (ap[i] == true)
            cout << i << " ";
}
 
// Driver program to test above function
int main()
{
    // Create graphs given in above diagrams
    cout << "\nArticulation points in first graph \n";
    Graph g1(5);
    g1.addEdge(1, 0);
    g1.addEdge(0, 2);
    g1.addEdge(2, 1);
    g1.addEdge(0, 3);
    g1.addEdge(3, 4);
    g1.AP();
 
    cout << "\nArticulation points in second graph \n";
    Graph g2(4);
    g2.addEdge(0, 1);
    g2.addEdge(1, 2);
    g2.addEdge(2, 3);
    g2.AP();
 
    cout << "\nArticulation points in third graph \n";
    Graph g3(7);
    g3.addEdge(0, 1);
    g3.addEdge(1, 2);
    g3.addEdge(2, 0);
    g3.addEdge(1, 3);
    g3.addEdge(1, 4);
    g3.addEdge(1, 6);
    g3.addEdge(3, 5);
    g3.addEdge(4, 5);
    g3.AP();
 
    return 0;
}

 

Java

 

// A Java program to find articulation points in an undirected graph
import java.io.*;
import java.util.*;
import java.util.LinkedList;
 
// This class represents an undirected graph using adjacency list
// representation
class Graph
{
    private int V;   // No. of vertices
 
    // Array  of lists for Adjacency List Representation
    private LinkedList<Integer> adj[];
    int time = 0;
    static final int NIL = -1;
 
    // Constructor
    Graph(int v)
    {
        V = v;
        adj = new LinkedList[v];
        for (int i=0; i<v; ++i)
            adj[i] = new LinkedList();
    }
 
    //Function to add an edge into the graph
    void addEdge(int v, int w)
    {
        adj[v].add(w);  // Add w to v's list.
        adj[w].add(v);  //Add v to w's list
    }
 
    // A recursive function that find articulation points using DFS
    // u --> The vertex to be visited next
    // visited[] --> keeps tract of visited vertices
    // disc[] --> Stores discovery times of visited vertices
    // parent[] --> Stores parent vertices in DFS tree
    // ap[] --> Store articulation points
    void APUtil(int u, boolean visited[], int disc[],
                int low[], int parent[], boolean ap[])
    {
 
        // Count of children in DFS Tree
        int children = 0;
 
        // Mark the current node as visited
        visited[u] = true;
 
        // Initialize discovery time and low value
        disc[u] = low[u] = ++time;
 
        // Go through all vertices aadjacent to this
        Iterator<Integer> i = adj[u].iterator();
        while (i.hasNext())
        {
            int v = i.next();  // v is current adjacent of u
 
            // If v is not visited yet, then make it a child of u
            // in DFS tree and recur for it
            if (!visited[v])
            {
                children++;
                parent[v] = u;
                APUtil(v, visited, disc, low, parent, ap);
 
                // Check if the subtree rooted with v has a connection to
                // one of the ancestors of u
                low[u]  = Math.min(low[u], low[v]);
 
                // u is an articulation point in following cases
 
                // (1) u is root of DFS tree and has two or more chilren.
                if (parent[u] == NIL && children > 1)
                    ap[u] = true;
 
                // (2) If u is not root and low value of one of its child
                // is more than discovery value of u.
                if (parent[u] != NIL && low[v] >= disc[u])
                    ap[u] = true;
            }
 
            // Update low value of u for parent function calls.
            else if (v != parent[u])
                low[u]  = Math.min(low[u], disc[v]);
        }
    }
 
    // The function to do DFS traversal. It uses recursive function APUtil()
    void AP()
    {
        // Mark all the vertices as not visited
        boolean visited[] = new boolean[V];
        int disc[] = new int[V];
        int low[] = new int[V];
        int parent[] = new int[V];
        boolean ap[] = new boolean[V]; // To store articulation points
 
        // Initialize parent and visited, and ap(articulation point)
        // arrays
        for (int i = 0; i < V; i++)
        {
            parent[i] = NIL;
            visited[i] = false;
            ap[i] = false;
        }
 
        // Call the recursive helper function to find articulation
        // points in DFS tree rooted with vertex 'i'
        for (int i = 0; i < V; i++)
            if (visited[i] == false)
                APUtil(i, visited, disc, low, parent, ap);
 
        // Now ap[] contains articulation points, print them
        for (int i = 0; i < V; i++)
            if (ap[i] == true)
                System.out.print(i+" ");
    }
 
    // Driver method
    public static void main(String args[])
    {
        // Create graphs given in above diagrams
        System.out.println("Articulation points in first graph ");
        Graph g1 = new Graph(5);
        g1.addEdge(1, 0);
        g1.addEdge(0, 2);
        g1.addEdge(2, 1);
        g1.addEdge(0, 3);
        g1.addEdge(3, 4);
        g1.AP();
        System.out.println();
 
        System.out.println("Articulation points in Second graph");
        Graph g2 = new Graph(4);
        g2.addEdge(0, 1);
        g2.addEdge(1, 2);
        g2.addEdge(2, 3);
        g2.AP();
        System.out.println();
 
        System.out.println("Articulation points in Third graph ");
        Graph g3 = new Graph(7);
        g3.addEdge(0, 1);
        g3.addEdge(1, 2);
        g3.addEdge(2, 0);
        g3.addEdge(1, 3);
        g3.addEdge(1, 4);
        g3.addEdge(1, 6);
        g3.addEdge(3, 5);
        g3.addEdge(4, 5);
        g3.AP();
    }
}
// This code is contributed by Aakash Hasija

 

Python

 

# Python program to find articulation points in an undirected graph
  
from collections import defaultdict
  
#This class represents an undirected graph 
#using adjacency list representation
class Graph:
  
    def __init__(self,vertices):
        self.V= vertices #No. of vertices
        self.graph = defaultdict(list) # default dictionary to store graph
        self.Time = 0
  
    # function to add an edge to graph
    def addEdge(self,u,v):
        self.graph[u].append(v)
        self.graph[v].append(u)
  
    '''A recursive function that find articulation points 
    using DFS traversal
    u --> The vertex to be visited next
    visited[] --> keeps tract of visited vertices
    disc[] --> Stores discovery times of visited vertices
    parent[] --> Stores parent vertices in DFS tree
    ap[] --> Store articulation points'''
    def APUtil(self,u, visited, ap, parent, low, disc):
 
        #Count of children in current node 
        children =0
 
        # Mark the current node as visited and print it
        visited[u]= True
 
        # Initialize discovery time and low value
        disc[u] = self.Time
        low[u] = self.Time
        self.Time += 1
 
        #Recur for all the vertices adjacent to this vertex
        for v in self.graph[u]:
            # If v is not visited yet, then make it a child of u
            # in DFS tree and recur for it
            if visited[v] == False :
                parent[v] = u
                children += 1
                self.APUtil(v, visited, ap, parent, low, disc)
 
                # Check if the subtree rooted with v has a connection to
                # one of the ancestors of u
                low[u] = min(low[u], low[v])
 
                # u is an articulation point in following cases
                # (1) u is root of DFS tree and has two or more chilren.
                if parent[u] == -1 and children > 1:
                    ap[u] = True
 
                #(2) If u is not root and low value of one of its child is more
                # than discovery value of u.
                if parent[u] != -1 and low[v] >= disc[u]:
                    ap[u] = True   
                     
                # Update low value of u for parent function calls   
            elif v != parent[u]: 
                low[u] = min(low[u], disc[v])
 
 
    #The function to do DFS traversal. It uses recursive APUtil()
    def AP(self):
  
        # Mark all the vertices as not visited 
        # and Initialize parent and visited, 
        # and ap(articulation point) arrays
        visited = [False] * (self.V)
        disc = [float("Inf")] * (self.V)
        low = [float("Inf")] * (self.V)
        parent = [-1] * (self.V)
        ap = [False] * (self.V) #To store articulation points
 
        # Call the recursive helper function
        # to find articulation points
        # in DFS tree rooted with vertex 'i'
        for i in range(self.V):
            if visited[i] == False:
                self.APUtil(i, visited, ap, parent, low, disc)
 
        for index, value in enumerate (ap):
            if value == True: print index,
 
 # Create a graph given in the above diagram
g1 = Graph(5)
g1.addEdge(1, 0)
g1.addEdge(0, 2)
g1.addEdge(2, 1)
g1.addEdge(0, 3)
g1.addEdge(3, 4)
  
print "\nArticulation points in first graph "
g1.AP()
 
g2 = Graph(4)
g2.addEdge(0, 1)
g2.addEdge(1, 2)
g2.addEdge(2, 3)
print "\nArticulation points in second graph "
g2.AP()
 
  
g3 = Graph (7)
g3.addEdge(0, 1)
g3.addEdge(1, 2)
g3.addEdge(2, 0)
g3.addEdge(1, 3)
g3.addEdge(1, 4)
g3.addEdge(1, 6)
g3.addEdge(3, 5)
g3.addEdge(4, 5)
print "\nArticulation points in third graph "
g3.AP()
 
#This code is contributed by Neelam Yadav

 

输出：

 

Articulation points in first graph
0 3
Articulation points in second graph
1 2
Articulation points in third graph
1

 

时间复杂度：上面的函数是一个含有额外数组的简单DFS。因此时间复杂度与DFS相同，对于用邻接表表示的图，为O(V+E)。

 

引用

 

https://www.cs.washington.edu/education/courses/421/04su/slides/artic.pdf

http://www.slideshare.net/TraianRebedea/algorithm-design-and-complexity-course-8

http://faculty.simpson.edu/lydia.sinapova/www/cmsc250/LN250_Weiss/L25-Connectivity.htm