使用拓扑排序检测有向图中的循环

给定一个由N个顶点和M 个边组成的有向图和一组Edges[][] ，任务是使用拓扑排序检查图是否包含循环。

Topological sort of directed graph is a linear ordering of its vertices such that, for every directed edge U -> V from vertex U to vertex V, U comes before V in the ordering. 
 

例子：

Input: N = 4, M = 6, Edges[][] = {{0, 1}, {1, 2}, {2, 0}, {0, 2}, {2, 3}, {3, 3}} 
Output: Yes 
Explanation: 
A cycle 0 -> 2 -> 0 exists in the given graph

Input: N = 4, M = 3, Edges[][] = {{0, 1}, {1, 2}, {2, 3}, {0, 2}} 
Output: No 

方法：
在拓扑排序中，思想是先访问父节点，然后再访问子节点。如果给定的图包含一个循环，那么至少有一个节点既是父节点又是子节点，因此这将破坏拓扑顺序。因此，在拓扑排序之后，检查每个有向边是否遵循顺序。

下面是上述方法的实现：

C++

// C++ Program to implement
// the above approach
#include 
using namespace std;
 
int t, n, m, a;
 
// Stack to store the
// visited vertices in
// the Topological Sort
stack s;
 
// Store Topological Order
vector tsort;
 
// Adjacency list to store edges
vector adj[int(1e5) + 1];
 
// To ensure visited vertex
vector visited(int(1e5) + 1);
 
// Function to perform DFS
void dfs(int u)
{
    // Set the vertex as visited
    visited[u] = 1;
 
    for (auto it : adj[u]) {
 
        // Visit connected vertices
        if (visited[it] == 0)
            dfs(it);
    }
 
    // Push into the stack on
    // complete visit of vertex
    s.push(u);
}
 
// Function to check and return
// if a cycle exists or not
bool check_cycle()
{
    // Stores the position of
    // vertex in topological order
    unordered_map pos;
    int ind = 0;
 
    // Pop all elements from stack
    while (!s.empty()) {
        pos[s.top()] = ind;
 
        // Push element to get
        // Topological Order
        tsort.push_back(s.top());
 
        ind += 1;
 
        // Pop from the stack
        s.pop();
    }
 
    for (int i = 0; i < n; i++) {
        for (auto it : adj[i]) {
 
            // If parent vertex
            // does not appear first
            if (pos[i] > pos[it]) {
 
                // Cycle exists
                return true;
            }
        }
    }
 
    // Return false if cycle
    // does not exist
    return false;
}
 
// Function to add edges
// from u to v
void addEdge(int u, int v)
{
    adj[u].push_back(v);
}
 
// Driver Code
int main()
{
    n = 4, m = 5;
 
    // Insert edges
    addEdge(0, 1);
    addEdge(0, 2);
    addEdge(1, 2);
    addEdge(2, 0);
    addEdge(2, 3);
 
    for (int i = 0; i < n; i++) {
        if (visited[i] == 0) {
            dfs(i);
        }
    }
 
    // If cycle exist
    if (check_cycle())
        cout << "Yes";
    else
        cout << "No";
 
    return 0;
}

Java

// Java program to implement
// the above approach
import java.util.*;
 
class GFG{
     
static int t, n, m, a;
 
// Stack to store the
// visited vertices in
// the Topological Sort
static Stack s;
 
// Store Topological Order
static ArrayList tsort;
 
// Adjacency list to store edges
static ArrayList> adj;
 
// To ensure visited vertex
static int[] visited = new int[(int)1e5 + 1];
 
// Function to perform DFS
static void dfs(int u)
{
     
    // Set the vertex as visited
    visited[u] = 1;
 
    for(Integer it : adj.get(u))
    {
         
        // Visit connected vertices
        if (visited[it] == 0)
            dfs(it);
    }
     
    // Push into the stack on
    // complete visit of vertex
    s.push(u);
}
 
// Function to check and return
// if a cycle exists or not
static boolean check_cycle()
{
     
    // Stores the position of
    // vertex in topological order
    Map pos = new HashMap<>();
     
    int ind = 0;
     
    // Pop all elements from stack
    while (!s.isEmpty())
    {
        pos.put(s.peek(), ind);
         
        // Push element to get
        // Topological Order
        tsort.add(s.peek());
 
        ind += 1;
         
        // Pop from the stack
        s.pop();
    }
 
    for(int i = 0; i < n; i++)
    {
        for(Integer it : adj.get(i))
        {
             
            // If parent vertex
            // does not appear first
            if (pos.get(i) > pos.get(it))
            {
                 
                // Cycle exists
                return true;
            }
        }
    }
 
    // Return false if cycle
    // does not exist
    return false;
}
 
// Function to add edges
// from u to v
static void addEdge(int u, int v)
{
    adj.get(u).add(v);
}
 
// Driver code   
public static void main (String[] args)
{
    n = 4; m = 5;
     
    s = new Stack<>();
    adj = new ArrayList<>();
    tsort = new ArrayList<>();
     
    for(int i = 0; i < 4; i++)
        adj.add(new ArrayList<>());
         
    // Insert edges
    addEdge(0, 1);
    addEdge(0, 2);
    addEdge(1, 2);
    addEdge(2, 0);
    addEdge(2, 3);
     
    for(int i = 0; i < n; i++)
    {
        if (visited[i] == 0)
        {
            dfs(i);
        }
    }
     
    // If cycle exist
    if (check_cycle())
       System.out.println("Yes");
    else
       System.out.println("No");
}
}
 
// This code is contributed by offbeat

Python3

# Python3 program to implement
# the above approach
t = 0
n = 0
m = 0
a = 0
  
# Stack to store the
# visited vertices in
# the Topological Sort
s = []
  
# Store Topological Order
tsort = []
  
# Adjacency list to store edges
adj = [[] for i in range(100001)]
  
# To ensure visited vertex
visited = [False for i in range(100001)]
  
# Function to perform DFS
def dfs(u):
 
    # Set the vertex as visited
    visited[u] = 1
     
    for it in adj[u]:
 
        # Visit connected vertices
        if (visited[it] == 0):
            dfs(it)
  
    # Push into the stack on
    # complete visit of vertex
    s.append(u)
 
# Function to check and return
# if a cycle exists or not
def check_cycle():
 
    # Stores the position of
    # vertex in topological order
    pos = dict()
     
    ind = 0
     
    # Pop all elements from stack
    while (len(s) != 0):
        pos[s[-1]] = ind
  
        # Push element to get
        # Topological Order
        tsort.append(s[-1])
  
        ind += 1
  
        # Pop from the stack
        s.pop()
     
    for i in range(n):
        for it in adj[i]:
            first = 0 if i not in pos else pos[i]
            second = 0 if it not in pos else pos[it]
 
            # If parent vertex
            # does not appear first
            if (first > second):
  
                # Cycle exists
                return True
  
    # Return false if cycle
    # does not exist
    return False
 
# Function to add edges
# from u to v
def addEdge(u, v):
     
    adj[u].append(v)
 
# Driver Code
if __name__ == "__main__":
     
    n = 4
    m = 5
  
    # Insert edges
    addEdge(0, 1)
    addEdge(0, 2)
    addEdge(1, 2)
    addEdge(2, 0)
    addEdge(2, 3)
     
    for i in range(n):
        if (visited[i] == False):
            dfs(i)
  
    # If cycle exist
    if (check_cycle()):
        print('Yes')
    else:
        print('No')
         
# This code is contributed by rutvik_56

C#

// C# program to implement
// the above approach
using System;
using System.Collections;
using System.Collections.Generic;
  
class GFG{
 
static int n;
  
// Stack to store the
// visited vertices in
// the Topological Sort
static Stack s;
  
// Store Topological Order
static ArrayList tsort;
  
// Adjacency list to store edges
static ArrayList adj;
  
// To ensure visited vertex
static int[] visited = new int[100001];
  
// Function to perform DFS
static void dfs(int u)
{
     
    // Set the vertex as visited
    visited[u] = 1;
  
    foreach(int it in (ArrayList)adj[u])
    {
         
        // Visit connected vertices
        if (visited[it] == 0)
            dfs(it);
    }
      
    // Push into the stack on
    // complete visit of vertex
    s.Push(u);
}
  
// Function to check and return
// if a cycle exists or not
static bool check_cycle()
{
     
    // Stores the position of
    // vertex in topological order
    Dictionary pos = new Dictionary();
      
    int ind = 0;
      
    // Pop all elements from stack
    while (s.Count != 0)
    {
        pos.Add(s.Peek(), ind);
         
        // Push element to get
        // Topological Order
        tsort.Add(s.Peek());
  
        ind += 1;
          
        // Pop from the stack
        s.Pop();
    }
  
    for(int i = 0; i < n; i++)
    {
        foreach(int it in (ArrayList)adj[i])
        {
             
            // If parent vertex
            // does not appear first
            if (pos[i] > pos[it])
            {
                 
                // Cycle exists
                return true;
            }
        }
    }
  
    // Return false if cycle
    // does not exist
    return false;
}
  
// Function to add edges
// from u to v
static void addEdge(int u, int v)
{
    ((ArrayList)adj[u]).Add(v);
}
  
// Driver code   
public static void Main(string[] args)
{
    n = 4;
      
    s = new Stack();
    adj = new ArrayList();
    tsort = new ArrayList();
      
    for(int i = 0; i < 4; i++)
        adj.Add(new ArrayList());
          
    // Insert edges
    addEdge(0, 1);
    addEdge(0, 2);
    addEdge(1, 2);
    addEdge(2, 0);
    addEdge(2, 3);
      
    for(int i = 0; i < n; i++)
    {
        if (visited[i] == 0)
        {
            dfs(i);
        }
    }
      
    // If cycle exist
    if (check_cycle())
       Console.WriteLine("Yes");
    else
       Console.WriteLine("No");
}
}
 
// This code is contributed by pratham76

Javascript

 
// JavaScript Program to implement
// the above approach
 
var t, n, m, a;
 
// Stack to store the
// visited vertices in
// the Topological Sort
var s = [];
 
// Store Topological Order
var tsort = [];
 
// Adjacency list to store edges
var adj = Array.from(Array(100001), ()=>Array());
 
// To ensure visited vertex
var visited = Array(100001).fill(0);
 
//( Function to perform )DFS
function dfs(u)
{
    // Set the vertex as visited
    visited[u] = 1;
 
    adj[u].forEach(it => {
         
 
        // Visit connected vertices
        if (visited[it] == 0)
            dfs(it);
    });
 
    // Push into the stack on
    // complete visit of vertex
    s.push(u);
}
 
// Function to check and return
// if a cycle exists or not
function check_cycle()
{
    // Stores the position of
    // vertex in topological order
    var pos = new Map();
    var ind = 0;
 
    // Pop all elements from stack
    while (s.length!=0) {
        pos.set(s[s.length-1], ind);
 
        // Push element to get
        // Topological Order
        tsort.push(s[s.length-1]);
 
        ind += 1;
 
        // Pop from the stack
        s.pop();
    }
    var ans = false;
    for (var i = 0; i < n; i++) {
        adj[i].forEach(it => {
             
            // If parent vertex
            // does not appear first
            if ((pos.has(i)?pos.get(i):0) >
            (pos.has(it)?pos.get(it):0))
            {
 
                // Cycle exists
                ans = true;
            }
        });
    };
 
    // Return false if cycle
    // does not exist
    return ans;
}
 
// Function to add edges
// from u to v
function addEdge(u, v)
{
    adj[u].push(v);
}
 
// Driver Code
 
n = 4, m = 5;
 
// Insert edges
addEdge(0, 1);
addEdge(0, 2);
addEdge(1, 2);
addEdge(2, 0);
addEdge(2, 3);
for (var i = 0; i < n; i++) {
    if (visited[i] == 0) {
        dfs(i);
    }
}
// If cycle exist
if (check_cycle())
    document.write( "Yes");
else
    document.write( "No");
 
 

输出：

Yes

时间复杂度： O(N + M)
辅助空间： O(N)

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