并集算法 | Set 2 (Union By Rank and Path Compression)

在上一篇文章中，我们介绍了并集查找算法并使用它来检测图中的循环。我们对子集使用了以下union()和find()操作。

C++

// Naive implementation of find
int find(int parent[], int i)
{
    if (parent[i] == -1)
        return i;
    return find(parent, parent[i]);
}
  
// Naive implementation of union()
void Union(int parent[], int x, int y)
{
    int xset = find(parent, x);
    int yset = find(parent, y);
    parent[xset] = yset;
}

Java

// Naive implementation of find
static int find(int parent[], int i)
{
    if (parent[i] == -1)
        return i;
    return find(parent, parent[i]);
}
   
// Naive implementation of union()
static void Union(int parent[], int x, int y)
{
    int xset = find(parent, x);
    int yset = find(parent, y);
    parent[xset] = yset;
}
 
// This code is contributed by divyesh072019

Python3

# Naive implementation of find
def find(parent, i):
     
    if (parent[i] == -1):
        return i
     
    return find(parent, parent[i])
 
# Naive implementation of union()
def Union(parent, x, y):
 
    xset = find(parent, x)
    yset = find(parent, y)
    parent[xset] = yset
 
# This code is contributed by rutvik_56

C#

// Naive implementation of find
static int find(int []parent, int i)
{
    if (parent[i] == -1)
        return i;
         
    return find(parent, parent[i]);
}
   
// Naive implementation of union()
static void Union(int []parent, int x, int y)
{
    int xset = find(parent, x);
    int yset = find(parent, y);
    parent[xset] = yset;
}
 
// This code is contributed by pratham76

Javascript

 
// Naive implementation of find
function find(parent, i)
{
    if (parent[i] == -1)
        return i;
         
    return find(parent, parent[i]);
}
   
// Naive implementation of union()
function Union(parent, x, y)
{
    let xset = find(parent, x);
    let yset = find(parent, y);
    parent[xset] = yset;
}
 
<script></code></pre></div><br><hr><h5 class="code">C++</h5><div class="hcb_wrap"><pre class="prism undefined-numbers lang-cpp" data-lang="C++"><code class="replace">// A C++ program to detect cycle in a graph using union by
// rank and path compression
#include <bits/stdc++.h>
using namespace std;
 
// a structure to represent an edge in the graph
struct Edge {
    int src, dest;
};
 
// a structure to represent a graph
struct Graph {
    // V-> Number of vertices, E-> Number of edges
    int V, E;
 
    // graph is represented as an array of edges
    struct Edge* edge;
};
 
struct subset {
    int parent;
    int rank;
};
 
// Creates a graph with V vertices and E edges
struct Graph* createGraph(int V, int E)
{
    struct Graph* graph = (struct Graph*)malloc(sizeof(struct Graph));
    graph->V = V;
    graph->E = E;
 
    graph->edge = (struct Edge*)malloc(graph->E * sizeof(struct Edge));
 
    return graph;
}
 
// A utility function to find set of an element i
// (uses path compression technique)
int find(struct subset subsets[], int i)
{
    // find root and make root as parent of i (path
    // compression)
    if (subsets[i].parent != i)
        subsets[i].parent = find(subsets, subsets[i].parent);
 
    return subsets[i].parent;
}
 
// A function that does union of two sets of x and y
// (uses union by rank)
void Union(struct subset subsets[], int xroot, int yroot)
{
 
    // Attach smaller rank tree under root of high rank tree
    // (Union by Rank)
    if (subsets[xroot].rank < subsets[yroot].rank)
        subsets[xroot].parent = yroot;
    else if (subsets[xroot].rank > subsets[yroot].rank)
        subsets[yroot].parent = xroot;
 
    // If ranks are same, then make one as root and
    // increment its rank by one
    else {
        subsets[yroot].parent = xroot;
        subsets[xroot].rank++;
    }
}
 
// The main function to check whether a given graph contains
// cycle or not
int isCycle(struct Graph* graph)
{
    int V = graph->V;
    int E = graph->E;
 
    // Allocate memory for creating V sets
    struct subset* subsets
        = (struct subset*)malloc(V * sizeof(struct subset));
 
    for (int v = 0; v < V; ++v) {
        subsets[v].parent = v;
        subsets[v].rank = 0;
    }
 
    // Iterate through all edges of graph, find sets of both
    // vertices of every edge, if sets are same, then there
    // is cycle in graph.
    for (int e = 0; e < E; ++e) {
        int x = find(subsets, graph->edge[e].src);
        int y = find(subsets, graph->edge[e].dest);
 
        if (x == y)
            return 1;
 
        Union(subsets, x, y);
    }
    return 0;
}
 
// Driver code
int main()
{
    /* Let us create the following graph
         0
        |  \
        |    \
        1-----2 */
 
    int V = 3, E = 3;
    struct Graph* graph = createGraph(V, E);
 
    // add edge 0-1
    graph->edge[0].src = 0;
    graph->edge[0].dest = 1;
 
    // add edge 1-2
    graph->edge[1].src = 1;
    graph->edge[1].dest = 2;
 
    // add edge 0-2
    graph->edge[2].src = 0;
    graph->edge[2].dest = 2;
 
    if (isCycle(graph))
        cout << "Graph contains cycle";
    else
        cout << "Graph doesn't contain cycle";
 
    return 0;
}
 
// This code is contributed by Aditya Kumar (adityakumar129)</code></pre></div><br><hr><h5 class="code">C</h5><div class="hcb_wrap"><pre class="prism undefined-numbers lang-c" data-lang="C"><code class="replace">// A C program to detect cycle in a graph using union by
// rank and path compression
#include <stdio.h>
#include <stdlib.h>
 
// a structure to represent an edge in the graph
typedef struct Edge {
    int src, dest;
}Edge;
 
// a structure to represent a graph
typedef struct Graph {
    // V-> Number of vertices, E-> Number of edges
    int V, E;
 
    // graph is represented as an array of edges
    struct Edge* edge;
}Graph;
 
typedef struct subset {
    int parent;
    int rank;
}subset;
 
// Creates a graph with V vertices and E edges
Graph* createGraph(int V, int E)
{
    Graph* graph = (Graph*)malloc(sizeof(Graph));
    graph->V = V;
    graph->E = E;
    graph->edge = (Edge*)malloc(graph->E * sizeof(Edge));
    return graph;
}
 
// A utility function to find set of an element i
// (uses path compression technique)
int find(subset subsets[], int i)
{
    // find root and make root as parent of i (path
    // compression)
    if (subsets[i].parent != i)
        subsets[i].parent = find(subsets, subsets[i].parent);
    return subsets[i].parent;
}
 
// A function that does union of two sets of x and y
// (uses union by rank)
void Union(subset subsets[], int xroot, int yroot)
{
    // Attach smaller rank tree under root of high rank tree
    // (Union by Rank)
    if (subsets[xroot].rank < subsets[yroot].rank)
        subsets[xroot].parent = yroot;
    else if (subsets[xroot].rank > subsets[yroot].rank)
        subsets[yroot].parent = xroot;
    // If ranks are same, then make one as root and
    // increment its rank by one
    else {
        subsets[yroot].parent = xroot;
        subsets[xroot].rank++;
    }
}
 
// The main function to check whether a given graph contains
// cycle or not
int isCycle(Graph* graph)
{
    int V = graph->V;
    int E = graph->E;
    // Allocate memory for creating V sets
    subset* subsets = (subset*)malloc(V * sizeof(subset));
    for (int v = 0; v < V; ++v) {
        subsets[v].parent = v;
        subsets[v].rank = 0;
    }
    // Iterate through all edges of graph, find sets of both
    // vertices of every edge, if sets are same, then there
    // is cycle in graph.
    for (int e = 0; e < E; ++e) {
        int x = find(subsets, graph->edge[e].src);
        int y = find(subsets, graph->edge[e].dest);
        if (x == y)
            return 1;
        Union(subsets, x, y);
    }
    return 0;
}
 
// Driver code
int main()
{
    /* Let us create the following graph
         0
        |  \
        |    \
        1-----2 */
 
    int V = 3, E = 3;
    Graph* graph = createGraph(V, E);
 
    // add edge 0-1
    graph->edge[0].src = 0;
    graph->edge[0].dest = 1;
 
    // add edge 1-2
    graph->edge[1].src = 1;
    graph->edge[1].dest = 2;
 
    // add edge 0-2
    graph->edge[2].src = 0;
    graph->edge[2].dest = 2;
 
    if (isCycle(graph))
        printf("Graph contains cycle");
    else
        printf("Graph doesn't contain cycle");
    return 0;
}
 
// This code is contributed by Aditya Kumar (adityakumar129)</code></pre></div><br><hr><h5 class="code">Java</h5><div class="hcb_wrap"><pre class="prism undefined-numbers lang-java" data-lang="java"><code class="replace">// A union by rank and path compression
// based program to detect cycle in a graph
class Graph
{
    int V, E;
    Edge[] edge;
 
    Graph(int nV, int nE)
    {
        V = nV;
        E = nE;
        edge = new Edge[E];
        for (int i = 0; i < E; i++)
        {
            edge[i] = new Edge();
        }
    }
 
    // class to represent edge
    class Edge
    {
        int src, dest;
    }
 
    // class to represent Subset
    class subset
    {
        int parent;
        int rank;
    }
 
    // A utility function to find
    // set of an element i (uses
    // path compression technique)
    int find(subset[] subsets, int i)
    {
        if (subsets[i].parent != i)
            subsets[i].parent
                = find(subsets, subsets[i].parent);
        return subsets[i].parent;
    }
 
    // A function that does union
    // of two sets of x and y
    // (uses union by rank)
    void Union(subset[] subsets, int x, int y)
    {
        int xroot = find(subsets, x);
        int yroot = find(subsets, y);
 
        if (subsets[xroot].rank < subsets[yroot].rank)
            subsets[xroot].parent = yroot;
        else if (subsets[yroot].rank < subsets[xroot].rank)
            subsets[yroot].parent = xroot;
        else {
            subsets[xroot].parent = yroot;
            subsets[yroot].rank++;
        }
    }
 
    // The main function to check whether
    // a given graph contains cycle or not
    int isCycle(Graph graph)
    {
        int V = graph.V;
        int E = graph.E;
 
        subset[] subsets = new subset[V];
        for (int v = 0; v < V; v++) {
 
            subsets[v] = new subset();
            subsets[v].parent = v;
            subsets[v].rank = 0;
        }
 
        for (int e = 0; e < E; e++) {
            int x = find(subsets, graph.edge[e].src);
            int y = find(subsets, graph.edge[e].dest);
            if (x == y)
                return 1;
            Union(subsets, x, y);
        }
        return 0;
    }
 
    // Driver Code
    public static void main(String[] args)
    {
        /* Let us create the following graph
            0
            | \
            | \
            1-----2 */
 
        int V = 3, E = 3;
        Graph graph = new Graph(V, E);
 
        // add edge 0-1
        graph.edge[0].src = 0;
        graph.edge[0].dest = 1;
 
        // add edge 1-2
        graph.edge[1].src = 1;
        graph.edge[1].dest = 2;
 
        // add edge 0-2
        graph.edge[2].src = 0;
        graph.edge[2].dest = 2;
 
        if (graph.isCycle(graph) == 1)
            System.out.println("Graph contains cycle");
        else
            System.out.println(
                "Graph doesn't contain cycle");
    }
}
 
// This code is contributed
// by ashwani khemani</code></pre></div><br><hr><h5 class="code">Python3</h5><div class="hcb_wrap"><pre class="prism undefined-numbers lang-python" data-lang="Python"><code class="replace"># A union by rank and path compression based
# program to detect cycle in a graph
from collections import defaultdict
 
# a structure to represent a graph
 
 
class Graph:
 
    def __init__(self, num_of_v):
        self.num_of_v = num_of_v
        self.edges = defaultdict(list)
 
    # graph is represented as an
    # array of edges
    def add_edge(self, u, v):
        self.edges[u].append(v)
 
 
class Subset:
    def __init__(self, parent, rank):
        self.parent = parent
        self.rank = rank
 
# A utility function to find set of an element
# node(uses path compression technique)
 
 
def find(subsets, node):
    if subsets[node].parent != node:
        subsets[node].parent = find(subsets, subsets[node].parent)
    return subsets[node].parent
 
# A function that does union of two sets
# of u and v(uses union by rank)
 
 
def union(subsets, u, v):
 
    # Attach smaller rank tree under root
    # of high rank tree(Union by Rank)
    if subsets[u].rank > subsets[v].rank:
        subsets[v].parent = u
    elif subsets[v].rank > subsets[u].rank:
        subsets[u].parent = v
 
    # If ranks are same, then make one as
    # root and increment its rank by one
    else:
        subsets[v].parent = u
        subsets[u].rank += 1
 
# The main function to check whether a given
# graph contains cycle or not
 
 
def isCycle(graph):
 
    # Allocate memory for creating sets
    subsets = []
 
    for u in range(graph.num_of_v):
        subsets.append(Subset(u, 0))
 
    # Iterate through all edges of graph,
    # find sets of both vertices of every
    # edge, if sets are same, then there
    # is cycle in graph.
    for u in graph.edges:
        u_rep = find(subsets, u)
 
        for v in graph.edges[u]:
            v_rep = find(subsets, v)
 
            if u_rep == v_rep:
                return True
            else:
                union(subsets, u_rep, v_rep)
 
 
# Driver Code
g = Graph(3)
 
# add edge 0-1
g.add_edge(0, 1)
 
# add edge 1-2
g.add_edge(1, 2)
 
# add edge 0-2
g.add_edge(0, 2)
 
if isCycle(g):
    print('Graph contains cycle')
else:
    print('Graph does not contain cycle')
 
# This code is contributed by
# Sampath Kumar Surine</code></pre></div><br><hr><h5 class="code">C#</h5><div class="hcb_wrap"><pre class="prism undefined-numbers lang-csharp" data-lang="C#"><code class="replace">// A union by rank and path compression
// based program to detect cycle in a graph
using System;
 
class Graph {
    public int V, E;
    public Edge[] edge;
 
    public Graph(int nV, int nE)
    {
        V = nV;
        E = nE;
        edge = new Edge[E];
        for (int i = 0; i < E; i++)
        {
            edge[i] = new Edge();
        }
    }
 
    // class to represent edge
    public class Edge {
        public int src, dest;
    }
 
    // class to represent Subset
    class subset
    {
        public int parent;
        public int rank;
    }
 
    // A utility function to find
    // set of an element i (uses
    // path compression technique)
    int find(subset[] subsets, int i)
    {
        if (subsets[i].parent != i)
            subsets[i].parent
                = find(subsets, subsets[i].parent);
        return subsets[i].parent;
    }
 
    // A function that does union
    // of two sets of x and y
    // (uses union by rank)
    void Union(subset[] subsets, int x, int y)
    {
        int xroot = find(subsets, x);
        int yroot = find(subsets, y);
 
        if (subsets[xroot].rank < subsets[yroot].rank)
            subsets[xroot].parent = yroot;
        else if (subsets[yroot].rank < subsets[xroot].rank)
            subsets[yroot].parent = xroot;
        else {
            subsets[xroot].parent = yroot;
            subsets[yroot].rank++;
        }
    }
 
    // The main function to check whether
    // a given graph contains cycle or not
    int isCycle(Graph graph)
    {
        int V = graph.V;
        int E = graph.E;
 
        subset[] subsets = new subset[V];
        for (int v = 0; v < V; v++)
        {
            subsets[v] = new subset();
            subsets[v].parent = v;
            subsets[v].rank = 0;
        }
 
        for (int e = 0; e < E; e++)
        {
            int x = find(subsets, graph.edge[e].src);
            int y = find(subsets, graph.edge[e].dest);
            if (x == y)
                return 1;
            Union(subsets, x, y);
        }
        return 0;
    }
 
    // Driver Code
    static public void Main(String[] args)
    {
        /* Let us create the following graph
            0
            | \
            | \
            1-----2 */
 
        int V = 3, E = 3;
        Graph graph = new Graph(V, E);
 
        // add edge 0-1
        graph.edge[0].src = 0;
        graph.edge[0].dest = 1;
 
        // add edge 1-2
        graph.edge[1].src = 1;
        graph.edge[1].dest = 2;
 
        // add edge 0-2
        graph.edge[2].src = 0;
        graph.edge[2].dest = 2;
 
        if (graph.isCycle(graph) == 1)
            Console.WriteLine("Graph contains cycle");
        else
            Console.WriteLine(
                "Graph doesn't contain cycle");
    }
}
 
// This code is contributed
// by Arnab Kundu</code></pre></div><br><hr><h5 class="code">Javascript</h5><div class="hcb_wrap"><pre class="prism undefined-numbers lang-js" data-lang="JavaScript"><code class="replace"><script>
// A union by rank and path compression
// based program to detect cycle in a graph
 
let V, E;
let edge;
 
function Graph(nV,nE)
{
    V = nV;
        E = nE;
        edge = new Array(E);
        for (let i = 0; i < E; i++)
        {
            edge[i] = new Edge();
        }
}
 
// class to represent edge
class Edge
{
    constructor()
    {
        this.src=0;
        this.dest=0;
    }
}
 
// class to represent Subset
class subset
{
    constructor()
    {
        this.parent=0;
        this.rank=0;
    }
}
 
// A utility function to find
    // set of an element i (uses
    // path compression technique)
function find(subsets,i)
{
    if (subsets[i].parent != i)
            subsets[i].parent
                = find(subsets, subsets[i].parent);
        return subsets[i].parent;
}
 
// A function that does union
    // of two sets of x and y
    // (uses union by rank)
function Union(subsets,x,y)
{
    let xroot = find(subsets, x);
        let yroot = find(subsets, y);
  
        if (subsets[xroot].rank < subsets[yroot].rank)
            subsets[xroot].parent = yroot;
        else if (subsets[yroot].rank < subsets[xroot].rank)
            subsets[yroot].parent = xroot;
        else {
            subsets[xroot].parent = yroot;
            subsets[yroot].rank++;
        }
}
 
// The main function to check whether
    // a given graph contains cycle or not
function isCycle()
{
     
  
        let subsets = new Array(V);
        for (let v = 0; v < V; v++) {
  
            subsets[v] = new subset();
            subsets[v].parent = v;
            subsets[v].rank = 0;
        }
  
        for (let e = 0; e < E; e++) {
            let x = find(subsets, edge[e].src);
            let y = find(subsets, edge[e].dest);
            if (x == y)
                return 1;
            Union(subsets, x, y);
        }
        return 0;
}
 
// Driver Code
/* Let us create the following graph
            0
            | \
            | \
            1-----2 */
  
        V = 3, E = 3;
        Graph(V, E);
  
        // add edge 0-1
        edge[0].src = 0;
        edge[0].dest = 1;
  
        // add edge 1-2
        edge[1].src = 1;
        edge[1].dest = 2;
  
        // add edge 0-2
        edge[2].src = 0;
        edge[2].dest = 2;
 
        if (isCycle() == 1)
            document.write("Graph contains cycle");
        else
            document.write(
                "Graph doesn't contain cycle");
 
 
// This code is contributed by avanitrachhadiya2155

上面的union()和find()是幼稚的，最坏情况的时间复杂度是线性的。为表示子集而创建的树可能是倾斜的，并且可能变得像一个链表。以下是最坏情况的示例。

Let there be 4 elements 0, 1, 2, 3

Initially, all elements are single element subsets.
0 1 2 3 

Do Union(0, 1)
   1   2   3  
  /
 0

Do Union(1, 2)
     2   3   
    /
   1
 /
0

Do Union(2, 3)
         3    
        /
      2
     /
   1
 /
0

在最坏的情况下，上述操作可以优化到O(Log n) 。这个想法是总是在更深的树的根下附加更小的深度树。这种技术称为union by rank 。优先使用术语rank而不是 height，因为如果使用路径压缩技术（我们在下面讨论过），那么rank并不总是等于高度。此外，树的大小（代替高度）也可以用作rank 。使用大小作为等级也会产生 O(Logn) 的最坏情况时间复杂度。

Let us see the above example with union by rank
Initially, all elements are single element subsets.
0 1 2 3 

Do Union(0, 1)
   1   2   3  
  /
 0

Do Union(1, 2)
   1    3
 /  \
0    2

Do Union(2, 3)
    1    
 /  |  \
0   2   3

朴素方法的第二个优化是路径压缩。这个想法是在调用find()时展平树。当为元素 x 调用find()时，将返回树的根。 find()操作从 x 向上遍历找到根。路径压缩的思想就是把找到的根作为x的父节点，这样我们就不用再遍历所有的中间节点了。如果 x 是子树的根，则 x 下所有节点的路径（到根）也会压缩。

Let the subset {0, 1, .. 9} be represented as below and find() is called
for element 3.
             9
         /   |   \  
        4    5    6
       /         /  \
      0         7    8
     /        
    3
   / \         
  1   2
When find() is called for 3, we traverse up and find 9 as representative
of this subset. With path compression, we also make 3 and 0 as the child of 9 so 
that when find() is called next time for 0, 1, 2 or 3, the path to root is reduced.

        --------9-------
      /   /    /  \      \
     0   4    5    6       3 
                  /  \    /  \
                 7    8   1   2

这两种技术相辅相成。每个操作的时间复杂度变得比 O(Logn) 还要小。事实上，摊销的时间复杂度实际上变成了小常数。

以下是基于秩和路径压缩的联合，用于在图中找到循环。

C++

// A C++ program to detect cycle in a graph using union by
// rank and path compression
#include 
using namespace std;
 
// a structure to represent an edge in the graph
struct Edge {
    int src, dest;
};
 
// a structure to represent a graph
struct Graph {
    // V-> Number of vertices, E-> Number of edges
    int V, E;
 
    // graph is represented as an array of edges
    struct Edge* edge;
};
 
struct subset {
    int parent;
    int rank;
};
 
// Creates a graph with V vertices and E edges
struct Graph* createGraph(int V, int E)
{
    struct Graph* graph = (struct Graph*)malloc(sizeof(struct Graph));
    graph->V = V;
    graph->E = E;
 
    graph->edge = (struct Edge*)malloc(graph->E * sizeof(struct Edge));
 
    return graph;
}
 
// A utility function to find set of an element i
// (uses path compression technique)
int find(struct subset subsets[], int i)
{
    // find root and make root as parent of i (path
    // compression)
    if (subsets[i].parent != i)
        subsets[i].parent = find(subsets, subsets[i].parent);
 
    return subsets[i].parent;
}
 
// A function that does union of two sets of x and y
// (uses union by rank)
void Union(struct subset subsets[], int xroot, int yroot)
{
 
    // Attach smaller rank tree under root of high rank tree
    // (Union by Rank)
    if (subsets[xroot].rank < subsets[yroot].rank)
        subsets[xroot].parent = yroot;
    else if (subsets[xroot].rank > subsets[yroot].rank)
        subsets[yroot].parent = xroot;
 
    // If ranks are same, then make one as root and
    // increment its rank by one
    else {
        subsets[yroot].parent = xroot;
        subsets[xroot].rank++;
    }
}
 
// The main function to check whether a given graph contains
// cycle or not
int isCycle(struct Graph* graph)
{
    int V = graph->V;
    int E = graph->E;
 
    // Allocate memory for creating V sets
    struct subset* subsets
        = (struct subset*)malloc(V * sizeof(struct subset));
 
    for (int v = 0; v < V; ++v) {
        subsets[v].parent = v;
        subsets[v].rank = 0;
    }
 
    // Iterate through all edges of graph, find sets of both
    // vertices of every edge, if sets are same, then there
    // is cycle in graph.
    for (int e = 0; e < E; ++e) {
        int x = find(subsets, graph->edge[e].src);
        int y = find(subsets, graph->edge[e].dest);
 
        if (x == y)
            return 1;
 
        Union(subsets, x, y);
    }
    return 0;
}
 
// Driver code
int main()
{
    /* Let us create the following graph
         0
        |  \
        |    \
        1-----2 */
 
    int V = 3, E = 3;
    struct Graph* graph = createGraph(V, E);
 
    // add edge 0-1
    graph->edge[0].src = 0;
    graph->edge[0].dest = 1;
 
    // add edge 1-2
    graph->edge[1].src = 1;
    graph->edge[1].dest = 2;
 
    // add edge 0-2
    graph->edge[2].src = 0;
    graph->edge[2].dest = 2;
 
    if (isCycle(graph))
        cout << "Graph contains cycle";
    else
        cout << "Graph doesn't contain cycle";
 
    return 0;
}
 
// This code is contributed by Aditya Kumar (adityakumar129)

C

// A C program to detect cycle in a graph using union by
// rank and path compression
#include 
#include 
 
// a structure to represent an edge in the graph
typedef struct Edge {
    int src, dest;
}Edge;
 
// a structure to represent a graph
typedef struct Graph {
    // V-> Number of vertices, E-> Number of edges
    int V, E;
 
    // graph is represented as an array of edges
    struct Edge* edge;
}Graph;
 
typedef struct subset {
    int parent;
    int rank;
}subset;
 
// Creates a graph with V vertices and E edges
Graph* createGraph(int V, int E)
{
    Graph* graph = (Graph*)malloc(sizeof(Graph));
    graph->V = V;
    graph->E = E;
    graph->edge = (Edge*)malloc(graph->E * sizeof(Edge));
    return graph;
}
 
// A utility function to find set of an element i
// (uses path compression technique)
int find(subset subsets[], int i)
{
    // find root and make root as parent of i (path
    // compression)
    if (subsets[i].parent != i)
        subsets[i].parent = find(subsets, subsets[i].parent);
    return subsets[i].parent;
}
 
// A function that does union of two sets of x and y
// (uses union by rank)
void Union(subset subsets[], int xroot, int yroot)
{
    // Attach smaller rank tree under root of high rank tree
    // (Union by Rank)
    if (subsets[xroot].rank < subsets[yroot].rank)
        subsets[xroot].parent = yroot;
    else if (subsets[xroot].rank > subsets[yroot].rank)
        subsets[yroot].parent = xroot;
    // If ranks are same, then make one as root and
    // increment its rank by one
    else {
        subsets[yroot].parent = xroot;
        subsets[xroot].rank++;
    }
}
 
// The main function to check whether a given graph contains
// cycle or not
int isCycle(Graph* graph)
{
    int V = graph->V;
    int E = graph->E;
    // Allocate memory for creating V sets
    subset* subsets = (subset*)malloc(V * sizeof(subset));
    for (int v = 0; v < V; ++v) {
        subsets[v].parent = v;
        subsets[v].rank = 0;
    }
    // Iterate through all edges of graph, find sets of both
    // vertices of every edge, if sets are same, then there
    // is cycle in graph.
    for (int e = 0; e < E; ++e) {
        int x = find(subsets, graph->edge[e].src);
        int y = find(subsets, graph->edge[e].dest);
        if (x == y)
            return 1;
        Union(subsets, x, y);
    }
    return 0;
}
 
// Driver code
int main()
{
    /* Let us create the following graph
         0
        |  \
        |    \
        1-----2 */
 
    int V = 3, E = 3;
    Graph* graph = createGraph(V, E);
 
    // add edge 0-1
    graph->edge[0].src = 0;
    graph->edge[0].dest = 1;
 
    // add edge 1-2
    graph->edge[1].src = 1;
    graph->edge[1].dest = 2;
 
    // add edge 0-2
    graph->edge[2].src = 0;
    graph->edge[2].dest = 2;
 
    if (isCycle(graph))
        printf("Graph contains cycle");
    else
        printf("Graph doesn't contain cycle");
    return 0;
}
 
// This code is contributed by Aditya Kumar (adityakumar129)

Java

// A union by rank and path compression
// based program to detect cycle in a graph
class Graph
{
    int V, E;
    Edge[] edge;
 
    Graph(int nV, int nE)
    {
        V = nV;
        E = nE;
        edge = new Edge[E];
        for (int i = 0; i < E; i++)
        {
            edge[i] = new Edge();
        }
    }
 
    // class to represent edge
    class Edge
    {
        int src, dest;
    }
 
    // class to represent Subset
    class subset
    {
        int parent;
        int rank;
    }
 
    // A utility function to find
    // set of an element i (uses
    // path compression technique)
    int find(subset[] subsets, int i)
    {
        if (subsets[i].parent != i)
            subsets[i].parent
                = find(subsets, subsets[i].parent);
        return subsets[i].parent;
    }
 
    // A function that does union
    // of two sets of x and y
    // (uses union by rank)
    void Union(subset[] subsets, int x, int y)
    {
        int xroot = find(subsets, x);
        int yroot = find(subsets, y);
 
        if (subsets[xroot].rank < subsets[yroot].rank)
            subsets[xroot].parent = yroot;
        else if (subsets[yroot].rank < subsets[xroot].rank)
            subsets[yroot].parent = xroot;
        else {
            subsets[xroot].parent = yroot;
            subsets[yroot].rank++;
        }
    }
 
    // The main function to check whether
    // a given graph contains cycle or not
    int isCycle(Graph graph)
    {
        int V = graph.V;
        int E = graph.E;
 
        subset[] subsets = new subset[V];
        for (int v = 0; v < V; v++) {
 
            subsets[v] = new subset();
            subsets[v].parent = v;
            subsets[v].rank = 0;
        }
 
        for (int e = 0; e < E; e++) {
            int x = find(subsets, graph.edge[e].src);
            int y = find(subsets, graph.edge[e].dest);
            if (x == y)
                return 1;
            Union(subsets, x, y);
        }
        return 0;
    }
 
    // Driver Code
    public static void main(String[] args)
    {
        /* Let us create the following graph
            0
            | \
            | \
            1-----2 */
 
        int V = 3, E = 3;
        Graph graph = new Graph(V, E);
 
        // add edge 0-1
        graph.edge[0].src = 0;
        graph.edge[0].dest = 1;
 
        // add edge 1-2
        graph.edge[1].src = 1;
        graph.edge[1].dest = 2;
 
        // add edge 0-2
        graph.edge[2].src = 0;
        graph.edge[2].dest = 2;
 
        if (graph.isCycle(graph) == 1)
            System.out.println("Graph contains cycle");
        else
            System.out.println(
                "Graph doesn't contain cycle");
    }
}
 
// This code is contributed
// by ashwani khemani

Python3

# A union by rank and path compression based
# program to detect cycle in a graph
from collections import defaultdict
 
# a structure to represent a graph
 
 
class Graph:
 
    def __init__(self, num_of_v):
        self.num_of_v = num_of_v
        self.edges = defaultdict(list)
 
    # graph is represented as an
    # array of edges
    def add_edge(self, u, v):
        self.edges[u].append(v)
 
 
class Subset:
    def __init__(self, parent, rank):
        self.parent = parent
        self.rank = rank
 
# A utility function to find set of an element
# node(uses path compression technique)
 
 
def find(subsets, node):
    if subsets[node].parent != node:
        subsets[node].parent = find(subsets, subsets[node].parent)
    return subsets[node].parent
 
# A function that does union of two sets
# of u and v(uses union by rank)
 
 
def union(subsets, u, v):
 
    # Attach smaller rank tree under root
    # of high rank tree(Union by Rank)
    if subsets[u].rank > subsets[v].rank:
        subsets[v].parent = u
    elif subsets[v].rank > subsets[u].rank:
        subsets[u].parent = v
 
    # If ranks are same, then make one as
    # root and increment its rank by one
    else:
        subsets[v].parent = u
        subsets[u].rank += 1
 
# The main function to check whether a given
# graph contains cycle or not
 
 
def isCycle(graph):
 
    # Allocate memory for creating sets
    subsets = []
 
    for u in range(graph.num_of_v):
        subsets.append(Subset(u, 0))
 
    # Iterate through all edges of graph,
    # find sets of both vertices of every
    # edge, if sets are same, then there
    # is cycle in graph.
    for u in graph.edges:
        u_rep = find(subsets, u)
 
        for v in graph.edges[u]:
            v_rep = find(subsets, v)
 
            if u_rep == v_rep:
                return True
            else:
                union(subsets, u_rep, v_rep)
 
 
# Driver Code
g = Graph(3)
 
# add edge 0-1
g.add_edge(0, 1)
 
# add edge 1-2
g.add_edge(1, 2)
 
# add edge 0-2
g.add_edge(0, 2)
 
if isCycle(g):
    print('Graph contains cycle')
else:
    print('Graph does not contain cycle')
 
# This code is contributed by
# Sampath Kumar Surine

C＃

// A union by rank and path compression
// based program to detect cycle in a graph
using System;
 
class Graph {
    public int V, E;
    public Edge[] edge;
 
    public Graph(int nV, int nE)
    {
        V = nV;
        E = nE;
        edge = new Edge[E];
        for (int i = 0; i < E; i++)
        {
            edge[i] = new Edge();
        }
    }
 
    // class to represent edge
    public class Edge {
        public int src, dest;
    }
 
    // class to represent Subset
    class subset
    {
        public int parent;
        public int rank;
    }
 
    // A utility function to find
    // set of an element i (uses
    // path compression technique)
    int find(subset[] subsets, int i)
    {
        if (subsets[i].parent != i)
            subsets[i].parent
                = find(subsets, subsets[i].parent);
        return subsets[i].parent;
    }
 
    // A function that does union
    // of two sets of x and y
    // (uses union by rank)
    void Union(subset[] subsets, int x, int y)
    {
        int xroot = find(subsets, x);
        int yroot = find(subsets, y);
 
        if (subsets[xroot].rank < subsets[yroot].rank)
            subsets[xroot].parent = yroot;
        else if (subsets[yroot].rank < subsets[xroot].rank)
            subsets[yroot].parent = xroot;
        else {
            subsets[xroot].parent = yroot;
            subsets[yroot].rank++;
        }
    }
 
    // The main function to check whether
    // a given graph contains cycle or not
    int isCycle(Graph graph)
    {
        int V = graph.V;
        int E = graph.E;
 
        subset[] subsets = new subset[V];
        for (int v = 0; v < V; v++)
        {
            subsets[v] = new subset();
            subsets[v].parent = v;
            subsets[v].rank = 0;
        }
 
        for (int e = 0; e < E; e++)
        {
            int x = find(subsets, graph.edge[e].src);
            int y = find(subsets, graph.edge[e].dest);
            if (x == y)
                return 1;
            Union(subsets, x, y);
        }
        return 0;
    }
 
    // Driver Code
    static public void Main(String[] args)
    {
        /* Let us create the following graph
            0
            | \
            | \
            1-----2 */
 
        int V = 3, E = 3;
        Graph graph = new Graph(V, E);
 
        // add edge 0-1
        graph.edge[0].src = 0;
        graph.edge[0].dest = 1;
 
        // add edge 1-2
        graph.edge[1].src = 1;
        graph.edge[1].dest = 2;
 
        // add edge 0-2
        graph.edge[2].src = 0;
        graph.edge[2].dest = 2;
 
        if (graph.isCycle(graph) == 1)
            Console.WriteLine("Graph contains cycle");
        else
            Console.WriteLine(
                "Graph doesn't contain cycle");
    }
}
 
// This code is contributed
// by Arnab Kundu

Javascript

// A union by rank and path compression
// based program to detect cycle in a graph
 
let V, E;
let edge;
 
function Graph(nV,nE)
{
    V = nV;
        E = nE;
        edge = new Array(E);
        for (let i = 0; i < E; i++)
        {
            edge[i] = new Edge();
        }
}
 
// class to represent edge
class Edge
{
    constructor()
    {
        this.src=0;
        this.dest=0;
    }
}
 
// class to represent Subset
class subset
{
    constructor()
    {
        this.parent=0;
        this.rank=0;
    }
}
 
// A utility function to find
    // set of an element i (uses
    // path compression technique)
function find(subsets,i)
{
    if (subsets[i].parent != i)
            subsets[i].parent
                = find(subsets, subsets[i].parent);
        return subsets[i].parent;
}
 
// A function that does union
    // of two sets of x and y
    // (uses union by rank)
function Union(subsets,x,y)
{
    let xroot = find(subsets, x);
        let yroot = find(subsets, y);
  
        if (subsets[xroot].rank < subsets[yroot].rank)
            subsets[xroot].parent = yroot;
        else if (subsets[yroot].rank < subsets[xroot].rank)
            subsets[yroot].parent = xroot;
        else {
            subsets[xroot].parent = yroot;
            subsets[yroot].rank++;
        }
}
 
// The main function to check whether
    // a given graph contains cycle or not
function isCycle()
{
     
  
        let subsets = new Array(V);
        for (let v = 0; v < V; v++) {
  
            subsets[v] = new subset();
            subsets[v].parent = v;
            subsets[v].rank = 0;
        }
  
        for (let e = 0; e < E; e++) {
            let x = find(subsets, edge[e].src);
            let y = find(subsets, edge[e].dest);
            if (x == y)
                return 1;
            Union(subsets, x, y);
        }
        return 0;
}
 
// Driver Code
/* Let us create the following graph
            0
            | \
            | \
            1-----2 */
  
        V = 3, E = 3;
        Graph(V, E);
  
        // add edge 0-1
        edge[0].src = 0;
        edge[0].dest = 1;
  
        // add edge 1-2
        edge[1].src = 1;
        edge[1].dest = 2;
  
        // add edge 0-2
        edge[2].src = 0;
        edge[2].dest = 2;
 
        if (isCycle() == 1)
            document.write("Graph contains cycle");
        else
            document.write(
                "Graph doesn't contain cycle");
 
 
// This code is contributed by avanitrachhadiya2155

输出

Graph contains cycle

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