Union-Find Algorithm | Set 1 (Detect Cycle in an Undirected Graph)-Graph cycle-A disjoint-set data structure is a data structure that keeps track of a set.

A *disjoint-set data structure* is a data structure that keeps track of a set of elements partitioned into a number of disjoint (non-overlapping) subsets. A *union-find algorithm* is an algorithm that performs two useful operations on such a data structure:

**Find:** Determine which subset a particular element is in. This can be used for determining if two elements are in the same subset.

**Union:** Join two subsets into a single subset.

In this post, we will discuss an application of Disjoint Set Data Structure. The application is to check whether a given graph contains a cycle or not.

*Union-Find Algorithm* can be used to check whether an undirected graph contains cycle or not. Note that we have discussed an algorithm to detect cycle. This is another method based on *Union-Find*. This method assumes that graph doesn’t contain any self-loops.

We can keeps track of the subsets in a 1D array, lets call it parent[].

Let us consider the following graph:

For each edge, make subsets using both the vertices of the edge. If both the vertices are in the same subset, a cycle is found.

Initially, all slots of parent array are initialized to -1 (means there is only one item in every subset).

0 1 2
-1 -1 -1

Now process all edges one by one.

*Edge 0-1:* Find the subsets in which vertices 0 and 1 are. Since they are in different subsets, we take the union of them. For taking the union, either make node 0 as parent of node 1 or vice-versa.

0 1 2 <----- 1 is made parent of 0 (1 is now representative of subset {0, 1})
1 -1 -1

*Edge 1-2:* 1 is in subset 1 and 2 is in subset 2. So, take union.

0 1 2 <----- 2 is made parent of 1 (2 is now representative of subset {0, 1, 2})
1 2 -1

*Edge 0-2:* 0 is in subset 2 and 2 is also in subset 2. Hence, including this edge forms a cycle.

How subset of 0 is same as 2?

0->1->2 // 1 is parent of 0 and 2 is parent of 1

Based on the above explanation, below are implementations:

```
// A union-find algorithm to detect cycle in a graph
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
// a structure to represent an edge in 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;
};
// 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 the subset of an element i
int find(int parent[], int i)
{
if (parent[i] == -1)
return i;
return find(parent, parent[i]);
}
// A utility function to do union of two subsets
void Union(int parent[], int x, int y)
{
int xset = find(parent, x);
int yset = find(parent, y);
parent[xset] = yset;
}
// The main function to check whether a given graph contains
// cycle or not
int isCycle( struct Graph* graph )
{
// Allocate memory for creating V subsets
int *parent = (int*) malloc( graph->V * sizeof(int) );
// Initialize all subsets as single element sets
memset(parent, -1, sizeof(int) * graph->V);
// Iterate through all edges of graph, find subset of both
// vertices of every edge, if both subsets are same, then
// there is cycle in graph.
for(int i = 0; i < graph->E; ++i)
{
int x = find(parent, graph->edge[i].src);
int y = find(parent, graph->edge[i].dest);
if (x == y)
return 1;
Union(parent, x, y);
}
return 0;
}
// Driver program to test above functions
int main()
{
/* Let us create 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))
printf( "graph contains cycle" );
else
printf( "graph doesn't contain cycle" );
return 0;
}
```

Output:

graph contains cycle