What is Minimum Spanning Tree?
Given a connected and undirected graph, a spanning tree of that graph is a subgraph that is a tree and connects all the vertices together. A single graph can have many different spanning trees. A minimum spanning tree (MST) or minimum weight spanning tree for a weighted, connected and undirected graph is a spanning tree with weight less than or equal to the weight of every other spanning tree. The weight of a spanning tree is the sum of weights given to each edge of the spanning tree.

Edges does a minimum spanning tree has ?

A minimum spanning tree has (V – 1) edges where V is the number of vertices in the given graph.

Below are the steps given for finding MST using Kruskal’s Minimum Spanning Tree Algorithm

```1. Sort all the edges in non-decreasing order of their weight.

2. Pick the smallest edge. Check if it forms a cycle with the spanning tree
formed so far. If cycle is not formed, include this edge. Else, discard it.

3. Repeat step#2 until there are (V-1) edges in the spanning tree.
```

The step#2 uses Union-Find algorithm to detect cycle.

The algorithm is a Greedy Algorithm. The Greedy Choice is to pick the smallest weight edge that doesn’t cause a cycle in the MST constructed so far. Let us understand it with an instance : Consider the below graph. The graph contains 9 vertices and 14 edges. So, the minimum spanning tree formed will be having (9 – 1) = 8 edges.

```After sorting:
Weight   Src    Dest
1         7      6
2         8      2
2         6      5
4         0      1
4         2      5
6         8      6
7         2      3
7         7      8
8         0      7
8         1      2
9         3      4
10        5      4
11        1      7
14        3      5```

Now pick all edges one by one from sorted list of edges
1. Pick edge 7-6: cycle is not formed, include it. 2. Pick edge 8-2: cycle is not formed, include it. 3. Pick edge 6-5: cycle is not formed, include it. 4. Pick edge 0-1: cycle is not formed, include it. 5. Pick edge 2-5: cycle is not formed, include it. 6. Pick edge 8-6: Since including this edge results in cycle will discard it.

7. Pick edge 2-3: cycle is not formed, include it. 8. Pick edge 7-8: Since including this edge results in cycle will discard it.

9. Pick edge 0-7: cycle is not formed, include it. 10. Pick edge 1-2: Since including this edge results in cycle will discard it.

11. Pick edge 3-4: cycle is not formed, include it. Since the number of edges included equals (V – 1), the algorithm stops by here itself.

// To find Minimum Spanning Tree of a given connected graph in Kruskal’s algortihm ,

#include <stdio.h>

#include <stdlib.h>

#include <string.h>

// a structure to represent a weighted edge in graph

struct Edge

{

int src, dest, weight;

};

// a structure to represent a connected, undirected and weighted graph

struct Graph

{

// V-> Number of vertices, E-> Number of edges

int V, E;

// graph is represented as an array of edges. Since the graph is

// undirected, the edge from src to dest is also edge from dest

// to src. Both are counted as 1 edge here.

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 structure to represent a subset for union-find

struct subset

{

int parent;

int rank;

};

// 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 x, int y)

{

int xroot = find(subsets, x);

int yroot = find(subsets, y);

// 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++;

}

}

// Compare two edges according to their weights.

// Used in qsort() for sorting an array of edges

int myComp(const void* a, const void* b)

{

struct Edge* a1 = (struct Edge*)a;

struct Edge* b1 = (struct Edge*)b;

return a1->weight > b1->weight;

}

// The main function to construct MST using Kruskal’s algorithm

void KruskalMST(struct Graph* graph)

{

int V = graph->V;

struct Edge result[V]; // Tnis will store the resultant MST

int e = 0; // An index variable, used for result[]

int i = 0; // An index variable, used for sorted edges

// Step 1: Sort all the edges in non-decreasing order of their weight

// If we are not allowed to change the given graph, we can create a copy of

// array of edges

qsort(graph->edge, graph->E, sizeof(graph->edge), myComp);

// Allocate memory for creating V ssubsets

struct subset *subsets =

(struct subset*) malloc( V * sizeof(struct subset) );

// Create V subsets with single elements

for (int v = 0; v < V; ++v)

{

subsets[v].parent = v;

subsets[v].rank = 0;

}

// Number of edges to be taken is equal to V-1

while (e < V – 1)

{

// Step 2: Pick the smallest edge. And increment the index

// for next iteration

struct Edge next_edge = graph->edge[i++];

int x = find(subsets, next_edge.src);

int y = find(subsets, next_edge.dest);

// If including this edge does’t cause cycle, include it

// in result and increment the index of result for next edge

if (x != y)

{

result[e++] = next_edge;

Union(subsets, x, y);

}

}

// print the contents of result[] to display the built MST

printf(“Following are the edges in the constructed MST\n”);

for (i = 0; i < e; ++i)

printf(“%d — %d == %d\n”, result[i].src, result[i].dest,

result[i].weight);

return;

}

// Driver program to test above functions

int main()

{

/* Let us create following weighted graph

10

0——–1

| \     |

6|   5\   |15

|     \ |

2——–3

4       */

int V = 4; // Number of vertices in graph

int E = 5; // Number of edges in graph

struct Graph* graph = createGraph(V, E);

graph->edge.src = 0;

graph->edge.dest = 1;

graph->edge.weight = 10;

graph->edge.src = 0;

graph->edge.dest = 2;

graph->edge.weight = 6;

graph->edge.src = 0;

graph->edge.dest = 3;

graph->edge.weight = 5;

graph->edge.src = 1;

graph->edge.dest = 3;

graph->edge.weight = 15;

graph->edge.src = 2;

graph->edge.dest = 3;

graph->edge.weight = 4;

KruskalMST(graph);

return 0;

}

```Following are the edges in the constructed MST
2 -- 3 == 4
0 -- 3 == 5
0 -- 1 == 10```

Time Complexity: O(ElogE) or O(ElogV). Sorting of edges takes O(ELogE) time. After sorting, we iterate through all edges and apply find-union algorithm. The find and union operations can take atmost O(LogV) time. So overall complexity is O(ELogE + ELogV) time. The value of E can be atmost V^2, so O(LogV) are O(LogE) same. Therefore, overall time complexity is O(ElogE) or O(ElogV)