You may refer to the main idea of MST in graph theory.
http://en.wikipedia.org/wiki/Minimum_spanning_tree
Here is my own interpretation
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.
How many 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.
Founding MST using Kruskal’s 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.
Analise
The algorithm is a Greedy Algorithm. The Greedy Choice is to pick the smallest weight edge that does not cause a cycle in the MST constructed so far. Let us understand it with an example: Consider the below input 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
1. Pick edge 7-6: No cycle is formed, include it.
6. Pick edge 8-6: Since including this edge results in cycle, discard it.
7. Pick edge 2-3: No cycle is formed, include it.
8. Pick edge 7-8: Since including this edge results in cycle, discard it.
9. Pick edge 0-7: No cycle is formed, include it.
10. Pick edge 1-2: Since including this edge results in cycle, discard it.
11. Pick edge 3-4: No cycle is formed, include it.
Since the number of edges included equals (V – 1), the algorithm stops here.
Here is the source code demonstrating the procedure.
#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[0]), 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); } // Else discard the next_edge } // print the contents of result[] to display the built MST printf("Following are the edges in the constructed MST "); for (i = 0; i < e; ++i) printf("%d -- %d == %d ", 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); // add edge 0-1 graph->edge[0].src = 0; graph->edge[0].dest = 1; graph->edge[0].weight = 10; // add edge 0-2 graph->edge[1].src = 0; graph->edge[1].dest = 2; graph->edge[1].weight = 6; // add edge 0-3 graph->edge[2].src = 0; graph->edge[2].dest = 3; graph->edge[2].weight = 5; // add edge 1-3 graph->edge[3].src = 1; graph->edge[3].dest = 3; graph->edge[3].weight = 15; // add edge 2-3 graph->edge[4].src = 2; graph->edge[4].dest = 3; graph->edge[4].weight = 4; KruskalMST(graph); return 0; }