Kruskal's Minimum Spanning Tree Algorithm - Greedy Algorithm in C++-C++-php.cn

Kruskal's Minimum Spanning Tree Algorithm - Greedy Algorithm in C++

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Release： 2023-08-28 15:05:07

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A spanning tree is a directed undirected graph subgraph that connects all vertices. There can be many spanning trees in a graph. The minimum spanning tree (MST) on each graph has the same or smaller weight than all other spanning trees. Weights are assigned to the edges of the spanning tree, and the sum is the weight assigned to each edge. Since V is the number of vertices in the graph, the number of edges of the minimum spanning tree is (V - 1), where V is the number of edges.

Use Kruskal's algorithm to find the minimum spanning tree

All edges should be sorted in non-descending order by weight.
Select the smallest edge. If no loop is formed, the edge is included.
Step 2 should be performed until the spanning tree has (V-1) edges.

In this case, we are told to use the greedy approach. The greedy option is to choose the edge with the smallest weight. For example: the minimum spanning tree of this graph is (9-1)= 8 edges.

Kruskals Minimum Spanning Tree Algorithm - Greedy Algorithm in C++

After sorting:

Weight  Src    Dest
21       27    26
22       28    22
22       26    25
24       20    21
24       22    25
26       28    26
27       22    23
27       27    28
28       20    27
28       21    22
29       23    24
30       25    24
31       21    27
34       23    25

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Now we need to select all edges based on sorting.

Contains edge 26-27->, because no loop is formed

Contains edge 28-22->, because no loop is formed

Includes edge 26- 25->, because no loop is formed.

Edge 20-21-> is included because no loop is formed

Edge 22-25-> is included because no loop is formed.

Edge 28-26-> dropped due to loop formation

Edge 22-23-> > included because no loop was formed

Edge 27-28- > Discarded due to loop formation

Edge 20-27-> Included because no loop was formed

Edge 21-22-> Discarded due to loop formation

Edge 23-24->Included because no cycle is formed

Since the number of edges is (V-1), the algorithm ends here.

Example

#include <stdio.h>
#include <stdlib.h>
#include <string.h>
struct Edge {
   int src, dest, weight;
};
struct Graph {
   int V, E;
   struct Edge* edge;
};
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(sizeof( struct Edge)*E);
   return graph;
}
struct subset {
   int parent;
   int rank;
};
int find(struct subset subsets[], int i){
   if (subsets[i].parent != i)
      subsets[i].parent
   = find(subsets, subsets[i].parent);
   return subsets[i].parent;
}
void Union(struct 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[xroot].rank > subsets[yroot].rank)
      subsets[yroot].parent = xroot;
   else{
      subsets[yroot].parent = xroot;
      subsets[xroot].rank++;
   }
}
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;
}
void KruskalMST(struct Graph* graph){
   int V = graph->V;
   struct Edge
   result[V];
   int e = 0;
   int i = 0;
   qsort(graph->edge, graph->E, sizeof(graph->edge[0]), myComp);
   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;
   }
   while (e < V - 1 && i < graph->E) {
      struct Edge next_edge = graph->edge[i++];
      int x = find(subsets, next_edge.src);
      int y = find(subsets, next_edge.dest);
      if (x != y) {
         result[e++] = next_edge;
         Union(subsets, x, y);
      }
   }
   printf("Following are the edges in the constructed MST\n");
   int minimumCost = 0;
   for (i = 0; i < e; ++i){
      printf("%d -- %d == %d\n", result[i].src,
      result[i].dest, result[i].weight);
      minimumCost += result[i].weight;
   }
   printf("Minimum Cost Spanning tree : %d",minimumCost);
   return;
}
int main(){
   /* Let us create the following weighted graph
   30
   0--------1
   | \       |
   26| 25\ |15
   | \ |
   22--------23
   24 */
   int V = 24;
   int E = 25;
   struct Graph* graph = createGraph(V, E);
   graph->edge[0].src = 20;
   graph->edge[0].dest = 21;
   graph->edge[0].weight = 30;
   graph->edge[1].src = 20;
   graph->edge[1].dest = 22;
   graph->edge[1].weight = 26;
   graph->edge[2].src = 20;
   graph->edge[2].dest = 23;
   graph->edge[2].weight = 25;
   graph->edge[3].src = 21;
   graph->edge[3].dest = 23;
   graph->edge[3].weight = 35;
   graph->edge[4].src = 22;
   graph->edge[4].dest = 23;
   graph->edge[4].weight = 24;
   KruskalMST(graph);
   return 0;
}

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Output

Following are the edges in the constructed MST
22 -- 23 == 24
20 -- 23 == 25
20 -- 21 == 30
Minimum Cost Spanning tree : 79

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Conclusion

This tutorial demonstrates how to use Kruskal's Minimum Spanning Tree Algorithm - Greedy method and C code to solve this question. We can also write this code in java, python and other languages. It is modeled after Kruskal's concept. This program finds the shortest spanning tree in a given graph. We hope you found this tutorial helpful.

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