Kruskal’s Algorithm

In the previous section, we considered the greedy algorithm that “grows” a minimum spanning tree through a greedy inclusion of the nearest vertex to the vertices already in the tree. Remarkably, there is another greedy algorithm for the minimum spanning tree problem that also always yields an optimal solution. It is named Kruskal’s algorithm after Joseph Kruskal, who discovered this algorithm when he was a second-year graduate student [Kru56]. Kruskal’s algorithm looks at a minimum spanning tree of a weighted connected graph G = V,E as an acyclic subgraph with |V | − 1 edges for which the sum of the edge weights is the smallest. (It is not difficult to prove that such a subgraph must be a tree.) Consequently,
the algorithm constructs a minimum spanning tree as an expanding sequence of subgraphs that are always acyclic but are not necessarily connected on the intermediate stages of the algorithm.

The algorithm begins by sorting the graph’s edges in nondecreasing order of
their weights. Then, starting with the empty subgraph, it scans this sorted list,adding the next edge on the list to the current subgraph if such an inclusion does not create a cycle and simply skipping the edge otherwise.

The correctness of Kruskal’s algorithm can be proved by repeating the essential steps of the proof of Prim’s algorithm given in the previous section. The fact that ET is actually a tree in Prim’s algorithm but generally just an acyclic subgraph in Kruskal’s algorithm turns out to be an obstacle that can be overcome.

Below are the steps for finding 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.

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

Now pick all edges one by one from sorted list of edges
1. Pick edge 7-6: No cycle is formed, include it.

2. Pick edge 8-2: No cycle is formed, include it.

3. Pick edge 6-5: No cycle is formed, include it.

4. Pick edge 0-1: No cycle is formed, include it.

5. Pick edge 2-5: 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.

Kruskal’s Program in C

// C program for Kruskal's algorithm to find Minimum Spanning Tree 
// of a given connected, undirected and weighted graph 
#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 = new Graph; 
    graph->V = V; 
    graph->E = E; 
  
    graph->edge = new Edge[E]; 
  
    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 && i < graph->E) 
    { 
        // 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\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); 
  
  
    // 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; 
} 

Output:

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 O(V²), so O(LogV) are O(LogE) same. Therefore, overall time complexity is O(ElogE) or O(ElogV)