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.
A minimum spanning tree has (V – 1) edges where V is the number of vertices in the given graph.
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 step#2 uses Union-Find algorithm to detect cycle. So we recommend to read following post as a prerequisite.
Union-Find Algorithm | Set 1 (Detect Cycle in a Graph)
Union-Find Algorithm | Set 2 (Union By Rank and Path Compression)
Union-Find Algorithm | Set 1 (Detect Cycle in a Graph)
Union-Find Algorithm | Set 2 (Union By Rank and Path Compression)
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.
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.
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