Given an undirected, connected and weighted graph, find Minimum Spanning Tree (MST) of the graph using Kruskal’s algorithm.
Input : Graph as an array of edges Output : Edges of MST are 6 - 7 2 - 8 5 - 6 0 - 1 2 - 5 2 - 3 0 - 7 3 - 4 Weight of MST is 37 Note : There are two possible MSTs, the other MST includes edge 1-2 in place of 0-7.
We have discussed below Kruskal’s MST implementations.
Below are the steps for finding MST using Kruskal’s algorithm
- Sort all the edges in non-decreasing order of their weight.
- 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.
- Repeat step#2 until there are (V-1) edges in the spanning tree.
Here are some key points which will be useful for us in implementing the Kruskal’s algorithm using STL.
- Use a vector of edges which consist of all the edges in the graph and each item of a vector will contain 3 parameters: source, destination and the cost of an edge between the source and destination.
vector<pair<int, pair<int, int> > > edges;
Here in the outer pair (i.e pair<int,pair<int,int> > ) the first element corresponds to the cost of a edge while the second element is itself a pair, and it contains two vertices of edge.
- Use the inbuilt std::sort to sort the edges in the non-decreasing order; by default the sort function sort in non-decreasing order.
- We use the Union Find Algorithm to check if it the current edge forms a cycle if it is added in the current MST. If yes discard it, else include it (union).
// Initialize result mst_weight = 0 // Create V single item sets for each vertex v parent[v] = v; rank[v] = 0; Sort all edges into non decreasing order by weight w for each (u, v) taken from the sorted list E do if FIND-SET(u) != FIND-SET(v) print edge(u, v) mst_weight += weight of edge(u, v) UNION(u, v)
Below is C++ implementation of above algorithm.
Edges of MST are 6 - 7 2 - 8 5 - 6 0 - 1 2 - 5 2 - 3 0 - 7 3 - 4 Weight of MST is 37
The above code can be optimized to stop the main loop of Kruskal when number of selected edges become V-1. We know that MST has V-1 edges and there is no point iterating after V-1 edges are selected. We have not added this optimization to keep code simple.
Time complexity and step by step illustration are discussed in previous post on Kruskal’s algorithm.
Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above