Kruskal's Minimum Spanning Tree using STL in C++

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. Greedy Algorithms | Set 2 (Kruskal’s Minimum Spanning Tree Algorithm) 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.

Here are some key points which will be useful for us in implementing the Kruskal’s algorithm using STL.

1. 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;

2. 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.
3. Use the inbuilt std::sort to sort the edges in the non-decreasing order; by default the sort function sort in non-decreasing order.
4. 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).

Pseudo Code: 

// 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

Time Complexity: O(E logV), here E is number of Edges and V is number of vertices in graph.
Auxiliary Space: O(V + E), here V is the number of vertices and E is the number of edges in the graph.

Optimization: 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.