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.
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- Prim’s Minimum Spanning Tree (MST) | Greedy Algo-5
- Kruskal’s Minimum Spanning Tree Algorithm | Greedy Algo-2
- Types of Spanning Tree Protocol (STP)
- Spanning Tree With Maximum Degree (Using Kruskal's Algorithm)
- Maximum Possible Edge Disjoint Spanning Tree From a Complete Graph
- Problem Solving for Minimum Spanning Trees (Kruskal’s and Prim’s)
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- Find the root of the sub-tree whose weighted sum is minimum
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