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
- 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.
- 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).
vector<pair<int, pair<int, int> > > edges;
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.
C++
// C++ program for Kruskal's algorithm to find Minimum // Spanning Tree of a given connected, undirected and // weighted graph #include<bits/stdc++.h> using namespace std; // Creating shortcut for an integer pair typedef pair<int, int> iPair; // Structure to represent a graph struct Graph { int V, E; vector< pair<int, iPair> > edges; // Constructor Graph(int V, int E) { this->V = V; this->E = E; } // Utility function to add an edge void addEdge(int u, int v, int w) { edges.push_back({w, {u, v}}); } // Function to find MST using Kruskal's // MST algorithm int kruskalMST(); }; // To represent Disjoint Sets struct DisjointSets { int *parent, *rnk; int n; // Constructor. DisjointSets(int n) { // Allocate memory this->n = n; parent = new int[n+1]; rnk = new int[n+1]; // Initially, all vertices are in // different sets and have rank 0. for (int i = 0; i <= n; i++) { rnk[i] = 0; //every element is parent of itself parent[i] = i; } } // Find the parent of a node 'u' // Path Compression int find(int u) { /* Make the parent of the nodes in the path from u--> parent[u] point to parent[u] */ if (u != parent[u]) parent[u] = find(parent[u]); return parent[u]; } // Union by rank void merge(int x, int y) { x = find(x), y = find(y); /* Make tree with smaller height a subtree of the other tree */ if (rnk[x] > rnk[y]) parent[y] = x; else // If rnk[x] <= rnk[y] parent[x] = y; if (rnk[x] == rnk[y]) rnk[y]++; } }; /* Functions returns weight of the MST*/ int Graph::kruskalMST() { int mst_wt = 0; // Initialize result // Sort edges in increasing order on basis of cost sort(edges.begin(), edges.end()); // Create disjoint sets DisjointSets ds(V); // Iterate through all sorted edges vector< pair<int, iPair> >::iterator it; for (it=edges.begin(); it!=edges.end(); it++) { int u = it->second.first; int v = it->second.second; int set_u = ds.find(u); int set_v = ds.find(v); // Check if the selected edge is creating // a cycle or not (Cycle is created if u // and v belong to same set) if (set_u != set_v) { // Current edge will be in the MST // so print it cout << u << " - " << v << endl; // Update MST weight mst_wt += it->first; // Merge two sets ds.merge(set_u, set_v); } } return mst_wt; } // Driver program to test above functions int main() { /* Let us create above shown weighted and undirected graph */ int V = 9, E = 14; Graph g(V, E); // making above shown graph g.addEdge(0, 1, 4); g.addEdge(0, 7, 8); g.addEdge(1, 2, 8); g.addEdge(1, 7, 11); g.addEdge(2, 3, 7); g.addEdge(2, 8, 2); g.addEdge(2, 5, 4); g.addEdge(3, 4, 9); g.addEdge(3, 5, 14); g.addEdge(4, 5, 10); g.addEdge(5, 6, 2); g.addEdge(6, 7, 1); g.addEdge(6, 8, 6); g.addEdge(7, 8, 7); cout << "Edges of MST are \n"; int mst_wt = g.kruskalMST(); cout << "\nWeight of MST is " << mst_wt; return 0; } |
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.
This article is contributed by Chirag Agrawal. If you like neveropen and would like to contribute, you can also write an article and mail your article to review-team@geeksforgeeks.org. See your article appearing on the neveropen main page and help other Geeks.
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