# Kruskal’s Minimum Spanning Tree using STL in C++

Kruskal’s Minimum Spanning Tree using STL in C++ - STL Implementation of Algorithms - Use a vector of edges which consist of all the edges in the graph.

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;`

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.

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

``````// 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 unidrected graph */
int V = 9, E = 14;
Graph g(V, E);

//  making above shown graph

cout << "Edges of MST are \n";
int mst_wt = g.kruskalMST();

cout << "\nWeight of MST is " << mst_wt;

return 0;
}``````

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```
READ  Java Algorithm - Maximum Bipartite Matching

#### Venkatesan Prabu

Wikitechy Founder, Author, International Speaker, and Job Consultant. My role as the CEO of Wikitechy, I help businesses build their next generation digital platforms and help with their product innovation and growth strategy. I'm a frequent speaker at tech conferences and events.

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