Given a bi-directed weighted (positive) graph without self-loops, the task is to generate the minimum spanning tree of the graph.
Examples:
Input: N = 9, E = 14, edges = {{0, 1, 4}, {0, 7, 8}, {1, 2, 8}, {1, 7, 11}, {2, 3, 7}, {2, 8, 2}, {2, 5, 4}, {3, 4, 9}, {3, 5, 14}, {4, 5, 10}, {5, 6, 2}, {6, 7, 1}, {6, 8, 6}, {7, 8, 7}}
Output:
((A, B), Cost)
((6, 7), 1)
((6, 5), 2)
((1, 0), 4)
((2, 3), 7)
((5, 2), 4)
((3, 4), 9)
((2, 1), 8)
((2, 8), 2)
An undirected graph consisting of all the vertices V and (V-1) edges has been generated
Input: N = 6, E = 14, edges = {{0, 2, 103}, {0, 1, 158}, {0, 2, 2}, {0, 5, 17}, {1, 3, 42}, {2, 4, 187}, {3, 0, 14}, {3, 2, 158}, {3, 5, 106}, {3, 4, 95}, {5, 1, 144}, {5, 2, 194}, {5, 3, 118}, {5, 3, 58}}
Output:
((A, B), Cost)
((0, 2), 2)
((0, 3), 14)
((0, 5), 17)
((3, 1), 42)
((3, 4), 95)
Approach
- First, the edge having minimum cost/weight is found in the given graph.
- The two initial vertices (vertex A, B of minimum cost edge) is added to visited/added set.
- Now, all the connected edges with newly added vertex are added to priority queue.
- The least cost vertex (add all the connected edges of pop vertex to priority queue) is popped from the priority queue and repeat until number of edges is equal to vertices-1.
- By using priority queue time complexity will be reduced to (O(E log V)) where E is the number of edges and V is the number of vertices.
- Pair class is also used to store the weights.
Below is the implementation of the above approach:
Java
// Java implementation of the approachimport java.io.*;
import java.util.*;
import java.lang.Comparable;
public class MST {
// Pair class with implemented comparable
static class Pair<U extends Comparable<U>,
V extends Comparable<V> >
implements Comparable<Pair<U, V> > {
public final U a;
public final V b;
private Pair(U a, V b)
{
this.a = a;
this.b = b;
}
@Override
public boolean equals(Object o)
{
if (this == o)
return true;
if (o == null || getClass() != o.getClass())
return false;
Pair<?, ?> pair = (Pair<?, ?>)o;
if (!a.equals(pair.a))
return false;
return b.equals(pair.b);
}
// Overriding so that objects in map
// could find the object key
@Override
public int hashCode()
{
return 31 * a.hashCode() + b.hashCode();
}
@Override
public String toString()
{
return "(" + a + ", " + b + ")";
}
@Override
public int compareTo(Pair<U, V> o)
{
return getV().compareTo(o.getV());
}
private U getU()
{
return a;
}
private V getV()
{
return b;
}
}
static class Graph {
int vertices;
ArrayList[] edges;
// This variable keeps the least cost edge
static Pair<Pair<Integer, Integer>,
Integer>
minCostEdge;
Graph(int vertices)
{
minCostEdge = new Pair<>(new Pair<>(1, 1),
Integer.MAX_VALUE);
this.vertices = vertices;
edges = new ArrayList[vertices + 1];
for (int i = 0; i <= vertices; i++) {
edges[i]
= new ArrayList<Pair<Integer, Integer> >();
}
}
void addEdge(int a, int b, int weight)
{
edges[a].add(new Pair<>(b, weight));
// Since its undirected, adding the
// edges to both the vertices
edges[b].add(new Pair<>(a, weight));
if (weight < minCostEdge.b) {
minCostEdge
= new Pair<>(new Pair<>(a, b), weight);
}
}
void MST()
{
// Priority queue for applying heap
PriorityQueue<Pair<Pair<Integer, Integer>,
Integer> >
priorityQueue
= new PriorityQueue<>();
// Adding all the connected vertices
// of MinCostEdge vertex A to PQ
Iterator<Pair<Integer, Integer> > iterator
= edges[minCostEdge.a.a].listIterator();
while (iterator.hasNext()) {
Pair<Integer, Integer> pair
= iterator.next();
priorityQueue.add(
new Pair<>(
new Pair<>(minCostEdge.a.a, pair.a),
pair.b));
}
// Adding all the connected vertices
// of MinCostEdge vertex B to PQ
iterator = edges[minCostEdge.a.b].listIterator();
while (iterator.hasNext()) {
Pair<Integer, Integer> pair = iterator.next();
priorityQueue.add(
new Pair<>(
new Pair<>(minCostEdge.a.b, pair.a),
pair.b));
}
// Set to check vertex is added or not
Set<Integer> addedVertices = new HashSet<>();
// Set contains all the added edges and cost from source
Set<Pair<Pair<Integer, Integer>, Integer> > addedEdges
= new HashSet<>();
// Using the greedy approach to find
// the least costing edge to the GRAPH
while (addedEdges.size() < vertices - 1) {
// Polling from priority queue
Pair<Pair<Integer, Integer>, Integer> pair
= priorityQueue.poll();
// Checking whether the vertex A is added or not
if (!addedVertices.contains(pair.a.a)) {
addedVertices.add(pair.a.a);
addedEdges.add(pair);
// Adding all the connected vertices with vertex A
iterator = edges[pair.a.a].listIterator();
while (iterator.hasNext()) {
Pair<Integer, Integer> pair1
= iterator.next();
priorityQueue.add(
new Pair<>(
new Pair<>(pair.a.a, pair1.a),
pair1.b));
}
}
// Checking whether the vertex B is added or not
if (!addedVertices.contains(pair.a.b)) {
addedVertices.add(pair.a.b);
addedEdges.add(pair);
// Adding all the connected vertices with vertex B
iterator = edges[pair.a.b].listIterator();
while (iterator.hasNext()) {
Pair<Integer, Integer> pair1
= iterator.next();
priorityQueue
.add(
new Pair<>(
new Pair<>(pair.a.b, pair1.a),
pair1.b));
}
}
}
// Printing the MST
Iterator<Pair<Pair<Integer, Integer>, Integer> > iterator1
= addedEdges.iterator();
System.out.println("((A"
+ ", "
+ "B)"
+ ", "
+ "Cost)");
while (iterator1.hasNext()) {
System.out.println(iterator1.next());
}
}
}
// Driver code
public static void main(String[] args) throws IOException
{
// Initializing the graph
Graph g = new Graph(9);
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);
// Applying MST
g.MST();
}
} |
Output:
((A, B), Cost) ((6, 7), 1) ((6, 5), 2) ((1, 0), 4) ((2, 3), 7) ((5, 2), 4) ((3, 4), 9) ((2, 1), 8) ((2, 8), 2)