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# Minimum Spanning Tree using Priority Queue and Array List

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 approach``import` `java.io.*;``import` `java.util.*;``import` `java.lang.Comparable;``public` `class` `MST {` `    ``// Pair class with implemented comparable``    ``static` `class` `Pair,``                                ``V ``extends` `Comparable >``        ``implements` `Comparable > {` `        ``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 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,``                    ``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 >();``            ``}``        ``}` `        ``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,``                               ``Integer> >``                ``priorityQueue``                ``= ``new` `PriorityQueue<>();` `            ``// Adding all the connected vertices``            ``// of MinCostEdge vertex A to PQ``            ``Iterator > iterator``                ``= edges[minCostEdge.a.a].listIterator();``            ``while` `(iterator.hasNext()) {``                ``Pair 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 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 addedVertices = ``new` `HashSet<>();` `            ``// Set contains all the added edges and cost from source``            ``Set, 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, 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 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 pair1``                            ``= iterator.next();``                        ``priorityQueue``                            ``.add(``                                ``new` `Pair<>(``                                    ``new` `Pair<>(pair.a.b, pair1.a),``                                    ``pair1.b));``                    ``}``                ``}``            ``}` `            ``// Printing the MST``            ``Iterator, 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)```

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