Kruskal.java

// Problem -> Connect all the edges with the minimum cost.
// Possible Solution -> Kruskal Algorithm (KA), KA finds the minimum-spanning-tree, which means, the
// group of edges with the minimum sum of their weights that connect the whole graph.
// The graph needs to be connected, because if there are nodes impossible to reach, there are no
// edges that could connect every node in the graph.
// KA is a Greedy Algorithm, because edges are analysed based on their weights, that is why a
// Priority Queue is used, to take first those less weighted.
// This implementations below has some changes compared to conventional ones, but they are explained
// all along the code.

import java.util.Comparator;
import java.util.HashSet;
import java.util.PriorityQueue;

public class Kruskal {

  // Complexity: O(E log V) time, where E is the number of edges in the graph and V is the number of
  // vertices

  private static class Edge {
    private int from;
    private int to;
    private int weight;

    public Edge(int from, int to, int weight) {
      this.from = from;
      this.to = to;
      this.weight = weight;
    }
  }

  private static void addEdge(HashSet<Edge>[] graph, int from, int to, int weight) {
    graph[from].add(new Edge(from, to, weight));
  }

  public static void main(String[] args) {
    HashSet<Edge>[] graph = new HashSet[7];
    for (int i = 0; i < graph.length; i++) {
      graph[i] = new HashSet<>();
    }
    addEdge(graph, 0, 1, 2);
    addEdge(graph, 0, 2, 3);
    addEdge(graph, 0, 3, 3);
    addEdge(graph, 1, 2, 4);
    addEdge(graph, 2, 3, 5);
    addEdge(graph, 1, 4, 3);
    addEdge(graph, 2, 4, 1);
    addEdge(graph, 3, 5, 7);
    addEdge(graph, 4, 5, 8);
    addEdge(graph, 5, 6, 9);

    System.out.println("Initial Graph: ");
    for (int i = 0; i < graph.length; i++) {
      for (Edge edge : graph[i]) {
        System.out.println(i + " <-- weight " + edge.weight + " --> " + edge.to);
      }
    }

    Kruskal k = new Kruskal();
    HashSet<Edge>[] solGraph = k.kruskal(graph);

    System.out.println("\nMinimal Graph: ");
    for (int i = 0; i < solGraph.length; i++) {
      for (Edge edge : solGraph[i]) {
        System.out.println(i + " <-- weight " + edge.weight + " --> " + edge.to);
      }
    }
  }

  public HashSet<Edge>[] kruskal(HashSet<Edge>[] graph) {
    int nodes = graph.length;
    int[] captain = new int[nodes];
    // captain of i, stores the set with all the connected nodes to i
    HashSet<Integer>[] connectedGroups = new HashSet[nodes];
    HashSet<Edge>[] minGraph = new HashSet[nodes];
    PriorityQueue<Edge> edges = new PriorityQueue<>((Comparator.comparingInt(edge -> edge.weight)));
    for (int i = 0; i < nodes; i++) {
      minGraph[i] = new HashSet<>();
      connectedGroups[i] = new HashSet<>();
      connectedGroups[i].add(i);
      captain[i] = i;
      edges.addAll(graph[i]);
    }
    int connectedElements = 0;
    // as soon as two sets merge all the elements, the algorithm must stop
    while (connectedElements != nodes && !edges.isEmpty()) {
      Edge edge = edges.poll();
      // This if avoids cycles
      if (!connectedGroups[captain[edge.from]].contains(edge.to)
          && !connectedGroups[captain[edge.to]].contains(edge.from)) {
        // merge sets of the captains of each point connected by the edge
        connectedGroups[captain[edge.from]].addAll(connectedGroups[captain[edge.to]]);
        // update captains of the elements merged
        connectedGroups[captain[edge.from]].forEach(i -> captain[i] = captain[edge.from]);
        // add Edge to minimal graph
        addEdge(minGraph, edge.from, edge.to, edge.weight);
        // count how many elements have been merged
        connectedElements = connectedGroups[captain[edge.from]].size();
      }
    }
    return minGraph;
  }
}