windNight · Jul 24, 2020 · May 18, 2020 · May 18, 2020 · May 21, 2020 · May 21, 2020
diff --git a/DataStructures/Graphs/Kruskal.java b/DataStructures/Graphs/Kruskal.java
@@ -0,0 +1,98 @@
+//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;
+    }
+}