104 lines
3.7 KiB
Java
104 lines
3.7 KiB
Java
// Problem -> Connect all the edges with the minimum cost.
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// Possible Solution -> Kruskal Algorithm (KA), KA finds the minimum-spanning-tree, which means, the
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// group of edges with the minimum sum of their weights that connect the whole graph.
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// The graph needs to be connected, because if there are nodes impossible to reach, there are no
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// edges that could connect every node in the graph.
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// KA is a Greedy Algorithm, because edges are analysed based on their weights, that is why a
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// Priority Queue is used, to take first those less weighted.
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// This implementations below has some changes compared to conventional ones, but they are explained
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// all along the code.
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import java.util.Comparator;
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import java.util.HashSet;
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import java.util.PriorityQueue;
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public class Kruskal {
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// Complexity: O(E log V) time, where E is the number of edges in the graph and V is the number of
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// vertices
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private static class Edge {
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private int from;
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private int to;
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private int weight;
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public Edge(int from, int to, int weight) {
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this.from = from;
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this.to = to;
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this.weight = weight;
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}
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}
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private static void addEdge(HashSet<Edge>[] graph, int from, int to, int weight) {
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graph[from].add(new Edge(from, to, weight));
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}
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public static void main(String[] args) {
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HashSet<Edge>[] graph = new HashSet[7];
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for (int i = 0; i < graph.length; i++) {
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graph[i] = new HashSet<>();
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}
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addEdge(graph, 0, 1, 2);
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addEdge(graph, 0, 2, 3);
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addEdge(graph, 0, 3, 3);
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addEdge(graph, 1, 2, 4);
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addEdge(graph, 2, 3, 5);
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addEdge(graph, 1, 4, 3);
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addEdge(graph, 2, 4, 1);
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addEdge(graph, 3, 5, 7);
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addEdge(graph, 4, 5, 8);
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addEdge(graph, 5, 6, 9);
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System.out.println("Initial Graph: ");
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for (int i = 0; i < graph.length; i++) {
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for (Edge edge : graph[i]) {
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System.out.println(i + " <-- weight " + edge.weight + " --> " + edge.to);
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}
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}
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Kruskal k = new Kruskal();
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HashSet<Edge>[] solGraph = k.kruskal(graph);
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System.out.println("\nMinimal Graph: ");
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for (int i = 0; i < solGraph.length; i++) {
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for (Edge edge : solGraph[i]) {
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System.out.println(i + " <-- weight " + edge.weight + " --> " + edge.to);
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}
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}
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}
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public HashSet<Edge>[] kruskal(HashSet<Edge>[] graph) {
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int nodes = graph.length;
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int[] captain = new int[nodes];
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// captain of i, stores the set with all the connected nodes to i
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HashSet<Integer>[] connectedGroups = new HashSet[nodes];
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HashSet<Edge>[] minGraph = new HashSet[nodes];
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PriorityQueue<Edge> edges = new PriorityQueue<>((Comparator.comparingInt(edge -> edge.weight)));
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for (int i = 0; i < nodes; i++) {
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minGraph[i] = new HashSet<>();
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connectedGroups[i] = new HashSet<>();
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connectedGroups[i].add(i);
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captain[i] = i;
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edges.addAll(graph[i]);
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}
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int connectedElements = 0;
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// as soon as two sets merge all the elements, the algorithm must stop
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while (connectedElements != nodes && !edges.isEmpty()) {
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Edge edge = edges.poll();
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// This if avoids cycles
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if (!connectedGroups[captain[edge.from]].contains(edge.to)
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&& !connectedGroups[captain[edge.to]].contains(edge.from)) {
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// merge sets of the captains of each point connected by the edge
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connectedGroups[captain[edge.from]].addAll(connectedGroups[captain[edge.to]]);
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// update captains of the elements merged
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connectedGroups[captain[edge.from]].forEach(i -> captain[i] = captain[edge.from]);
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// add Edge to minimal graph
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addEdge(minGraph, edge.from, edge.to, edge.weight);
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// count how many elements have been merged
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connectedElements = connectedGroups[captain[edge.from]].size();
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}
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}
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return minGraph;
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}
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}
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