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Add Johnson's algorithm (#5712)
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Saahil Mahato
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package com.thealgorithms.datastructures.graphs;
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import java.util.ArrayList;
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import java.util.Arrays;
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import java.util.List;
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/**
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* This class implements Johnson's algorithm for finding all-pairs shortest paths in a weighted,
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* directed graph that may contain negative edge weights.
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*
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* Johnson's algorithm works by using the Bellman-Ford algorithm to compute a transformation of the
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* input graph that removes all negative weights, allowing Dijkstra's algorithm to be used for
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* efficient shortest path computations.
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*
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* Time Complexity: O(V^2 * log(V) + V*E)
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* Space Complexity: O(V^2)
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*
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* Where V is the number of vertices and E is the number of edges in the graph.
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*
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* For more information, please visit {@link https://en.wikipedia.org/wiki/Johnson%27s_algorithm}
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*/
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public final class JohnsonsAlgorithm {
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// Constant representing infinity
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private static final double INF = Double.POSITIVE_INFINITY;
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/**
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* A private constructor to hide the implicit public one.
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*/
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private JohnsonsAlgorithm() {
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}
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/**
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* Executes Johnson's algorithm on the given graph.
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*
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* @param graph The input graph represented as an adjacency matrix.
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* @return A 2D array representing the shortest distances between all pairs of vertices.
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*/
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public static double[][] johnsonAlgorithm(double[][] graph) {
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int numVertices = graph.length;
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double[][] edges = convertToEdgeList(graph);
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// Step 1: Add a new vertex and run Bellman-Ford
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double[] modifiedWeights = bellmanFord(edges, numVertices);
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// Step 2: Reweight the graph
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double[][] reweightedGraph = reweightGraph(graph, modifiedWeights);
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// Step 3: Run Dijkstra's algorithm for each vertex
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double[][] shortestDistances = new double[numVertices][numVertices];
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for (int source = 0; source < numVertices; source++) {
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shortestDistances[source] = dijkstra(reweightedGraph, source, modifiedWeights);
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}
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return shortestDistances;
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}
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/**
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* Converts the adjacency matrix representation of the graph to an edge list.
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*
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* @param graph The input graph as an adjacency matrix.
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* @return An array of edges, where each edge is represented as [from, to, weight].
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*/
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public static double[][] convertToEdgeList(double[][] graph) {
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int numVertices = graph.length;
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List<double[]> edgeList = new ArrayList<>();
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for (int i = 0; i < numVertices; i++) {
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for (int j = 0; j < numVertices; j++) {
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if (i != j && !Double.isInfinite(graph[i][j])) {
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// Only add edges that are not self-loops and have a finite weight
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edgeList.add(new double[] {i, j, graph[i][j]});
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}
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}
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}
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// Convert the List to a 2D array
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return edgeList.toArray(new double[0][]);
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}
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/**
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* Implements the Bellman-Ford algorithm to compute the shortest paths from a new vertex
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* to all other vertices. This is used to calculate the weight function h(v) for reweighting.
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*
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* @param edges The edge list of the graph.
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* @param numVertices The number of vertices in the original graph.
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* @return An array of modified weights for each vertex.
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*/
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private static double[] bellmanFord(double[][] edges, int numVertices) {
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double[] dist = new double[numVertices + 1];
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Arrays.fill(dist, INF);
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dist[numVertices] = 0; // Distance to the new source vertex is 0
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// Add edges from the new vertex to all original vertices
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double[][] allEdges = Arrays.copyOf(edges, edges.length + numVertices);
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for (int i = 0; i < numVertices; i++) {
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allEdges[edges.length + i] = new double[] {numVertices, i, 0};
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}
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// Relax all edges V times
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for (int i = 0; i < numVertices; i++) {
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for (double[] edge : allEdges) {
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int u = (int) edge[0];
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int v = (int) edge[1];
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double weight = edge[2];
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if (dist[u] != INF && dist[u] + weight < dist[v]) {
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dist[v] = dist[u] + weight;
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}
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}
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}
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// Check for negative weight cycles
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for (double[] edge : allEdges) {
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int u = (int) edge[0];
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int v = (int) edge[1];
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double weight = edge[2];
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if (dist[u] + weight < dist[v]) {
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throw new IllegalArgumentException("Graph contains a negative weight cycle");
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}
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}
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return Arrays.copyOf(dist, numVertices);
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}
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/**
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* Reweights the graph using the modified weights computed by Bellman-Ford.
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*
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* @param graph The original graph.
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* @param modifiedWeights The modified weights from Bellman-Ford.
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* @return The reweighted graph.
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*/
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public static double[][] reweightGraph(double[][] graph, double[] modifiedWeights) {
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int numVertices = graph.length;
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double[][] reweightedGraph = new double[numVertices][numVertices];
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for (int i = 0; i < numVertices; i++) {
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for (int j = 0; j < numVertices; j++) {
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if (graph[i][j] != 0) {
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// New weight = original weight + h(u) - h(v)
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reweightedGraph[i][j] = graph[i][j] + modifiedWeights[i] - modifiedWeights[j];
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}
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}
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}
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return reweightedGraph;
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}
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/**
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* Implements Dijkstra's algorithm for finding shortest paths from a source vertex.
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*
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* @param reweightedGraph The reweighted graph to run Dijkstra's on.
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* @param source The source vertex.
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* @param modifiedWeights The modified weights from Bellman-Ford.
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* @return An array of shortest distances from the source to all other vertices.
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*/
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public static double[] dijkstra(double[][] reweightedGraph, int source, double[] modifiedWeights) {
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int numVertices = reweightedGraph.length;
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double[] dist = new double[numVertices];
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boolean[] visited = new boolean[numVertices];
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Arrays.fill(dist, INF);
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dist[source] = 0;
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for (int count = 0; count < numVertices - 1; count++) {
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int u = minDistance(dist, visited);
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visited[u] = true;
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for (int v = 0; v < numVertices; v++) {
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if (!visited[v] && reweightedGraph[u][v] != 0 && dist[u] != INF && dist[u] + reweightedGraph[u][v] < dist[v]) {
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dist[v] = dist[u] + reweightedGraph[u][v];
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}
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}
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}
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// Adjust distances back to the original graph weights
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for (int i = 0; i < numVertices; i++) {
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if (dist[i] != INF) {
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dist[i] = dist[i] - modifiedWeights[source] + modifiedWeights[i];
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}
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}
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return dist;
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}
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/**
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* Finds the vertex with the minimum distance value from the set of vertices
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* not yet included in the shortest path tree.
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*
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* @param dist Array of distances.
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* @param visited Array of visited vertices.
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* @return The index of the vertex with minimum distance.
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*/
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public static int minDistance(double[] dist, boolean[] visited) {
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double min = INF;
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int minIndex = -1;
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for (int v = 0; v < dist.length; v++) {
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if (!visited[v] && dist[v] <= min) {
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min = dist[v];
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minIndex = v;
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}
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}
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return minIndex;
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}
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}
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@ -0,0 +1,136 @@
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package com.thealgorithms.datastructures.graphs;
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import static org.junit.jupiter.api.Assertions.assertArrayEquals;
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import static org.junit.jupiter.api.Assertions.assertEquals;
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import static org.junit.jupiter.api.Assertions.assertThrows;
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import org.junit.jupiter.api.Test;
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/**
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* Unit tests for {@link JohnsonsAlgorithm} class. This class
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* contains test cases to verify the correct implementation of
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* various methods used in Johnson's Algorithm such as shortest path
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* calculations, graph reweighting, and more.
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*/
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class JohnsonsAlgorithmTest {
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// Constant representing infinity
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private static final double INF = Double.POSITIVE_INFINITY;
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/**
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* Tests the Johnson's Algorithm with a simple graph without negative edges.
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* Verifies that the algorithm returns the correct shortest path distances.
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*/
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@Test
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void testSimpleGraph() {
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// Test case for a simple graph without negative edges
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double[][] graph = {{0, 4, INF, INF}, {INF, 0, 1, INF}, {INF, INF, 0, 2}, {INF, INF, INF, 0}};
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double[][] result = JohnsonsAlgorithm.johnsonAlgorithm(graph);
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double[][] expected = {{0, 4, 5, 7}, {INF, 0, 1, 3}, {INF, INF, 0, 2}, {INF, INF, INF, 0}};
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assertArrayEquals(expected, result);
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}
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/**
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* Tests Johnson's Algorithm on a graph with negative edges but no
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* negative weight cycles. Verifies the algorithm handles negative
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* edge weights correctly.
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*/
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@Test
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void testGraphWithNegativeEdges() {
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// Graph with negative edges but no negative weight cycles
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double[][] graph = {{0, -1, 4}, {INF, 0, 3}, {INF, INF, 0}};
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double[][] result = JohnsonsAlgorithm.johnsonAlgorithm(graph);
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double[][] expected = {{0, INF, 4}, {INF, 0, 3}, {INF, INF, 0}};
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assertArrayEquals(expected, result);
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}
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/**
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* Tests the behavior of Johnson's Algorithm on a graph with a negative
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* weight cycle. Expects an IllegalArgumentException to be thrown
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* due to the presence of the cycle.
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*/
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@Test
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void testNegativeWeightCycle() {
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// Graph with a negative weight cycle
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double[][] graph = {{0, 1, INF}, {INF, 0, -1}, {-1, INF, 0}};
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// Johnson's algorithm should throw an exception when a negative cycle is detected
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assertThrows(IllegalArgumentException.class, () -> { JohnsonsAlgorithm.johnsonAlgorithm(graph); });
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}
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/**
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* Tests Dijkstra's algorithm as a part of Johnson's algorithm implementation
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* on a small graph. Verifies that the shortest path is correctly calculated.
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*/
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@Test
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void testDijkstra() {
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// Testing Dijkstra's algorithm with a small graph
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double[][] graph = {{0, 1, 2}, {INF, 0, 3}, {INF, INF, 0}};
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double[] modifiedWeights = {0, 0, 0}; // No reweighting in this simple case
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double[] result = JohnsonsAlgorithm.dijkstra(graph, 0, modifiedWeights);
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double[] expected = {0, 1, 2};
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assertArrayEquals(expected, result);
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}
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/**
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* Tests the conversion of an adjacency matrix to an edge list.
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* Verifies that the conversion process generates the correct edge list.
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*/
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@Test
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void testEdgeListConversion() {
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// Test the conversion of adjacency matrix to edge list
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double[][] graph = {{0, 5, INF}, {INF, 0, 2}, {INF, INF, 0}};
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// Running convertToEdgeList
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double[][] edges = JohnsonsAlgorithm.convertToEdgeList(graph);
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// Expected edge list: (0 -> 1, weight 5), (1 -> 2, weight 2)
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double[][] expected = {{0, 1, 5}, {1, 2, 2}};
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// Verify the edge list matches the expected values
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assertArrayEquals(expected, edges);
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}
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/**
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* Tests the reweighting of a graph as a part of Johnson's Algorithm.
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* Verifies that the reweighted graph produces correct results.
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*/
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@Test
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void testReweightGraph() {
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// Test reweighting of the graph
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double[][] graph = {{0, 2, 9}, {INF, 0, 1}, {INF, INF, 0}};
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double[] modifiedWeights = {1, 2, 3}; // Arbitrary weight function
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double[][] reweightedGraph = JohnsonsAlgorithm.reweightGraph(graph, modifiedWeights);
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// Expected reweighted graph:
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double[][] expected = {{0, 1, 7}, {INF, 0, 0}, {INF, INF, 0}};
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assertArrayEquals(expected, reweightedGraph);
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}
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/**
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* Tests the minDistance method used in Dijkstra's algorithm to find
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* the vertex with the minimum distance that has not yet been visited.
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*/
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@Test
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void testMinDistance() {
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// Test minDistance method
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double[] dist = {INF, 3, 1, INF};
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boolean[] visited = {false, false, false, false};
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int minIndex = JohnsonsAlgorithm.minDistance(dist, visited);
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// The vertex with minimum distance is vertex 2 with a distance of 1
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assertEquals(2, minIndex);
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}
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}
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