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Add Johnson's algorithm (#5712)

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Saahil Mahato
2024-10-12 13:29:41 +05:45
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src/main/java/com/thealgorithms/datastructures/graphs/JohnsonsAlgorithm.java Normal file
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package com.thealgorithms.datastructures.graphs;
import java.util.ArrayList;
import java.util.Arrays;
import java.util.List;
/**
* This class implements Johnson's algorithm for finding all-pairs shortest paths in a weighted,
* directed graph that may contain negative edge weights.
*
* Johnson's algorithm works by using the Bellman-Ford algorithm to compute a transformation of the
* input graph that removes all negative weights, allowing Dijkstra's algorithm to be used for
* efficient shortest path computations.
*
* Time Complexity: O(V^2 * log(V) + V*E)
* Space Complexity: O(V^2)
*
* Where V is the number of vertices and E is the number of edges in the graph.
*
* For more information, please visit {@link https://en.wikipedia.org/wiki/Johnson%27s_algorithm}
*/
public final class JohnsonsAlgorithm {
// Constant representing infinity
private static final double INF = Double.POSITIVE_INFINITY;
/**
* A private constructor to hide the implicit public one.
*/
private JohnsonsAlgorithm() {
}
/**
* Executes Johnson's algorithm on the given graph.
*
* @param graph The input graph represented as an adjacency matrix.
* @return A 2D array representing the shortest distances between all pairs of vertices.
*/
public static double[][] johnsonAlgorithm(double[][] graph) {
int numVertices = graph.length;
double[][] edges = convertToEdgeList(graph);
// Step 1: Add a new vertex and run Bellman-Ford
double[] modifiedWeights = bellmanFord(edges, numVertices);
// Step 2: Reweight the graph
double[][] reweightedGraph = reweightGraph(graph, modifiedWeights);
// Step 3: Run Dijkstra's algorithm for each vertex
double[][] shortestDistances = new double[numVertices][numVertices];
for (int source = 0; source < numVertices; source++) {
shortestDistances[source] = dijkstra(reweightedGraph, source, modifiedWeights);
}
return shortestDistances;
}
/**
* Converts the adjacency matrix representation of the graph to an edge list.
*
* @param graph The input graph as an adjacency matrix.
* @return An array of edges, where each edge is represented as [from, to, weight].
*/
public static double[][] convertToEdgeList(double[][] graph) {
int numVertices = graph.length;
List<double[]> edgeList = new ArrayList<>();
for (int i = 0; i < numVertices; i++) {
for (int j = 0; j < numVertices; j++) {
if (i != j && !Double.isInfinite(graph[i][j])) {
// Only add edges that are not self-loops and have a finite weight
edgeList.add(new double[] {i, j, graph[i][j]});
}
}
}
// Convert the List to a 2D array
return edgeList.toArray(new double[0][]);
}
/**
* Implements the Bellman-Ford algorithm to compute the shortest paths from a new vertex
* to all other vertices. This is used to calculate the weight function h(v) for reweighting.
*
* @param edges The edge list of the graph.
* @param numVertices The number of vertices in the original graph.
* @return An array of modified weights for each vertex.
*/
private static double[] bellmanFord(double[][] edges, int numVertices) {
double[] dist = new double[numVertices + 1];
Arrays.fill(dist, INF);
dist[numVertices] = 0; // Distance to the new source vertex is 0
// Add edges from the new vertex to all original vertices
double[][] allEdges = Arrays.copyOf(edges, edges.length + numVertices);
for (int i = 0; i < numVertices; i++) {
allEdges[edges.length + i] = new double[] {numVertices, i, 0};
}
// Relax all edges V times
for (int i = 0; i < numVertices; i++) {
for (double[] edge : allEdges) {
int u = (int) edge[0];
int v = (int) edge[1];
double weight = edge[2];
if (dist[u] != INF && dist[u] + weight < dist[v]) {
dist[v] = dist[u] + weight;
}
}
}
// Check for negative weight cycles
for (double[] edge : allEdges) {
int u = (int) edge[0];
int v = (int) edge[1];
double weight = edge[2];
if (dist[u] + weight < dist[v]) {
throw new IllegalArgumentException("Graph contains a negative weight cycle");
}
}
return Arrays.copyOf(dist, numVertices);
}
/**
* Reweights the graph using the modified weights computed by Bellman-Ford.
*
* @param graph The original graph.
* @param modifiedWeights The modified weights from Bellman-Ford.
* @return The reweighted graph.
*/
public static double[][] reweightGraph(double[][] graph, double[] modifiedWeights) {
int numVertices = graph.length;
double[][] reweightedGraph = new double[numVertices][numVertices];
for (int i = 0; i < numVertices; i++) {
for (int j = 0; j < numVertices; j++) {
if (graph[i][j] != 0) {
// New weight = original weight + h(u) - h(v)
reweightedGraph[i][j] = graph[i][j] + modifiedWeights[i] - modifiedWeights[j];
}
}
}
return reweightedGraph;
}
/**
* Implements Dijkstra's algorithm for finding shortest paths from a source vertex.
*
* @param reweightedGraph The reweighted graph to run Dijkstra's on.
* @param source The source vertex.
* @param modifiedWeights The modified weights from Bellman-Ford.
* @return An array of shortest distances from the source to all other vertices.
*/
public static double[] dijkstra(double[][] reweightedGraph, int source, double[] modifiedWeights) {
int numVertices = reweightedGraph.length;
double[] dist = new double[numVertices];
boolean[] visited = new boolean[numVertices];
Arrays.fill(dist, INF);
dist[source] = 0;
for (int count = 0; count < numVertices - 1; count++) {
int u = minDistance(dist, visited);
visited[u] = true;
for (int v = 0; v < numVertices; v++) {
if (!visited[v] && reweightedGraph[u][v] != 0 && dist[u] != INF && dist[u] + reweightedGraph[u][v] < dist[v]) {
dist[v] = dist[u] + reweightedGraph[u][v];
}
}
}
// Adjust distances back to the original graph weights
for (int i = 0; i < numVertices; i++) {
if (dist[i] != INF) {
dist[i] = dist[i] - modifiedWeights[source] + modifiedWeights[i];
}
}
return dist;
}
/**
* Finds the vertex with the minimum distance value from the set of vertices
* not yet included in the shortest path tree.
*
* @param dist Array of distances.
* @param visited Array of visited vertices.
* @return The index of the vertex with minimum distance.
*/
public static int minDistance(double[] dist, boolean[] visited) {
double min = INF;
int minIndex = -1;
for (int v = 0; v < dist.length; v++) {
if (!visited[v] && dist[v] <= min) {
min = dist[v];
minIndex = v;
}
}
return minIndex;
}
}

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src/test/java/com/thealgorithms/datastructures/graphs/JohnsonsAlgorithmTest.java Normal file
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package com.thealgorithms.datastructures.graphs;
import static org.junit.jupiter.api.Assertions.assertArrayEquals;
import static org.junit.jupiter.api.Assertions.assertEquals;
import static org.junit.jupiter.api.Assertions.assertThrows;
import org.junit.jupiter.api.Test;
/**
* Unit tests for {@link JohnsonsAlgorithm} class. This class
* contains test cases to verify the correct implementation of
* various methods used in Johnson's Algorithm such as shortest path
* calculations, graph reweighting, and more.
*/
class JohnsonsAlgorithmTest {
// Constant representing infinity
private static final double INF = Double.POSITIVE_INFINITY;
/**
* Tests the Johnson's Algorithm with a simple graph without negative edges.
* Verifies that the algorithm returns the correct shortest path distances.
*/
@Test
void testSimpleGraph() {
// Test case for a simple graph without negative edges
double[][] graph = {{0, 4, INF, INF}, {INF, 0, 1, INF}, {INF, INF, 0, 2}, {INF, INF, INF, 0}};
double[][] result = JohnsonsAlgorithm.johnsonAlgorithm(graph);
double[][] expected = {{0, 4, 5, 7}, {INF, 0, 1, 3}, {INF, INF, 0, 2}, {INF, INF, INF, 0}};
assertArrayEquals(expected, result);
}
/**
* Tests Johnson's Algorithm on a graph with negative edges but no
* negative weight cycles. Verifies the algorithm handles negative
* edge weights correctly.
*/
@Test
void testGraphWithNegativeEdges() {
// Graph with negative edges but no negative weight cycles
double[][] graph = {{0, -1, 4}, {INF, 0, 3}, {INF, INF, 0}};
double[][] result = JohnsonsAlgorithm.johnsonAlgorithm(graph);
double[][] expected = {{0, INF, 4}, {INF, 0, 3}, {INF, INF, 0}};
assertArrayEquals(expected, result);
}
/**
* Tests the behavior of Johnson's Algorithm on a graph with a negative
* weight cycle. Expects an IllegalArgumentException to be thrown
* due to the presence of the cycle.
*/
@Test
void testNegativeWeightCycle() {
// Graph with a negative weight cycle
double[][] graph = {{0, 1, INF}, {INF, 0, -1}, {-1, INF, 0}};
// Johnson's algorithm should throw an exception when a negative cycle is detected
assertThrows(IllegalArgumentException.class, () -> { JohnsonsAlgorithm.johnsonAlgorithm(graph); });
}
/**
* Tests Dijkstra's algorithm as a part of Johnson's algorithm implementation
* on a small graph. Verifies that the shortest path is correctly calculated.
*/
@Test
void testDijkstra() {
// Testing Dijkstra's algorithm with a small graph
double[][] graph = {{0, 1, 2}, {INF, 0, 3}, {INF, INF, 0}};
double[] modifiedWeights = {0, 0, 0}; // No reweighting in this simple case
double[] result = JohnsonsAlgorithm.dijkstra(graph, 0, modifiedWeights);
double[] expected = {0, 1, 2};
assertArrayEquals(expected, result);
}
/**
* Tests the conversion of an adjacency matrix to an edge list.
* Verifies that the conversion process generates the correct edge list.
*/
@Test
void testEdgeListConversion() {
// Test the conversion of adjacency matrix to edge list
double[][] graph = {{0, 5, INF}, {INF, 0, 2}, {INF, INF, 0}};
// Running convertToEdgeList
double[][] edges = JohnsonsAlgorithm.convertToEdgeList(graph);
// Expected edge list: (0 -> 1, weight 5), (1 -> 2, weight 2)
double[][] expected = {{0, 1, 5}, {1, 2, 2}};
// Verify the edge list matches the expected values
assertArrayEquals(expected, edges);
}
/**
* Tests the reweighting of a graph as a part of Johnson's Algorithm.
* Verifies that the reweighted graph produces correct results.
*/
@Test
void testReweightGraph() {
// Test reweighting of the graph
double[][] graph = {{0, 2, 9}, {INF, 0, 1}, {INF, INF, 0}};
double[] modifiedWeights = {1, 2, 3}; // Arbitrary weight function
double[][] reweightedGraph = JohnsonsAlgorithm.reweightGraph(graph, modifiedWeights);
// Expected reweighted graph:
double[][] expected = {{0, 1, 7}, {INF, 0, 0}, {INF, INF, 0}};
assertArrayEquals(expected, reweightedGraph);
}
/**
* Tests the minDistance method used in Dijkstra's algorithm to find
* the vertex with the minimum distance that has not yet been visited.
*/
@Test
void testMinDistance() {
// Test minDistance method
double[] dist = {INF, 3, 1, INF};
boolean[] visited = {false, false, false, false};
int minIndex = JohnsonsAlgorithm.minDistance(dist, visited);
// The vertex with minimum distance is vertex 2 with a distance of 1
assertEquals(2, minIndex);
}
}