Merge branch 'master' into master

Author: BYT-Bender

src/main/java/com/thealgorithms/datastructures/graphs/JohnsonsAlgorithm.java

@@ -0,0 +1,203 @@
+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;
+    }
+}

src/main/java/com/thealgorithms/maths/FastExponentiation.java

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+package com.thealgorithms.maths;
+
+/**
+ * This class provides a method to perform fast exponentiation (exponentiation by squaring),
+ * which calculates (base^exp) % mod efficiently.
+ *
+ * <p>The algorithm works by repeatedly squaring the base and reducing the exponent
+ * by half at each step. It exploits the fact that:
+ * <ul>
+ *   <li>If exp is even, (base^exp) = (base^(exp/2))^2</li>
+ *   <li>If exp is odd, (base^exp) = base * (base^(exp-1))</li>
+ * </ul>
+ * The result is computed modulo `mod` at each step to avoid overflow and keep the result within bounds.
+ * </p>
+ *
+ * <p><strong>Time complexity:</strong> O(log(exp)) — much faster than naive exponentiation (O(exp)).</p>
+ *
+ * For more information, please visit {@link https://en.wikipedia.org/wiki/Exponentiation_by_squaring}
+ */
+public final class FastExponentiation {
+
+    /**
+     * Private constructor to hide the implicit public one.
+     */
+    private FastExponentiation() {
+    }
+
+    /**
+     * Performs fast exponentiation to calculate (base^exp) % mod using the method
+     * of exponentiation by squaring.
+     *
+     * <p>This method efficiently computes the result by squaring the base and halving
+     * the exponent at each step. It multiplies the base to the result when the exponent is odd.
+     *
+     * @param base the base number to be raised to the power of exp
+     * @param exp the exponent to which the base is raised
+     * @param mod the modulus to ensure the result does not overflow
+     * @return (base^exp) % mod
+     * @throws IllegalArgumentException if the modulus is less than or equal to 0
+     * @throws ArithmeticException if the exponent is negative (not supported in this implementation)
+     */
+    public static long fastExponentiation(long base, long exp, long mod) {
+        if (mod <= 0) {
+            throw new IllegalArgumentException("Modulus must be positive.");
+        }
+
+        if (exp < 0) {
+            throw new ArithmeticException("Negative exponent is not supported.");
+        }
+
+        long result = 1;
+        base = base % mod; // Take the modulus of the base to handle large base values
+
+        // Fast exponentiation by squaring algorithm
+        while (exp > 0) {
+            // If exp is odd, multiply the base to the result
+            if ((exp & 1) == 1) { // exp & 1 checks if exp is odd
+                result = result * base % mod;
+            }
+            // Square the base and halve the exponent
+            base = base * base % mod; // base^2 % mod to avoid overflow
+            exp >>= 1; // Right shift exp to divide it by 2
+        }
+
+        return result;
+    }
+}