[FEAT] Implement Dials Algorithm (Graph)

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

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+package com.thealgorithms.datastructures.graphs;
+
+import java.util.ArrayList;
+import java.util.Arrays;
+import java.util.HashSet;
+import java.util.List;
+import java.util.Set;
+
+/**
+ * An implementation of Dial's Algorithm for the single-source shortest path problem.
+ * This algorithm is an optimization of Dijkstra's algorithm and is particularly
+ * efficient for graphs with small, non-negative integer edge weights.
+ *
+ * It uses a bucket queue (implemented here as a List of HashSets) to store vertices,
+ * where each bucket corresponds to a specific distance from the source. This is more
+ * efficient than a standard priority queue when the range of edge weights is small.
+ *
+ * Time Complexity: O(E + W * V), where E is the number of edges, V is the number
+ * of vertices, and W is the maximum weight of any edge.
+ *
+ * @see <a href="https://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#Dial's_algorithm">Wikipedia - Dial's Algorithm</a>
+ */
+public final class DialsAlgorithm {
+    /**
+     * Private constructor to prevent instantiation of this utility class.
+     */
+    private DialsAlgorithm() {
+    }
+    /**
+     * Represents an edge in the graph, connecting to a destination vertex with a given weight.
+     */
+    public static class Edge {
+        private final int destination;
+        private final int weight;
+
+        public Edge(int destination, int weight) {
+            this.destination = destination;
+            this.weight = weight;
+        }
+
+        public int getDestination() {
+            return destination;
+        }
+
+        public int getWeight() {
+            return weight;
+        }
+    }
+    /**
+     * Finds the shortest paths from a source vertex to all other vertices in a weighted graph.
+     *
+     * @param graph The graph represented as an adjacency list.
+     * @param source The source vertex to start from (0-indexed).
+     * @param maxEdgeWeight The maximum weight of any single edge in the graph.
+     * @return An array of integers where the value at each index `i` is the
+     * shortest distance from the source to vertex `i`. Unreachable vertices
+     * will have a value of Integer.MAX_VALUE.
+     * @throws IllegalArgumentException if the source vertex is out of bounds.
+     */
+    public static int[] run(List<List<Edge>> graph, int source, int maxEdgeWeight) {
+        int numVertices = graph.size();
+        if (source < 0 || source >= numVertices) {
+            throw new IllegalArgumentException("Source vertex is out of bounds.");
+        }
+
+        // Initialize distances array
+        int[] distances = new int[numVertices];
+        Arrays.fill(distances, Integer.MAX_VALUE);
+        distances[source] = 0;
+
+        // The bucket queue. Size is determined by the max possible path length.
+        int maxPathWeight = maxEdgeWeight * (numVertices > 0 ? numVertices - 1 : 0);
+        List<Set<Integer>> buckets = new ArrayList<>(maxPathWeight + 1);
+        for (int i = 0; i <= maxPathWeight; i++) {
+            buckets.add(new HashSet<>());
+        }
+
+        // Add the source vertex to the first bucket
+        buckets.get(0).add(source);
+
+        // Process buckets in increasing order of distance
+        for (int d = 0; d <= maxPathWeight; d++) {
+            // Process all vertices in the current bucket
+            while (!buckets.get(d).isEmpty()) {
+                // Get and remove a vertex from the current bucket
+                int u = buckets.get(d).iterator().next();
+                buckets.get(d).remove(u);
+
+                // If we've found a shorter path already, skip
+                if (d > distances[u]) {
+                    continue;
+                }
+
+                // Relax all adjacent edges
+                for (Edge edge : graph.get(u)) {
+                    int v = edge.getDestination();
+                    int weight = edge.getWeight();
+
+                    // If a shorter path to v is found
+                    if (distances[u] != Integer.MAX_VALUE && distances[u] + weight < distances[v]) {
+                        // If v was already in a bucket, remove it from the old one
+                        if (distances[v] != Integer.MAX_VALUE) {
+                            buckets.get(distances[v]).remove(v);
+                        }
+                        // Update distance and move v to the new bucket
+                        distances[v] = distances[u] + weight;
+                        buckets.get(distances[v]).add(v);
+                    }
+                }
+            }
+        }
+        return distances;
+    }
+}

src/test/java/com/thealgorithms/datastructures/graphs/DialsAlgorithmTest.java

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+package com.thealgorithms.datastructures.graphs;
+
+import static org.junit.jupiter.api.Assertions.assertArrayEquals;
+import static org.junit.jupiter.api.Assertions.assertThrows;
+
+import java.util.ArrayList;
+import java.util.List;
+import org.junit.jupiter.api.BeforeEach;
+import org.junit.jupiter.api.DisplayName;
+import org.junit.jupiter.api.Test;
+
+final class DialsAlgorithmTest {
+
+    private List<List<DialsAlgorithm.Edge>> graph;
+    private static final int NUM_VERTICES = 6;
+    private static final int MAX_EDGE_WEIGHT = 10;
+
+    @BeforeEach
+    void setUp() {
+        graph = new ArrayList<>();
+        for (int i = 0; i < NUM_VERTICES; i++) {
+            graph.add(new ArrayList<>());
+        }
+    }
+
+    private void addEdge(int u, int v, int weight) {
+        graph.get(u).add(new DialsAlgorithm.Edge(v, weight));
+    }
+
+    @Test
+    @DisplayName("Test with a simple connected graph")
+    void testSimpleGraph() {
+        // Build graph from a standard example
+        addEdge(0, 1, 2);
+        addEdge(0, 2, 4);
+        addEdge(1, 2, 1);
+        addEdge(1, 3, 7);
+        addEdge(2, 4, 3);
+        addEdge(3, 5, 1);
+        addEdge(4, 3, 2);
+        addEdge(4, 5, 5);
+
+        int[] expectedDistances = {0, 2, 3, 8, 6, 9};
+        int[] actualDistances = DialsAlgorithm.run(graph, 0, MAX_EDGE_WEIGHT);
+        assertArrayEquals(expectedDistances, actualDistances);
+    }
+
+    @Test
+    @DisplayName("Test with a disconnected node")
+    void testDisconnectedNode() {
+        addEdge(0, 1, 5);
+        addEdge(1, 2, 5);
+        // Node 3, 4, 5 are disconnected
+
+        int[] expectedDistances = {0, 5, 10, Integer.MAX_VALUE, Integer.MAX_VALUE, Integer.MAX_VALUE};
+        int[] actualDistances = DialsAlgorithm.run(graph, 0, MAX_EDGE_WEIGHT);
+        assertArrayEquals(expectedDistances, actualDistances);
+    }
+
+    @Test
+    @DisplayName("Test with source as destination")
+    void testSourceIsDestination() {
+        addEdge(0, 1, 10);
+        int[] expectedDistances = {0, 10, Integer.MAX_VALUE, Integer.MAX_VALUE, Integer.MAX_VALUE, Integer.MAX_VALUE};
+        // Run with source 0
+        int[] actualDistances = DialsAlgorithm.run(graph, 0, MAX_EDGE_WEIGHT);
+        assertArrayEquals(expectedDistances, actualDistances);
+    }
+
+    @Test
+    @DisplayName("Test graph with multiple paths to a node")
+    void testMultiplePaths() {
+        addEdge(0, 1, 10);
+        addEdge(0, 2, 3);
+        addEdge(2, 1, 2); // Shorter path to 1 is via 2 (3+2=5)
+
+        int[] expectedDistances = {0, 5, 3, Integer.MAX_VALUE, Integer.MAX_VALUE, Integer.MAX_VALUE};
+        int[] actualDistances = DialsAlgorithm.run(graph, 0, MAX_EDGE_WEIGHT);
+        assertArrayEquals(expectedDistances, actualDistances);
+    }
+
+    @Test
+    @DisplayName("Test with an invalid source vertex")
+    void testInvalidSource() {
+        assertThrows(IllegalArgumentException.class, () -> DialsAlgorithm.run(graph, -1, MAX_EDGE_WEIGHT));
+        assertThrows(IllegalArgumentException.class, () -> DialsAlgorithm.run(graph, NUM_VERTICES, MAX_EDGE_WEIGHT));
+    }
+}