Hierholzer path algorithm

src/main/java/com/thealgorithms/graph/HierholzerEulerianPath.java

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+package com.thealgorithms.graph;
+
+import java.util.ArrayDeque;
+import java.util.ArrayList;
+import java.util.Collections;
+import java.util.Deque;
+import java.util.List;
+
+/**
+ * Implementation of Hierholzer's Algorithm for finding an Eulerian Path or Circuit
+ * in a directed graph.
+ *
+ * <p>
+ * An <b>Eulerian Circuit</b> is a path that starts and ends at the same vertex
+ * and visits every edge exactly once.
+ * </p>
+ *
+ * <p>
+ * An <b>Eulerian Path</b> visits every edge exactly once but may start and end
+ * at different vertices.
+ * </p>
+ *
+ * <p>
+ * <b>Algorithm Summary:</b><br>
+ * 1. Compute indegree and outdegree for all vertices.<br>
+ * 2. Check if the graph satisfies Eulerian path or circuit conditions.<br>
+ * 3. Verify that all vertices with non-zero degree are weakly connected (undirected connectivity).<br>
+ * 4. Use Hierholzer’s algorithm to build the path by exploring unused edges iteratively.
+ * </p>
+ *
+ * <p>
+ * <b>Time Complexity:</b> O(E + V).<br>
+ * <b>Space Complexity:</b> O(V + E).
+ * </p>
+ *
+ * @author <a href="https://en.wikipedia.org/wiki/Eulerian_path#Hierholzer's_algorithm">Wikipedia: Hierholzer algorithm</a>
+ */
+public class HierholzerEulerianPath {
+
+    /**
+     * Simple directed graph represented by adjacency lists.
+     */
+    public static class Graph {
+        private final List<List<Integer>> adjacencyList;
+
+        /**
+         * Constructs a graph with a given number of vertices.
+         *
+         * @param numNodes number of vertices
+         */
+        public Graph(int numNodes) {
+            adjacencyList = new ArrayList<>();
+            for (int i = 0; i < numNodes; i++) {
+                adjacencyList.add(new ArrayList<>());
+            }
+        }
+
+        /**
+         * Adds a directed edge from vertex {@code from} to vertex {@code to}.
+         *
+         * @param from source vertex
+         * @param to   destination vertex
+         */
+        public void addEdge(int from, int to) {
+            adjacencyList.get(from).add(to);
+        }
+
+        /**
+         * Returns a list of outgoing edges from the given vertex.
+         *
+         * @param node vertex index
+         * @return list of destination vertices
+         */
+        public List<Integer> getEdges(int node) {
+            return adjacencyList.get(node);
+        }
+
+        /**
+         * Returns the number of vertices in the graph.
+         *
+         * @return number of vertices
+         */
+        public int getNumNodes() {
+            return adjacencyList.size();
+        }
+    }
+
+    private final Graph graph;
+
+    /**
+     * Creates a Hierholzer solver for the given graph.
+     *
+     * @param graph directed graph
+     */
+    public HierholzerEulerianPath(Graph graph) {
+        this.graph = graph;
+    }
+
+    /**
+     * Finds an Eulerian Path or Circuit using Hierholzer’s Algorithm.
+     *
+     * @return list of vertices representing the Eulerian Path/Circuit,
+     *         or an empty list if none exists
+     */
+    public List<Integer> findEulerianPath() {
+        int n = graph.getNumNodes();
+
+        // empty graph -> no path
+        if (n == 0) {
+            return new ArrayList<>();
+        }
+
+        int[] inDegree = new int[n];
+        int[] outDegree = new int[n];
+        int edgeCount = computeDegrees(inDegree, outDegree);
+
+        // no edges -> single vertex response requested by tests: [0]
+        if (edgeCount == 0) {
+            return Collections.singletonList(0);
+        }
+
+        int startNode = determineStartNode(inDegree, outDegree);
+        if (startNode == -1) {
+            return new ArrayList<>();
+        }
+
+        if (!allNonZeroDegreeVerticesWeaklyConnected(startNode, n, outDegree, inDegree)) {
+            return new ArrayList<>();
+        }
+
+        List<Integer> path = buildHierholzerPath(startNode, n);
+        if (path.size() != edgeCount + 1) {
+            return new ArrayList<>();
+        }
+
+        return rotateEulerianCircuitIfNeeded(path, outDegree, inDegree);
+    }
+
+    private int computeDegrees(int[] inDegree, int[] outDegree) {
+        int edgeCount = 0;
+        for (int u = 0; u < graph.getNumNodes(); u++) {
+            for (int v : graph.getEdges(u)) {
+                outDegree[u]++;
+                inDegree[v]++;
+                edgeCount++;
+            }
+        }
+        return edgeCount;
+    }
+
+    private int determineStartNode(int[] inDegree, int[] outDegree) {
+        int n = graph.getNumNodes();
+        int startNode = -1;
+        int startCount = 0;
+        int endCount = 0;
+
+        for (int i = 0; i < n; i++) {
+            int diff = outDegree[i] - inDegree[i];
+            if (diff == 1) {
+                startNode = i;
+                startCount++;
+            } else if (diff == -1) {
+                endCount++;
+            } else if (Math.abs(diff) > 1) {
+                return -1;
+            }
+        }
+
+        if (!((startCount == 1 && endCount == 1) || (startCount == 0 && endCount == 0))) {
+            return -1;
+        }
+
+        if (startNode == -1) {
+            for (int i = 0; i < n; i++) {
+                if (outDegree[i] > 0) {
+                    startNode = i;
+                    break;
+                }
+            }
+        }
+        return startNode;
+    }
+
+    private List<Integer> buildHierholzerPath(int startNode, int n) {
+        List<Deque<Integer>> tempAdj = new ArrayList<>();
+        for (int i = 0; i < n; i++) {
+            tempAdj.add(new ArrayDeque<>(graph.getEdges(i)));
+        }
+
+        Deque<Integer> stack = new ArrayDeque<>();
+        List<Integer> path = new ArrayList<>();
+        stack.push(startNode);
+
+        while (!stack.isEmpty()) {
+            int u = stack.peek();
+            if (!tempAdj.get(u).isEmpty()) {
+                stack.push(tempAdj.get(u).pollFirst());
+            } else {
+                path.add(stack.pop());
+            }
+        }
+
+        Collections.reverse(path);
+        return path;
+    }
+
+    private List<Integer> rotateEulerianCircuitIfNeeded(List<Integer> path, int[] outDegree, int[] inDegree) {
+        int startCount = 0;
+        int endCount = 0;
+        for (int i = 0; i < outDegree.length; i++) {
+            int diff = outDegree[i] - inDegree[i];
+            if (diff == 1) {
+                startCount++;
+            } else if (diff == -1) {
+                endCount++;
+            }
+        }
+
+        if (startCount == 0 && endCount == 0 && !path.isEmpty()) {
+            int preferredStart = -1;
+            for (int i = 0; i < outDegree.length; i++) {
+                if (outDegree[i] > 0) {
+                    preferredStart = i;
+                    break;
+                }
+            }
+
+            if (preferredStart != -1 && path.get(0) != preferredStart) {
+                int idx = 0;
+                for (Integer node : path) { // replaced indexed loop
+                    if (node == preferredStart) {
+                        break;
+                    }
+                    idx++;
+                }
+
+                if (idx > 0) {
+                    List<Integer> rotated = new ArrayList<>();
+                    int currentIndex = 0;
+                    for (Integer node : path) { // replaced indexed loop
+                        if (currentIndex >= idx) {
+                            rotated.add(node);
+                        }
+                        currentIndex++;
+                    }
+                    currentIndex = 0;
+                    for (Integer node : path) { // replaced indexed loop
+                        if (currentIndex < idx) {
+                            rotated.add(node);
+                        }
+                        currentIndex++;
+                    }
+                    path = rotated;
+                }
+            }
+        }
+        return path;
+    }
+
+    /**
+     * Checks weak connectivity (undirected) among vertices that have non-zero degree.
+     *
+     * @param startNode node to start DFS from (must be a vertex with non-zero degree)
+     * @param n number of vertices
+     * @param outDegree out-degree array
+     * @param inDegree in-degree array
+     * @return true if all vertices having non-zero degree belong to a single weak component
+     */
+    private boolean allNonZeroDegreeVerticesWeaklyConnected(int startNode, int n, int[] outDegree, int[] inDegree) {
+        boolean[] visited = new boolean[n];
+        Deque<Integer> stack = new ArrayDeque<>();
+        stack.push(startNode);
+        visited[startNode] = true;
+
+        while (!stack.isEmpty()) {
+            int u = stack.pop();
+            for (int v : graph.getEdges(u)) {
+                if (!visited[v]) {
+                    visited[v] = true;
+                    stack.push(v);
+                }
+            }
+            for (int x = 0; x < n; x++) {
+                if (!visited[x]) {
+                    for (int y : graph.getEdges(x)) {
+                        if (y == u) {
+                            visited[x] = true;
+                            stack.push(x);
+                            break;
+                        }
+                    }
+                }
+            }
+        }
+
+        for (int i = 0; i < n; i++) {
+            if (outDegree[i] + inDegree[i] > 0 && !visited[i]) {
+                return false;
+            }
+        }
+        return true;
+    }
+}