Merge branch 'main' of github.com:4ndrelim/data-structures-and-algorithms

Author: 4ndrelim

docs/assets/images/MSTProperty3.png

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+# Minimum Spanning Tree Algorithms
+
+## Background
+
+Minimum Spanning Tree (MST) algorithms are used to find the minimum spanning tree of a weighted, connected graph. A
+spanning tree of a graph is a connected, acyclic subgraph that includes all the vertices of the original graph. An MST 
+is a spanning tree with the minimum possible total edge weight.
+
+### 4 Properties of MST
+1. An MST should not have any cycles
+2. If you cut an MST at any single edge, the two pieces will also be MSTs
+3. **Cycle Property:** For every cycle, the maximum weight edge is not in the MST
+
+![MST Property 3](../../../../../docs/assets/images/MSTProperty3.png)
+
+Image Source: CS2040S 22/23 Sem 2 Lecture Slides
+
+4. **Cut Property:** For every partition of the nodes, the minimum weight edge across the cut is in the MST
+
+![MST Property 4](../../../../../docs/assets/images/MSTProperty4.png)
+   
+Image Source: CS2040S 22/23 Sem 2 Lecture Slides
+
+Note that the other edges across the partition may or may not be in the MST.
+
+## Prim's Algorithm and Kruskal's Algorithm
+
+We will discuss more implementation-specific details and complexity analysis in the respective folders. In short,
+1. [Prim's Algorithm](prim) is a greedy algorithm that finds the minimum spanning tree of a graph by starting from an
+arbitrary node (vertex) and adding the edge with the minimum weight that connects the current tree to a new node, adding
+the node to the current tree, until all nodes are included in the tree.
+<<<<<<< HEAD
+2. [Kruskal's Algorithm](kruskal) is a greedy algorithm that finds the minimum spanning tree of a graph by sorting the
+edges by weight and adding the edge with the minimum weight that does not form a cycle into the current tree.
+
+## Notes
+
+### Difference between Minimum Spanning Tree and Shortest Path
+It is important to note that a Minimum Spanning Tree of a graph does not represent the shortest path between all the
+nodes. See below for an example:
+
+The below graph is a weighted, connected graph with 5 nodes and 6 edges:
+![original graph img](../../../../../docs/assets/images/originalGraph.jpg)
+
+The following is the Minimum Spanning Tree of the above graph:
+![MST img](../../../../../docs/assets/images/MST.jpg)
+
+Taking node A and D into consideration, the shortest path between them is A -> D, with a total weight of 4.
+![SPOriginal img](../../../../../docs/assets/images/SPOriginal.jpg)
+
+However, the shortest path between A and D in the Minimum Spanning Tree is A -> C -> D, with a total weight of 5, which
+is not the shortest path in the original graph.
+![SPMST img](../../../../../docs/assets/images/SPMST.jpg)

docs/assets/images/MSTProperty4.png

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+package algorithms.minimumSpanningTree.kruskal;
+
+import java.util.ArrayList;
+import java.util.List;
+
+import dataStructures.disjointSet.weightedUnion.DisjointSet;
+
+/**
+ * Implementation of Kruskal's Algorithm to find MSTs
+ * Idea:
+ *  Sort all edges by weight in non-decreasing order. Consider the edges in this order. If an edge does not form a cycle
+ *  with the edges already in the MST, add it to the MST. Repeat until all nodes are in the MST.
+ * Actual implementation:
+ *  An Edge class is implemented for easier sorting of edges by weight and for identifying the source and destination.
+ *  A Node class is implemented for easier tracking of nodes in the graph for the disjoint set.
+ *  A DisjointSet class is used to track the nodes in the graph and to determine if adding an edge will form a cycle.
+ */
+public class Kruskal {
+    public static int[][] getKruskalMST(Node[] nodes, int[][] adjacencyMatrix) {
+        int numOfNodes = nodes.length;
+        List<Edge> edges = new ArrayList<>();
+
+        // Convert adjacency matrix to list of edges
+        for (int i = 0; i < numOfNodes; i++) {
+            for (int j = i + 1; j < numOfNodes; j++) {
+                if (adjacencyMatrix[i][j] != Integer.MAX_VALUE) {
+                    edges.add(new Edge(nodes[i], nodes[j], adjacencyMatrix[i][j]));
+                }
+            }
+        }
+
+        // Sort edges by weight
+        edges.sort(Edge::compareTo);
+
+        // Initialize Disjoint Set for vertex tracking
+        DisjointSet<Node> ds = new DisjointSet<>(nodes);
+
+        // MST adjacency matrix to be returned
+        int[][] mstMatrix = new int[numOfNodes][numOfNodes];
+
+        // Initialize the MST matrix to represent no edges with Integer.MAX_VALUE and 0 for self loops
+        for (int i = 0; i < nodes.length; i++) {
+            for (int j = 0; j < nodes.length; j++) {
+                mstMatrix[i][j] = (i == j) ? 0 : Integer.MAX_VALUE;
+            }
+        }
+
+        // Process edges to build MST
+        for (Edge edge : edges) {
+            Node source = edge.getSource();
+            Node destination = edge.getDestination();
+            if (!ds.find(source, destination)) {
+                mstMatrix[source.getIndex()][destination.getIndex()] = edge.getWeight();
+                mstMatrix[destination.getIndex()][source.getIndex()] = edge.getWeight();
+                ds.union(source, destination);
+            }
+        }
+
+        return mstMatrix;
+    }
+
+    /**
+     * Node class to represent a node in the graph
+     * Note: In our Node class, we do not allow the currMinWeight to be updated after initialization to prevent any
+     * reference issues in the PriorityQueue.
+     */
+    static class Node {
+        private final int index; // Index of this node in the adjacency matrix
+        private final String identifier;
+
+        /**
+         * Constructor for Node
+         * @param identifier
+         * @param index
+         */
+        public Node(String identifier, int index) {
+            this.identifier = identifier;
+            this.index = index;
+        }
+
+        /**
+         * Getter for identifier
+         * @return identifier
+         */
+        public String getIdentifier() {
+            return identifier;
+        }
+
+        public int getIndex() {
+            return index;
+        }
+
+        @Override
+        public String toString() {
+            return "Node{" + "identifier='" + identifier + '\'' + ", index=" + index + '}';
+        }
+    }
+
+    /**
+     * Edge class to represent an edge in the graph
+     */
+    static class Edge implements Comparable<Edge> {
+        private final Node source;
+        private final Node destination;
+        private final int weight;
+
+        /**
+         * Constructor for Edge
+         */
+        public Edge(Node source, Node destination, int weight) {
+            this.source = source;
+            this.destination = destination;
+            this.weight = weight;
+        }
+
+        public int getWeight() {
+            return weight;
+        }
+
+        public Node getSource() {
+            return source;
+        }
+
+        public Node getDestination() {
+            return destination;
+        }
+
+        @Override
+        public int compareTo(Edge other) {
+            return Integer.compare(this.weight, other.weight);
+        }
+    }
+}
+

src/main/java/algorithms/minimumSpanningTree/README.md

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+# Kruskal's Algorithm
+
+## Background
+Kruskal's Algorithm is a greedy algorithm used to find the minimum spanning tree (MST) of a connected, weighted graph.
+It works by sorting all the edges in the graph by their weight in non-decreasing order and then adding the smallest edge
+to the MST, provided it does not form a cycle with the already included edges. This is repeated until all vertices are
+included in the MST.
+
+## Implementation Details
+Kruskal's Algorithm uses a simple `ArrayList` to sort the edges by weight. 
+
+A [`DisjointSet`](/dataStructures/disjointSet/weightedUnion) data structure is also used to keep track of the
+connectivity of vertices and detect cycles.
+
+## Complexity Analysis
+
+**Time Complexity:**
+Sorting the edges by weight: O(E log E) = O(E log V), where V and E is the number of vertices and edges respectively.
+Union-Find operations: O(E α(V)), where α is the inverse Ackermann function.
+Overall complexity: O(E log V)
+
+**Space Complexity:**
+O(V + E) for the storage of vertices in the disjoint set and edges in the priority queue.

src/main/java/algorithms/minimumSpanningTree/kruskal/Kruskal.java

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+package algorithms.minimumSpanningTree.prim;
+
+import java.util.Arrays;
+import java.util.PriorityQueue;
+
+/**
+ * Implementation of Prim's Algorithm to find MSTs
+ * Idea:
+ *  Starting from any source (this will be the first node to be in the MST), pick the lightest outgoing edge, and
+ *  include the node at the other end as part of a set of nodes S. Now repeatedly do the above by picking the lightest
+ *  outgoing edge adjacent to any node in the MST (ensure the other end of the node is not already in the MST).
+ *  Repeat until S contains all nodes in the graph. S is the MST.
+ * Actual implementation:
+ *  No Edge class was implemented. Instead, the weights of the edges are stored in a 2D array adjacency matrix. An
+ *  adjacency list may be used instead
+ *  A Node class is implemented to encapsulate the current minimum weight to reach the node.
+ */
+public class Prim {
+    public static int[][] getPrimsMST(Node[] nodes, int[][] adjacencyMatrix) {
+        // Recall that PriorityQueue is a min heap by default
+        PriorityQueue<Node> pq = new PriorityQueue<>((a, b) -> a.getCurrMinWeight() - b.getCurrMinWeight());
+        int[][] mstMatrix = new int[nodes.length][nodes.length]; // MST adjacency matrix
+
+        int[] parent = new int[nodes.length]; // To track the parent node of each node in the MST
+        Arrays.fill(parent, -1); // Initialize parent array with -1, indicating no parent
+
+        boolean[] visited = new boolean[nodes.length]; // To track visited nodes
+        Arrays.fill(visited, false); // Initialize visited array with false, indicating not visited
+
+        // Initialize the MST matrix to represent no edges with Integer.MAX_VALUE and 0 for self loops
+        for (int i = 0; i < nodes.length; i++) {
+            for (int j = 0; j < nodes.length; j++) {
+                mstMatrix[i][j] = (i == j) ? 0 : Integer.MAX_VALUE;
+            }
+        }
+
+        // Add all nodes to the priority queue, with each node's curr min weight already set to Integer.MAX_VALUE
+        pq.addAll(Arrays.asList(nodes));
+
+        while (!pq.isEmpty()) {
+            Node current = pq.poll();
+
+            int currentIndex = current.getIndex();
+
+            if (visited[currentIndex]) { // Skip if node is already visited
+                continue;
+            }
+
+            visited[currentIndex] = true;
+
+            for (int i = 0; i < nodes.length; i++) {
+                if (adjacencyMatrix[currentIndex][i] != Integer.MAX_VALUE && !visited[nodes[i].getIndex()]) {
+                    int weight = adjacencyMatrix[currentIndex][i];
+
+                    if (weight < nodes[i].getCurrMinWeight()) {
+                        Node newNode = new Node(nodes[i].getIdentifier(), nodes[i].getIndex(), weight);
+                        parent[i] = currentIndex; // Set current node as parent of adjacent node
+                        pq.add(newNode);
+                    }
+                }
+            }
+        }
+
+        // Build MST matrix based on parent array
+        for (int i = 1; i < nodes.length; i++) {
+            int p = parent[i];
+            if (p != -1) {
+                int weight = adjacencyMatrix[p][i];
+                mstMatrix[p][i] = weight;
+                mstMatrix[i][p] = weight; // For undirected graphs
+            }
+        }
+
+        return mstMatrix;
+    }
+
+    /**
+     * Node class to represent a node in the graph
+     * Note: In our Node class, we do not allow the currMinWeight to be updated after initialization to prevent any
+     * reference issues in the PriorityQueue.
+     */
+    static class Node {
+        private final int currMinWeight; // Current minimum weight to get to this node
+        private int index; // Index of this node in the adjacency matrix
+        private final String identifier;
+
+        /**
+         * Constructor for Node
+         * @param identifier
+         * @param index
+         * @param currMinWeight
+         */
+        public Node(String identifier, int index, int currMinWeight) {
+            this.identifier = identifier;
+            this.index = index;
+            this.currMinWeight = currMinWeight;
+        }
+
+        /**
+         * Constructor for Node with default currMinWeight
+         * @param identifier
+         * @param index
+         */
+        public Node(String identifier, int index) {
+            this.identifier = identifier;
+            this.index = index;
+            this.currMinWeight = Integer.MAX_VALUE;
+        }
+
+        /**
+         * Getter and setter for currMinWeight
+         */
+        public int getCurrMinWeight() {
+            return currMinWeight;
+        }
+
+        /**
+         * Getter for identifier
+         * @return identifier
+         */
+        public String getIdentifier() {
+            return identifier;
+        }
+
+        public int getIndex() {
+            return index;
+        }
+
+        @Override
+        public String toString() {
+            return "Node{" + "identifier='" + identifier + '\'' + ", index=" + index + '}';
+        }
+    }
+}
+