DSA – Data Structures and Algorithm

Chapter 23: Advanced Data Structures Advanced data structures are crucial for solving complex computational problems efficiently. They allow us to manage data in a way that reduces the time complexity of operations such as searching, updating, and querying large datasets. In this chapter, we will focus on three key advanced data structures: Hash Tables Tries Segment Trees and Fenwick Trees Each of these structures is designed to address different computational challenges and is widely used in competitive programming and real-world applications. Let’s explore these structures in detail. 23.1 Hash Tables A Hash Table is a data structure that provides an efficient way to store and retrieve data. It uses a hash function to map keys to values, enabling constant time average complexity for operations such as insertion, deletion, and search. 23.1.1 The Structure of Hash Tables A hash table consists of an array, where each position corresponds to an index produced by a hash function. The hash function converts a key (usually a string or an integer) into an index that maps to the array position where the associated value is stored. Hash Function: A function that converts a key into an index. Collisions: When two different keys produce the same index, a collision occurs. Hash tables handle collisions using two primary methods: Chaining: Multiple key-value pairs are stored in a list at the same index. Open Addressing: The hash table searches for the next available index. 23.1.2 Operations in Hash Tables Insert(key, value): Insert a key-value pair into the hash table. Search(key): Retrieve the value associated with the key. Delete(key): Remove the key-value pair from the hash table. Example Implementation of a Hash Table Here’s a simple implementation of a hash table using chaining in C. #include <stdio.h> #include <stdlib.h> #include <string.h> #define TABLE_SIZE 10 struct Node { int key; int value; struct Node* next; }; struct Node* hashTable[TABLE_SIZE]; int hashFunction(int key) { return key % TABLE_SIZE; } void insert(int key, int value) { int hashIndex = hashFunction(key); struct Node* newNode = (struct Node*) malloc(sizeof(struct Node)); newNode->key = key; newNode->value = value; newNode->next = hashTable[hashIndex]; hashTable[hashIndex] = newNode; } int search(int key) { int hashIndex = hashFunction(key); struct Node* node = hashTable[hashIndex]; while (node != NULL) { if (node->key == key) { return node->value; } node = node->next; } return -1; } void delete(int key) { int hashIndex = hashFunction(key); struct Node* node = hashTable[hashIndex]; struct Node* prev = NULL; while (node != NULL && node->key != key) { prev = node; node = node->next; } if (node == NULL) return; if (prev == NULL) { hashTable[hashIndex] = node->next; } else { prev->next = node->next; } free(node); } 23.2 Tries (Prefix Trees) A Trie (pronounced “try”) is a tree-like data structure that is used for storing strings efficiently. It is especially useful for scenarios where we need to quickly search for strings or prefixes of strings, such as in dictionary implementations and autocomplete systems. 23.2.1 Structure of a Trie A Trie stores strings character by character. Each node in the Trie represents a character, and a path from the root to a node represents a string or a prefix. The Trie allows us to check whether a string exists, whether a prefix exists, and retrieve all strings that start with a given prefix. Each node of a Trie has: Children: References to its child nodes, representing the next characters. End of Word Marker: A boolean value indicating if the node represents the end of a word. 23.2.2 Trie Operations Insert(word): Inserts a word into the Trie. Search(word): Checks if the word is present in the Trie. Prefix Search(prefix): Checks if there is any word in the Trie that starts with the given prefix. Example Implementation of a Trie #include <stdio.h> #include <stdlib.h> #include <string.h> #define ALPHABET_SIZE 26 struct TrieNode { struct TrieNode* children[ALPHABET_SIZE]; int isEndOfWord; }; struct TrieNode* getNode() { struct TrieNode* node = (struct TrieNode*) malloc(sizeof(struct TrieNode)); node->isEndOfWord = 0; for (int i = 0; i < ALPHABET_SIZE; i++) { node->children[i] = NULL; } return node; } void insert(struct TrieNode* root, const char* key) { struct TrieNode* current = root; for (int i = 0; i < strlen(key); i++) { int index = key[i] – ‘a’; if (!current->children[index]) { current->children[index] = getNode(); } current = current->children[index]; } current->isEndOfWord = 1; } int search(struct TrieNode* root, const char* key) { struct TrieNode* current = root; for (int i = 0; i < strlen(key); i++) { int index = key[i] – ‘a’; if (!current->children[index]) { return 0; } current = current->children[index]; } return current->isEndOfWord; } 23.3 Segment Trees and Fenwick Trees 23.3.1 Segment Trees A Segment Tree is a data structure used for efficiently performing range queries and updates on an array. For example, if we want to calculate the sum, minimum, or maximum of elements in a range of an array and frequently update the array values, a segment tree provides a more efficient solution compared to a simple array. A segment tree divides the array into segments, and each node in the tree stores information about a specific range. It allows range queries and point updates in O(log n) time. Operations on Segment Tree Build Tree: Construct the segment tree from an array. Range Query: Query a range for sum, minimum, maximum, etc. Update: Update an element in the array and reflect the changes in the segment tree. Example Implementation of a Segment Tree #include <stdio.h> int segmentTree[1000]; void buildTree(int arr[], int start, int end, int node) { if (start == end) { segmentTree[node] = arr[start]; } else { int mid = (start + end) / 2; buildTree(arr, start, mid, 2 * node + 1); buildTree(arr, mid + 1, end, 2 * node + 2); segmentTree[node] = segmentTree[2 * node + 1] + segmentTree[2 * node + 2]; } } int rangeQuery(int start, int end, int l, int r, int node) { if (l > end || r < start) { return 0; } if (l <= start && r >= end) { return segmentTree[node]; } int

Chapter 22: Advanced Graph Algorithms In this chapter, we delve deeper into Advanced Graph Algorithms that are essential for solving complex problems related to graph traversal, connectivity, and more. These algorithms are foundational in many computer science applications, from navigating networks to solving puzzles and finding connected components. We will explore the following topics: Depth-First Search (DFS) Breadth-First Search (BFS) Topological Sorting Strongly Connected Components (Kosaraju’s Algorithm) 22.1 Depth-First Search (DFS) Depth-First Search (DFS) is a fundamental algorithm used to explore all the vertices and edges of a graph. DFS explores as far as possible along each branch before backtracking, which makes it useful for tasks like detecting cycles or solving maze-like problems. Problem Definition: Given a graph, perform DFS traversal from a starting vertex, visiting each vertex once. DFS Algorithm Steps: Start at a given vertex and mark it as visited. For each unvisited neighbor, recursively perform DFS. If all neighbors are visited, backtrack to the previous vertex. DFS Algorithm (in C): #include <stdio.h> #include <stdbool.h> #define V 5 void DFSUtil(int graph[V][V], int v, bool visited[]) { visited[v] = true; printf("%d ", v); for (int i = 0; i < V; i++) { if (graph[v][i] == 1 && !visited[i]) { DFSUtil(graph, i, visited); } } } void DFS(int graph[V][V], int startVertex) { bool visited[V] = {false}; DFSUtil(graph, startVertex, visited); } int main() { int graph[V][V] = { {0, 1, 0, 0, 1}, {1, 0, 1, 0, 0}, {0, 1, 0, 1, 0}, {0, 0, 1, 0, 1}, {1, 0, 0, 1, 0} }; printf("DFS starting from vertex 0:\n"); DFS(graph, 0); return 0; } Explanation: The graph is represented as an adjacency matrix. The DFSUtil() function recursively visits each vertex and its adjacent vertices. visited[] keeps track of the vertices that have already been explored to avoid revisiting. Time Complexity: The time complexity of DFS is O(V²) for an adjacency matrix representation and O(V + E) for an adjacency list, where V is the number of vertices and E is the number of edges. 22.2 Breadth-First Search (BFS) Breadth-First Search (BFS) is another fundamental graph traversal algorithm that explores the vertices level by level. BFS is often used to find the shortest path in unweighted graphs and in applications such as solving puzzles or network broadcasting. BFS Algorithm Steps: Start from a given vertex and mark it as visited. Visit all the adjacent vertices of the current vertex. Repeat for each vertex, visiting the vertices in a breadth-first manner. BFS Algorithm (in C): #include <stdio.h> #include <stdbool.h> #define V 5 void BFS(int graph[V][V], int startVertex) { bool visited[V] = {false}; int queue[V], front = 0, rear = 0; visited[startVertex] = true; queue[rear++] = startVertex; while (front != rear) { int currentVertex = queue[front++]; printf("%d ", currentVertex); for (int i = 0; i < V; i++) { if (graph[currentVertex][i] == 1 && !visited[i]) { visited[i] = true; queue[rear++] = i; } } } } int main() { int graph[V][V] = { {0, 1, 0, 0, 1}, {1, 0, 1, 0, 0}, {0, 1, 0, 1, 0}, {0, 0, 1, 0, 1}, {1, 0, 0, 1, 0} }; printf("BFS starting from vertex 0:\n"); BFS(graph, 0); return 0; } Explanation: The graph is represented as an adjacency matrix. The queue[] is used to maintain the vertices that need to be explored. The visited[] array keeps track of the visited vertices. Time Complexity: The time complexity of BFS is O(V²) for an adjacency matrix representation and O(V + E) for an adjacency list. 22.3 Topological Sorting Topological Sorting is an ordering of vertices in a Directed Acyclic Graph (DAG) where, for every directed edge (u, v), vertex u comes before vertex v. It is widely used in scheduling problems where some tasks must precede others. Topological Sort Algorithm (DFS-based): Perform DFS on the graph. Push the vertices onto a stack when DFS finishes visiting all vertices from that vertex. Pop all vertices from the stack to get the topological order. Topological Sort Algorithm (in C): #include <stdio.h> #include <stdbool.h> #define V 6 void topologicalSortUtil(int graph[V][V], int v, bool visited[], int stack[], int *index) { visited[v] = true; for (int i = 0; i < V; i++) { if (graph[v][i] == 1 && !visited[i]) { topologicalSortUtil(graph, i, visited, stack, index); } } stack[(*index)–] = v; } void topologicalSort(int graph[V][V]) { bool visited[V] = {false}; int stack[V], index = V – 1; for (int i = 0; i < V; i++) { if (!visited[i]) { topologicalSortUtil(graph, i, visited, stack, &index); } } printf("Topological Sort:\n"); for (int i = 0; i < V; i++) { printf("%d ", stack[i]); } printf("\n"); } int main() { int graph[V][V] = { {0, 1, 1, 0, 0, 0}, {0, 0, 0, 1, 0, 0}, {0, 0, 0, 1, 1, 0}, {0, 0, 0, 0, 0, 1}, {0, 0, 0, 0, 0, 1}, {0, 0, 0, 0, 0, 0} }; topologicalSort(graph); return 0; } Explanation: The graph is represented as an adjacency matrix. The vertices are stored in the stack in a topologically sorted order. The algorithm makes use of DFS to ensure that all dependencies are visited before the vertex itself. Time Complexity: The time complexity of Topological Sort is O(V²) for an adjacency matrix and O(V + E) for an adjacency list. 22.4 Strongly Connected Components (Kosaraju’s Algorithm) In directed graphs, a Strongly Connected Component (SCC) is a maximal subgraph where every vertex is reachable from every other vertex in the subgraph. Kosaraju’s Algorithm is an efficient method to find SCCs in a directed graph. Kosaraju’s Algorithm Steps: Perform DFS on the original graph and push the vertices onto a stack based on their finish times. Reverse the graph. Perform DFS again, using the vertices in the order defined by the stack. Kosaraju’s Algorithm (in C): #include <stdio.h> #include <stdbool.h> #define V 5 void DFSUtil(int graph[V][V], int v, bool visited[]) { visited[v] = true; printf("%d ", v); for (int i = 0; i < V; i++) { if (graph[v][i] == 1 && !visited[i]) { DFSUtil(graph, i, visited); } }

Chapter 21: Graph Algorithms Welcome back! In this chapter, we’ll explore Graph Algorithms, which are crucial in solving problems related to networks, connections, and paths. These algorithms help us navigate through graphs and solve problems like finding the shortest path, traversing graphs, and much more. We’ll focus on two widely-used algorithms for solving problems involving graphs: Shortest Path Algorithms Minimum Spanning Tree (MST) Algorithms Let’s dive right in! 21.1 Shortest Path Algorithms In many real-world applications like GPS navigation, computer networks, and transportation systems, we are often asked to find the shortest path from one point to another in a graph. 21.1.1 Dijkstra’s Algorithm Dijkstra’s Algorithm is used to find the shortest path from a source node to all other nodes in a graph with non-negative edge weights. Problem Definition: Given a graph and a source node, find the shortest path from the source to all other nodes. Algorithm Steps: Create a set of all nodes in the graph. Assign a tentative distance value to every node: 0 for the source node and infinity for all other nodes. While there are unvisited nodes: Pick the unvisited node with the smallest tentative distance. Update the tentative distances of its neighboring nodes. Mark the node as visited. Dijkstra’s Algorithm (in C): #include <stdio.h> #include <limits.h> #include <stdbool.h> #define V 9 int minDistance(int dist[], bool sptSet[]) { int min = INT_MAX, min_index; for (int v = 0; v < V; v++) if (sptSet[v] == false && dist[v] <= min) min = dist[v], min_index = v; return min_index; } void printSolution(int dist[]) { printf("Vertex \t Distance from Source\n"); for (int i = 0; i < V; i++) printf("%d \t %d\n", i, dist[i]); } void dijkstra(int graph[V][V], int src) { int dist[V]; bool sptSet[V]; for (int i = 0; i < V; i++) dist[i] = INT_MAX, sptSet[i] = false; dist[src] = 0; for (int count = 0; count < V – 1; count++) { int u = minDistance(dist, sptSet); sptSet[u] = true; for (int v = 0; v < V; v++) if (!sptSet[v] && graph[u][v] && dist[u] != INT_MAX && dist[u] + graph[u][v] < dist[v]) dist[v] = dist[u] + graph[u][v]; } printSolution(dist); } int main() { int graph[V][V] = { {0, 4, 0, 0, 0, 0, 0, 8, 0}, {4, 0, 8, 0, 0, 0, 0, 11, 0}, {0, 8, 0, 7, 0, 4, 0, 0, 2}, {0, 0, 7, 0, 9, 14, 0, 0, 0}, {0, 0, 0, 9, 0, 10, 0, 0, 0}, {0, 0, 4, 14, 10, 0, 2, 0, 0}, {0, 0, 0, 0, 0, 2, 0, 1, 6}, {8, 11, 0, 0, 0, 0, 1, 0, 7}, {0, 0, 2, 0, 0, 0, 6, 7, 0} }; dijkstra(graph, 0); return 0; } Explanation: The graph is represented by an adjacency matrix where graph[i][j] represents the weight of the edge between vertices i and j. If there is no edge, the value is 0. We maintain an array dist[] where dist[i] holds the shortest distance from the source to vertex i. sptSet[] keeps track of vertices for which we have finalized the shortest distance. Time Complexity: The time complexity of this implementation is O(V²), where V is the number of vertices. 21.2 Minimum Spanning Tree (MST) Algorithms Next up, we’ll look at Minimum Spanning Trees (MST). This is a fundamental problem in network design. Problem Definition: Given a connected, undirected graph, a Minimum Spanning Tree is a subset of the edges that connects all the vertices together without any cycles and with the minimum possible total edge weight. Two well-known algorithms to find MSTs: Prim’s Algorithm Kruskal’s Algorithm 21.2.1 Prim’s Algorithm Prim’s Algorithm finds a minimum spanning tree by starting from a single vertex and growing the tree one edge at a time, always choosing the smallest edge that connects a new vertex to the tree. Algorithm Steps: Initialize the minimum spanning tree (MST) with a single vertex. Repeatedly add the smallest edge that connects a vertex inside the MST to one outside it. Stop when all vertices are in the MST. Prim’s Algorithm (in C): #include <stdio.h> #include <limits.h> #include <stdbool.h> #define V 5 int minKey(int key[], bool mstSet[]) { int min = INT_MAX, min_index; for (int v = 0; v < V; v++) if (mstSet[v] == false && key[v] < min) min = key[v], min_index = v; return min_index; } void printMST(int parent[], int graph[V][V]) { printf("Edge \tWeight\n"); for (int i = 1; i < V; i++) printf("%d – %d \t%d \n", parent[i], i, graph[i][parent[i]]); } void primMST(int graph[V][V]) { int parent[V]; int key[V]; bool mstSet[V]; for (int i = 0; i < V; i++) key[i] = INT_MAX, mstSet[i] = false; key[0] = 0; parent[0] = -1; for (int count = 0; count < V – 1; count++) { int u = minKey(key, mstSet); mstSet[u] = true; for (int v = 0; v < V; v++) if (graph[u][v] && mstSet[v] == false && graph[u][v] < key[v]) parent[v] = u, key[v] = graph[u][v]; } printMST(parent, graph); } int main() { int graph[V][V] = { {0, 2, 0, 6, 0}, {2, 0, 3, 8, 5}, {0, 3, 0, 0, 7}, {6, 8, 0, 0, 9}, {0, 5, 7, 9, 0} }; primMST(graph); return 0; } Explanation: We represent the graph using an adjacency matrix. graph[i][j] contains the weight of the edge between vertex i and j. key[] keeps track of the minimum edge weights that connect vertices to the MST. parent[] stores the constructed MST. Time Complexity: The time complexity of Prim’s algorithm is O(V²). 21.2.2 Kruskal’s Algorithm Kruskal’s Algorithm works by sorting all the edges of the graph in non-decreasing order of their weight. It then repeatedly adds the smallest edge to the MST unless doing so would form a cycle. Algorithm Steps: Sort all edges in non-decreasing order of their weight. Pick the smallest edge. If it doesn’t form a cycle, add it to the MST. Repeat until there are exactly V-1 edges in the MST. Kruskal’s Algorithm (in C): #include <stdio.h> #include <stdlib.h> #define V 4 #define

Chapter 19: Advanced Graph Algorithms Welcome to Chapter 19, where we’ll explore some of the more advanced algorithms used in graph theory. These algorithms are crucial for solving complex problems related to networks, paths, and flows. In this chapter, we will cover: Shortest Path Algorithms: Dijkstra’s Algorithm Bellman-Ford Algorithm Maximum Flow Algorithms: Ford-Fulkerson Method Edmonds-Karp Algorithm 19.1 Shortest Path Algorithms In graph theory, finding the shortest path between two nodes is a common problem. Depending on the type of graph (directed, undirected, weighted, unweighted), different algorithms can be used to find the shortest paths efficiently. 19.1.1 Dijkstra’s Algorithm Definition: Dijkstra’s Algorithm finds the shortest path from a source node to all other nodes in a graph with non-negative edge weights. It follows a greedy approach by choosing the shortest path at each step. How Dijkstra’s Algorithm Works: Initialize Distances: Start by assigning an infinite distance to all nodes except the source node, which gets a distance of 0. Priority Queue: Use a priority queue to select the node with the smallest known distance. Relaxation: For each neighboring node, check if a shorter path can be found through the current node. If so, update the distance. Repeat: Continue this process until all nodes have been visited and their shortest paths are known. Dijkstra’s Algorithm Example: Consider the graph below: A –1–> B A –4–> C B –2–> C B –6–> D C –3–> D To find the shortest path from A to all other nodes: Initialization: Set the distance to A as 0, and all others as infinity: Distances: A = 0, B = ∞, C = ∞, D = ∞ Priority Queue: Start with A (0). Relaxation: From A, update B (0 + 1 = 1) and C (0 + 4 = 4). New distances: A = 0, B = 1, C = 4, D = ∞ Next Node: Select B (1), update C (1 + 2 = 3) and D (1 + 6 = 7). New distances: A = 0, B = 1, C = 3, D = 7 Next Node: Select C (3), update D (3 + 3 = 6). Final distances: A = 0, B = 1, C = 3, D = 6 The shortest paths from A are: A to B: 1 A to C: 3 A to D: 6 Dijkstra’s Algorithm Implementation in C++: #include <iostream> #include <vector> #include <queue> #include <limits.h> using namespace std; #define INF INT_MAX // Structure for a graph edge struct Edge { int dest, weight; }; // Dijkstra’s algorithm to find the shortest path void dijkstra(vector<vector<Edge>>& graph, int src) { int V = graph.size(); vector<int> dist(V, INF); dist[src] = 0; priority_queue<pair<int, int>, vector<pair<int, int>>, greater<pair<int, int>>> pq; pq.push({0, src}); while (!pq.empty()) { int u = pq.top().second; pq.pop(); for (auto edge : graph[u]) { int v = edge.dest; int weight = edge.weight; if (dist[u] + weight < dist[v]) { dist[v] = dist[u] + weight; pq.push({dist[v], v}); } } } cout << "Vertex\tDistance from Source\n"; for (int i = 0; i < V; ++i) cout << i << "\t\t" << dist[i] << "\n"; } int main() { int V = 5; vector<vector<Edge>> graph(V); graph[0].push_back({1, 10}); graph[0].push_back({4, 5}); graph[1].push_back({2, 1}); graph[2].push_back({3, 4}); graph[4].push_back({1, 3}); graph[4].push_back({2, 9}); dijkstra(graph, 0); return 0; } This code illustrates Dijkstra’s algorithm for a graph with 5 vertices. 19.1.2 Bellman-Ford Algorithm Definition: The Bellman-Ford Algorithm is another method for finding the shortest path from a source node to all other nodes, but unlike Dijkstra’s algorithm, it works with graphs that may contain negative weight edges. How Bellman-Ford Algorithm Works: Initialize Distances: Set the distance of the source node to 0 and all other nodes to infinity. Relax All Edges: For each edge, update the distance to its destination if a shorter path is found through the source. Repeat: Perform the relaxation process V-1 times, where V is the number of vertices. Negative Cycle Detection: In the final pass, check for any further distance reductions. If found, a negative weight cycle exists. Example: Consider the graph: A –1–> B B –2–> C C –(-1)–> A B –3–> D To find the shortest path from A: Initialize distances: A = 0, B = ∞, C = ∞, D = ∞. After the first pass: A to B: 1 B to C: 1 + 2 = 3 B to D: 1 + 3 = 4 After the second pass, no changes occur, confirming the final shortest paths. Bellman-Ford Algorithm Implementation in C++: #include <iostream> #include <vector> using namespace std; struct Edge { int u, v, weight; }; void bellmanFord(int V, int E, vector<Edge>& edges, int src) { vector<int> dist(V, INT_MAX); dist[src] = 0; for (int i = 1; i <= V – 1; i++) { for (int j = 0; j < E; j++) { int u = edges[j].u; int v = edges[j].v; int weight = edges[j].weight; if (dist[u] != INT_MAX && dist[u] + weight < dist[v]) { dist[v] = dist[u] + weight; } } } // Check for negative-weight cycles for (int i = 0; i < E; i++) { int u = edges[i].u; int v = edges[i].v; int weight = edges[i].weight; if (dist[u] != INT_MAX && dist[u] + weight < dist[v]) { cout << "Graph contains a negative-weight cycle\n"; return; } } cout << "Vertex\tDistance from Source\n"; for (int i = 0; i < V; i++) cout << i << "\t\t" << dist[i] << "\n"; } int main() { int V = 5, E = 8; vector<Edge> edges = {{0, 1, -1}, {0, 2, 4}, {1, 2, 3}, {1, 3, 2}, {1, 4, 2}, {3, 2, 5}, {3, 1, 1}, {4, 3, -3}}; bellmanFord(V, E, edges, 0); return 0; } The Bellman-Ford algorithm is more versatile than Dijkstra’s because it can handle negative weights but is less efficient in practice. 19.2 Maximum Flow Algorithms In network flow problems, we aim to send as much flow as possible from a source node to a destination node, subject to capacity limits on the edges. Two classic algorithms for finding the maximum

Chapter 18: Greedy Algorithms Welcome to Chapter 18, where we’ll dive into the fascinating world of Greedy Algorithms. A Greedy Algorithm works by making the best decision at each step, aiming to reach an optimal solution. The idea is simple: take the immediate advantage without worrying about the future. While this may sound risky, greedy algorithms often lead to optimal solutions in many real-world problems. In this chapter, we will explore two popular applications of greedy algorithms: Huffman Coding — for data compression. Minimum Spanning Trees — using Kruskal’s and Prim’s algorithms for efficient network design. Let’s get started with Huffman Coding! 18.1 Huffman Coding Definition: Huffman Coding is a greedy algorithm used for lossless data compression. It assigns variable-length codes to input characters, with shorter codes assigned to more frequent characters and longer codes to less frequent ones. This minimizes the overall size of the encoded data. How Huffman Coding Works: Here’s how you can visualize Huffman Coding: Frequency Calculation: Start by calculating the frequency of each character in the input data. Build a Priority Queue: Insert all characters (or symbols) into a priority queue, where the element with the lowest frequency has the highest priority. Construct the Huffman Tree: Repeatedly extract the two characters with the smallest frequencies from the queue, combine them into a new node (with the combined frequency), and reinsert this node back into the queue. Keep doing this until there’s only one node left – the root of the Huffman tree. Generate Codes: Starting from the root of the tree, assign 0 to the left edge and 1 to the right edge of each node. Traverse the tree to generate the variable-length codes for each character. Example: Let’s say we want to encode the string "AABACD". Character frequencies: A: 3 B: 1 C: 1 D: 1 Create a priority queue: Insert A(3), B(1), C(1), D(1). Building the tree: Combine B and C (1 + 1 = 2), now we have A(3), BC(2), D(1). Combine D and BC (1 + 2 = 3), now we have A(3), DBC(3). Combine A and DBC (3 + 3 = 6), this becomes the root. Generate Huffman Codes: A = 0 B = 101 C = 100 D = 11 So, the encoded string for "AABACD" becomes: 0 0 101 0 100 11. Huffman Coding Implementation in C: #include <stdio.h> #include <stdlib.h> // A Huffman tree node struct MinHeapNode { char data; unsigned freq; struct MinHeapNode *left, *right; }; // Function to create a new min heap node struct MinHeapNode* newNode(char data, unsigned freq) { struct MinHeapNode* temp = (struct MinHeapNode*) malloc(sizeof(struct MinHeapNode)); temp->data = data; temp->freq = freq; temp->left = temp->right = NULL; return temp; } // A utility function to print an array of size n void printArray(int arr[], int n) { for (int i = 0; i < n; ++i) printf("%d", arr[i]); printf("\n"); } // Print Huffman codes from the root of the tree void printHuffmanCodes(struct MinHeapNode* root, int arr[], int top) { if (root->left) { arr[top] = 0; printHuffmanCodes(root->left, arr, top + 1); } if (root->right) { arr[top] = 1; printHuffmanCodes(root->right, arr, top + 1); } if (!(root->left && root->right)) { printf("%c: ", root->data); printArray(arr, top); } } // This function builds the Huffman Tree and prints the codes by traversing the built tree void HuffmanCodes(char data[], int freq[], int size) { // Create a simple example of a Huffman Tree (for demonstration purposes) struct MinHeapNode* root = newNode(‘$’, 0); root->left = newNode(‘A’, freq[0]); root->right = newNode(‘B’, freq[1]); int arr[100], top = 0; printHuffmanCodes(root, arr, top); } int main() { char arr[] = {‘A’, ‘B’}; int freq[] = {5, 9}; int size = sizeof(arr) / sizeof(arr[0]); HuffmanCodes(arr, freq, size); return 0; } This code gives a basic illustration of Huffman Coding. You would need a priority queue or min-heap to fully implement the algorithm, which can be a great exercise for you to try! Pros and Cons of Huffman Coding: Pros: Reduces the size of data significantly, leading to efficient storage. Works well for characters with varying frequencies. Cons: Requires prior knowledge of character frequencies. Can be time-consuming to build for very large datasets. 18.2 Minimum Spanning Trees Now let’s move to the Minimum Spanning Tree (MST) problem. An MST is a subgraph of a graph that connects all vertices with the minimum possible total edge weight, without forming any cycles. There are two popular algorithms for finding the MST: Kruskal’s Algorithm Prim’s Algorithm Let’s start with Kruskal’s Algorithm. Kruskal’s Algorithm Kruskal’s Algorithm is a greedy approach that treats the graph as a forest, where each vertex is its own tree. It keeps adding the smallest edge that connects two separate trees until all vertices are connected. How Kruskal’s Algorithm Works: Sort all the edges in increasing order based on their weights. Initialize an empty forest (collection of trees), where each vertex is its own tree. Pick the smallest edge. If it connects two different trees, add it to the MST and merge the trees. Repeat this until all vertices are connected in a single tree. Kruskal’s Algorithm Example: Let’s say we have the following graph with edges: Edge Weight A-B 1 A-C 4 B-C 3 B-D 2 C-D 5 Sort edges: A-B (1), B-D (2), B-C (3), A-C (4), C-D (5). Start by adding A-B to the MST. Add B-D next. Add B-C. Skip C-D (since it would create a cycle). The MST includes edges: A-B, B-D, B-C with a total weight of 6. Kruskal’s Algorithm Implementation in C: #include <stdio.h> #include <stdlib.h> // Structure to represent an edge struct Edge { int src, dest, weight; }; // Structure to represent a graph struct Graph { int V, E; struct Edge* edges; }; // Create a new graph struct Graph* createGraph(int V, int E) { struct Graph* graph = (struct Graph*) malloc(sizeof(struct Graph)); graph->V = V; graph->E = E; graph->edges = (struct Edge*) malloc(graph->E * sizeof(struct Edge)); return graph; } // A utility function to find the subset of an element int find(int

Chapter 17: Advanced Sorting Algorithms Welcome back, dear reader! Now that we’ve dipped our toes into the world of simple sorting algorithms, it’s time to dive a little deeper. In this chapter, we’ll explore some more powerful sorting algorithms that are designed to handle larger datasets efficiently. Before we jump in, take a moment to appreciate how far you’ve come! Sorting is one of the most important concepts in data structures, and understanding it thoroughly gives you a solid foundation for tackling more complex problems. So, what’s next? Let’s begin with a sorting algorithm that’s fast, efficient, and widely used in real-world applications — Quick Sort. 17.1 Quick Sort Definition: Quick Sort is a divide-and-conquer algorithm. It works by selecting a "pivot" element from the array and partitioning the other elements into two sub-arrays: those smaller than the pivot and those larger. The pivot is then placed in its correct position, and the process is recursively applied to the sub-arrays. How It Works: Quick Sort might sound a little tricky, but let’s break it down step-by-step: Pick an element from the array. This is called the pivot. Partition the array into two parts: elements less than the pivot go to the left, and elements greater than the pivot go to the right. Recursively apply Quick Sort to the left and right sub-arrays. Combine the results, and you have a sorted array! Let’s walk through an example to see how Quick Sort works in action. Quick Sort Example: Imagine we have the following array: [30, 10, 50, 20, 60, 40] We choose 30 as our pivot (you can choose any element, but for simplicity, we’ll take the first one). Elements less than 30: [10, 20] Elements greater than 30: [50, 60, 40] Now we recursively sort these two sub-arrays. The left sub-array [10, 20] is already sorted, so no further action is needed there. But the right sub-array [50, 60, 40] needs sorting. We choose 50 as the pivot for this sub-array: Elements less than 50: [40] Elements greater than 50: [60] Now, both [40] and [60] are sorted individually. We combine everything, and voila! The final sorted array is: [10, 20, 30, 40, 50, 60] See how it works? It’s like dividing the problem into smaller pieces until it’s easy to solve, and then putting everything back together. Time Complexity: Best case: O(n log n) — This happens when the pivot divides the array into two nearly equal parts each time. Worst case: O(n²) — This occurs when the pivot is always the smallest or largest element, leading to unbalanced partitions. Average case: O(n log n) — Generally, Quick Sort is quite efficient, and this is the expected performance for most datasets. Quick Sort Implementation in C: Let’s write a Quick Sort program in C, but instead of using just arrays, we’ll also show you how to use pointers. Pointers are powerful tools in C, and they make managing memory much easier. #include <stdio.h> // Function to swap two elements using pointers void swap(int *a, int *b) { int temp = *a; *a = *b; *b = temp; } // Partition function that returns the index of the pivot int partition(int arr[], int low, int high) { int pivot = arr[high]; // Choosing the last element as the pivot int i = (low – 1); // Index of the smaller element for (int j = low; j <= high – 1; j++) { if (arr[j] < pivot) { i++; // Increment the index of the smaller element swap(&arr[i], &arr[j]); } } swap(&arr[i + 1], &arr[high]); return (i + 1); } // Quick Sort function that uses recursion void quickSort(int arr[], int low, int high) { if (low < high) { // Get the pivot element in its correct position int pi = partition(arr, low, high); // Recursively sort the left and right sub-arrays quickSort(arr, low, pi – 1); quickSort(arr, pi + 1, high); } } // Function to print the array void printArray(int arr[], int size) { for (int i = 0; i < size; i++) { printf("%d ", arr[i]); } printf("\n"); } int main() { int arr[] = {30, 10, 50, 20, 60, 40}; int n = sizeof(arr) / sizeof(arr[0]); printf("Original array: \n"); printArray(arr, n); quickSort(arr, 0, n – 1); printf("Sorted array: \n"); printArray(arr, n); return 0; } Explanation: The partition() function is the heart of the Quick Sort algorithm. It rearranges the array by moving smaller elements to the left of the pivot and larger elements to the right. We use pointers in the swap() function to directly modify the elements of the array in memory. The quickSort() function is recursive and sorts the array by dividing it into smaller sub-arrays around the pivot. Why Use Quick Sort? Quick Sort is fast and works well for most real-world data. It’s not only good for small datasets but also scales efficiently for larger ones. And since it’s a divide-and-conquer algorithm, it can be optimized for parallel processing. But wait! We’re not done yet. Next, we’ll look at another very important sorting algorithm — Merge Sort. This one is also based on the divide-and-conquer approach, but it works in a slightly different way. Should we continue with Merge Sort? I promise it’s super interesting! Alright! Let’s keep the momentum going and dive into Merge Sort — another powerful sorting algorithm that you’ll often encounter in the real world. If you liked the divide-and-conquer strategy of Quick Sort, you’re going to love Merge Sort because it takes this strategy to the next level with guaranteed performance. 17.2 Merge Sort Definition: Merge Sort is a divide-and-conquer algorithm that breaks the array into smaller sub-arrays, sorts those sub-arrays, and then merges them back together in the correct order. The key idea in Merge Sort is that merging two sorted arrays is easier than sorting a large unsorted array. So instead of sorting the entire array in one go, we: Break the array into two halves. Recursively sort both halves. Merge the two