Answer: Merge Sort and Quick Sort are fundamental comparison-based sorting algorithms taught in courses like BCS-040, both employing the Divide and Conquer paradigm. While their theoretical average-case time complexity is `O(n log n)`, their practical performance can differ due to constant factors, memory access patterns, and worst-case scenarios. This comparison will illustrate their behavior on randomly generated arrays of sizes 10 and 50, focusing on loop counts and execution times. Merge Sort opera...

Answer: The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones, usually starting with 0 and 1. Formally, F(0) = 0, F(1) = 1, and F(n) = F(n-1) + F(n-2) for n > 1. Generating this sequence is a classic problem used to illustrate fundamental programming concepts like recursion and iteration, and to analyze algorithmic efficiency. ### Recursive Implementation The recursive approach directly translates the mathematical definition of the Fibonacci sequence into...

Answer: Calculating the power of a number, `b` raised to the exponent `n` (bⁿ), is a fundamental operation in computer science. While seemingly straightforward, the efficiency of this calculation becomes critical for large exponents. This question requires developing C programs for two distinct methods – the Naive method and Binary Exponentiation – and comparing their multiplication counts, a key metric for algorithmic efficiency. **1. The Naive Method for Power Calculation** The Naive method, also k...

Answer: Dijkstra's algorithm is a renowned greedy algorithm used to find the shortest paths from a single source vertex to all other vertices in a graph with non-negative edge weights. As discussed in BCS-040, Unit 5 on Graph Algorithms, it systematically explores the graph, always extending the shortest path found so far. This iterative process guarantees an optimal solution by making locally optimal choices at each step. For this implementation, we will consider a graph with 5 nodes, labeled 0, 1, 2...

Answer: Graph traversal algorithms are fundamental techniques used to visit every node (vertex) and edge in a graph, starting from a specified initial node. Depth-First Search (DFS) and Breadth-First Search (BFS) are two primary methods, each with distinct exploration strategies and applications. DFS explores as deeply as possible along each branch before backtracking, while BFS explores all neighbors at the current depth level before moving to the next level. To implement and visualize these traversa...