Explain the divide and conquer apprach of multiplying two matrices of large size. Also, explain the Strassen’s matric multiplication algorithm. Find the time complexity of both these approaches.
### Divide and Conquer Approach for Matrix Multiplication The divide and conquer approach for multiplying two large matrices follows these steps: 1. Divide: Split each \( n \times n \) matrix into four \( \frac{n}{2} \times \frac{n}{2} \) submatrices. 2. Conquer: Compute the intermediate matrix products recursively. 3. Combine: Sum up the results to get the final matrix. For two matrices \( A \) and \( B \), we split them into four submatrices: \[ A = \begin{bmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{bmatrix}, \quad B = \begin{bmatrix} B_{11} ______ _________ ______ ___ ______ ______ _________ __________.
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What is a Greedy Approach to Problem-solving? Formulate the fractional Knapsack Problem as an optimisation problem and write a greedy algorithm to solve this problem. Solve the following fractional Knapsack problem using this algorithm. Show all the steps. Suppose there is a knapsack of capacity 15 Kg and 6 items are to packed in it. The weight and profit of the items are as under:
(p1, p2,…, p6) = (3, 2, 4, 5, 1, 6)
(w1, w2,…, w6) = (2, 1, 2, 1, 5, 1)
Graph Colouring Problem
Using Horner's rule, evaluate the polynomialAnalyse the computation time required for polynomial evaluation using Horner’s rule against the Brute force method.
3-CNF SAT
What is the use of Topological sorting? Write and explain the Topological sorting algorithm. Also, compute the time complexity for the topological sorting algorithm.
Explain binary exponentiation for computing the value 519. Write the right-toleft binary exponentiation algorithm and show its working for the exponentiation 519. Also, find the worst-case complexity of this algorithm
Write and explain the linear search algorithm and discuss its best and worstcase time complexity. Show the working of the linear search algorithm for the data: 12, 11, 3, 9, 15, 20, 18, 19, 13, 8, 2, 1, 16.