**3. Matrix Multiplication Algorithm**Assume that we have a method for multiplying two \(2 \times 2\) matrices using only 7 scalar multiplications. Use this method to design a divide and conquer algorithm for multiplying two \(n \times n\) matrices. What is the time complexity of the algorithm? You should start with a description of your algorithm, then write a recurrence relation which you should solve.**4. Shortest Path in Directed Graph**Let \(G = (V, E)\) be a directed graph. Assume that each edge \(ij \in E\) has a non-negative cost \(c(i, j)\) associated with it. Design a dynamic programming algorithm for computing the shortest path between any vertex pair. *(Floyd-Warshall Algorithm)* You should define all necessary terms and write a recurrence relation. What is the time complexity of your algorithm? Finally, write a short pseudo-code for your algorithm that uses only one matrix. 1. **Prove the following known result on minimum spanning trees.** **Theorem.** Let \( G = (V, E) \) be a graph and let \( w : E \rightarrow \mathbb{R}^+ \) be a weight function. Let \( A \) be a subset of edges that is contained in some minimum spanning tree, and let \( (S, \bar{S}) \) be a cut so that \( (S, \bar{S}) \cap A = \emptyset \). Prove that if \( e \) is a light edge in \( (S, \bar{S}) \), then \( e \) is safe for \( A \).2. Let \( S = \{1, 2, \ldots, n\} \) be a set activities, where activity \( i \in S \) has a start time \( s_i \), a finish time \( f_i \) and a profit \( p_i \) associated with it. Design a dynamic programming algorithm for computing a set of compatible activities of maximum profit. You should start with defining some terms and then write a recurrence relation for solving the problem. What is the time complexity of your algorithm?

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Assume that we have a method for multiplying two 2 x 2 matrices using only 7 scalar mul- tiplications. Use this method to design a divide and conquer algorithm for multiplying two n x n matrices. What is the time complexity of the algorithm? You should start with a description of your algorithm, then write a recurrence relation which you should solve.

Assume that we have a method for multiplying two 2 x 2 matrices using only 7 scalar mul- tiplications. Use this method to design a divide and conquer algorithm for multiplying two n x n matrices. What is the time complexity of the algorithm? You should start with a description of your algorithm, then write a recurrence relation which you should solve.

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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