Please answer both questions. Using the master theorem, what is the solution to the recurrence relation \( T(1) = 42, T(n) = 4T\left(\frac{n}{2}\right) + n^2\log^2(n) \)?- a. \(\Theta(n \log(n))\)- b. \(\Theta(n^2)\)- c. \(\Theta(n^2 \log^2(n))\)- d. \(\Theta(n^2 \log^3(n))\)- e. \(\Theta(n^3)\) **Question 6 Explanation**If \( L \) is a list of size \( n \), what is the asymptotic runtime of the following function?```function f(L) if length(L) < 3 return L A = [] B = [] for i from 0 to length(L) - 1 if i % 3 <= 1 A.append(3 * L[i]) else B.append(L[i]) return f(A) + f(B) # A concatenated with B```### Optionsa. \( \Theta(n) \)b. \( \Theta(n \log^2(n)) \)c. \( \Theta(n^{\log_{3/2}(2)} \log(n)) \)d. \( \Theta(n^2) \)e. \( \Theta(n^{\log_{3/2}(2)}) \)### AnalysisThis function is a recursive function where the list \( L \) is split into two lists, \( A \) and \( B \). The splitting is determined by the index modulo 3. After splitting, the function \( f \) is recursively called on both \( A \) and \( B \).- **Base Case**: If the length of \( L \) is less than 3, the function returns \( L \) immediately.- **Splitting Logic**: For each element in \( L \), if its index modulo 3 is less than or equal to 1, it's added to list \( A \) (and multiplied by 3), otherwise, it's added to list \( B \).- **Recursive Call**: The function then calls itself on the newly formed lists \( A \) and \( B \), and concatenates their results.The question asks for the asymptotic runtime, which is a measure of how the runtime grows with input size. By understanding the splitting logic and recursive nature of the calls, we can determine the complexity.The correct answer involves analyzing the recursive breakdown and size reduction pattern, common in algorithms evaluated using the Master Theorem or other recursive complexity analysis techniques.

Answered: Please answer both questions.

Please answer both questions.

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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