a$)$ Design and analyze a recursive version of the LOGICALAND$($A$,$B$,$C$,$p$,$r$)$ algorithm that uscs a divide$-$and$-$conquer approach in which the algorithm divides the array elements $p.r$ in half and calls itself recursively on each half. Specifically, give pseudocode for your recursive LOGICALAND$($A$,$B$,$C$,$p$,$r$)$ algorithm. The initial call to your algorithm would pass $1$ in for p and n for r $.$ Don't use global variables.

$($b$)$ Design and analyze an iterative version of the LOGICALAND$($A$,$B$,$C$,$p$,$r$)$ algorithm.

$($c$)$ Perform a line$-$by$-$line analysis of your algorithms in $($a$)$ and $($b$)$ and derive a recurrence $T(n)$ for the running time, where $n=r-p+1.$ You may assume that each linc of pseudocode takes unit time $($i$.$e$.,$ ci $=1),$ except, of course, those lines containing nonprimitive operations

$($d$)$ Compare the order of growth of the two versions and determine which one is better in terms of order of growth and practical use.

$2.$ Solve the following recurrences:

a$)T(n)=3T(n-1)+4T(n-2)+2,n>0,$ where $T(0)=T(1)=1$

b$)T(n)=2T(\frac{n}{4})+n,n>1,$ where $T(1)=1$

c$)T(n)=5T(\frac{n}{16})+n\frac{1}{2}+n,n>0,$ where $T(0)=2$

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LOGICALAND Algorithm

Understanding the Problem:

Before we dive into the solutions, let's clarify the problem. We're assuming that:

Iogicaland $(A,B,C,p,P)$ is a function that takes three arrays $A,B,$ and C of equal length and two indices p and r $.$

It computes C$[$i$]=A[i]$ AND B$[$i$]$ for all $i$ in the range $p$ to $r.$

Solution:

$($a$)$ Recursive LOGICALAND

LOGICAIAND $(A,B,C,p,q$