$1.$ Theoretical Analysis: Determine the asymptotic time and space complexities of both

$$algorithms:

$$To determine the asymptotic time and space complexities of both algorithms, let's analyze each algorithm separately:

$$Naive Algorithm:

$$Time Complexity:

$$The naive algorithm has a time complexity of O$($n$^2),$ where n is the number of people in the queue. This is because for each person, there is a nested loop that iterates over all previous people to decide which ATM to assign them to$.$

$$Space Complexity:

$$The space complexity of the naive algorithm is O$($n$),$ where n is the number of people in the queue. This is because it only requires space for the queues representing the two ATMs.

$$Optimized Algorithm: Time Complexity:

$$The optimized algorithm has a time complexity of O$($n$),$ where n is the number of people in the queue. This is because it iterates through the list of people once, assigning each person to the ATM with the shorter total transaction time.

$$Space Complexity:

$$The space complexity of the optimized algorithm is O$($n$),$ similar to the naive algorithm, as it also requires space for the queues representing the two ATMs.

$$In summary:

$$Naive Algorithm:

$$Time Complexity: O$($n$^2)$

$$Space Complexity: O$($n$)$ Optimized Algorithm:

$$Time Complexity: O$($n$)$

$$Space Complexity: O$($n$)$

$$These complexities represent the theoretical performance of each algorithm as the input size $($number of people$)$ grows.

I did point $1,$ now you do the point $2$ and $3$ :

$2-$ Empirical Analysis: Test both algorithms with various input sizes and document

$$performance:

$3-$ Results Comparison: Compare theoretical and empirical results, and explain any

$$discrepancies$.$

This are two codes :

$$Naive algorithm :

$$#include

$$#include

$$#include

$$struct Person $\{$

$$ int id;

$$ int transactionTime; $//$ in minutes

$\}$;

$$void naiveATMQueue$($std::queue& ATM$1,$ std::queue& ATM$2,$ const std::vector& people$)\{$

$$ for $($int i $=0$; i $<$ people.size$()$; i$++)\{$

$$ bool assignToATM$1=$ true; $//$ Default assignment to ATM$1$

$//$ Recalculate assignment for each person up to the current person

$$ for $($int j $=0$; j $<=$ i; j$++)\{$

$$ if $($j $\%2==0)\{$

$$ assignToATM$1=$ true;

$\}$ else $\{$

$$ assignToATM$1=$ false;

$\}$

$\}$

$$ if $($assignToATM$1)\{$

$$ ATM$1.$push$($people$[$i$])$;

$$ std::cout $<<$ "Person $"<<$ people$[$i$].$id $<<"$ assigned to ATM$1$ with transaction time $"$

$<<$ people$[$i$].$transactionTime $<<"$ minutes.

$"$;

$\}$ else $\{$

$$ ATM$2.$push$($people$[$i$])$;

$$ std::cout $<<$ "Person $"<<$ people$[$i$].$id $<<"$ assigned to ATM$2$ with transaction time $"$

$<<$ people$[$i$].$transactionTime $<<"$ minutes.

$"$;

$\}$

$\}$

$\}$

$$int main$()\{$

$$ std::queue ATM$1,$ ATM$2$;

$$ std::vector people $=\{\{1,5\},\{2,3\},\{3,7\},\{4,4\},\{5,6\},\{6,2\}\}$;

$$ naiveATMQueue$($ATM$1,$ ATM$2,$ people$)$;

$$ return $0$;

$\}$

$$#include

$$#include

$$struct Person $\{$

$$ int id;

$$ int transactionTime; $//$ in minutes

$\}$;

$$ Optimized algorithm:

$$void optimizedATMQueue$($std::queue& ATM$1,$ std::queue& ATM$2,$ const std::vector& people$)\{$

$$ int ATM$1\_$time $=0,$ ATM$2\_$time $=0$; $//$ Total transaction times for each ATM

$$ for $($const auto& person : people$)\{$

$$ if $($ATM$1\_$time $<=$ ATM$2\_$time$)\{$

$$ ATM$1.$push$($person$)$;

$$ ATM$1\_$time $+=$ person.transactionTime;

$$ std::cout $<<$ "Person $"<<$ person.id $<<"$ assigned to ATM$1.$ Total time now: $"<<$ ATM$1\_$time $<<"$ minutes.

$"$;

$\}$ else $\{$

$$ ATM$2.$push$($person$)$;

$$ ATM$2\_$time $+=$ person.transactionTime;

$$ std::cout $<<$ "Person $"<<$ person.id $<<"$ assigned to ATM$2.$ Total time now: $"<<$ ATM$2\_$time $<<"$ minutes.

$"$;

$\}$

$\}$

$\}$

$$int main$()\{$

$$ std::queue ATM$1,$ ATM$2$;

$$ std::vector people $=\{\{1,5\},\{2,3\},\{3,7\},\{4,4\},\{5,6\},\{6,2\}\}$;

$$ optimizedATMQueue$($ATM$1,$ ATM$2,$ people$)$;

$$ return $0$;

$\}$