$(4).$ After the midterm was graded, your instructor tries to return the exam paper to you during the class

meeting. At the day, while all $n$ enrolled students had taken the exam, only $k$ students have attended. The

longest name in the class contains $k$ characters. Your instructor was not good at reading the names. So$,$

he had you come up with your ID card to retrieve your exam paper. Unfortunately, the grader forgot to

sort the papers. Therefore, your instructor had to compare the name listed on your ID card with the names

written on each of the exam papers to retrieve yours. Consider a simplified model: your instructor needed

$t_1$ time to recognize a letter and an additional $t_2$ time to compare two letters $($to verify whether they are

identical or determine their alphabetical order$).$ For example, the instructor could spend $t_1$ time to read

the first letter from your ID$,$ another $t_1$ time to read the first letter from an exam paper, and $t_2$ time to

decide whether the two letters are identical. Assume that no one is allowed to help the instructor $($to avoid

unexpected leak of grades$).$ Also, assume English alphabet $(26$ letters$),$ and that the letters in the names

are distributed uniformly. Finally, note that your instructor is a human being, who cannot manually

perform complicated $O(nlogn)$ sorting algorithms. Devise a strategy to help your instructor distribute

the papers to the $k$ attending students as fast as possible.

Describe strategies that can potentially help this case. $(8$pts$)$

Analyze how much time does each strategy take in terms of $n,k,t_1,$ and $t_2.(10$pts$)$

Make your recommendation based on $n,k,t_1,$ and $t_2.(2$pts$)$