Question $1.[15$ points$]$

Scenario:

A market research company is conducting a study on consumer behavior. They have collected a large dataset, represented as an array ConsumerData, consisting of n elements. Each element in the array represents detailed information about a single consumer. To analyze this data, the company uses a unique algorithm, "Algorithm MarketScan." This algorithm randomly selects log$($n$)$ consumers from the ConsumerData array. For each selected consumer, it performs a comprehensive analysis that takes O$($n$)$ time $($where n is the total number of consumers in the dataset$).$

Assignment Question:

a$)$ Analytical Task: $(7\mathrm{.}5$ points$)$

a$.$ Determine the worst$-$case running time of "Algorithm MarketScan" for the entire market research process. Explain how did you get your answer.

b$)$ After analyzing the performance and implications of "Algorithm MarketScan" in market research, you are now hired as a consultant to enhance the efficiency and effectiveness of this system.

a$.$ To improve the time complexity of the "Algorithm MarketScan" but also keeping it is efficiency, which part of the algorithm you should improve? $(5$ points$)$

b$.$ Suggest a way to improve it$.(2\mathrm{.}5$ points$)$