$($c$).$ I entered the following text $($from problem $3($b$)$ below$)$ into ChatGPT $($GPT$-3\mathrm{.}5)$ Devadoss$-$O$$Rourke: Problem $2\mathrm{.}20$: $$Describe a point set with n points that constitutes the best$-$ case for the gift$-$wrapping algorithm. What is its time complexity in this case?$$

and it returned the text below

$$The best$-$case scenario for the gift$-$wrapping algorithm $($also known as the Jarvis march$)$ occurs when the algorithm performs the minimal number of operations to find the convex hull.

Best$-$Case Point Set: The best$-$case configuration is when the set of points is arranged such that: All Points on the Convex Hull: The simplest best$-$case scenario is when all n points are on the convex hull. This means that each point in the set is a vertex of the convex polygon formed by the convex hull. In this configuration, the gift$-$wrapping algorithm will need to perform exactly n iterations to wrap around the entire hull.

E$$icient Point Selection: Even though the set of points might not be in a regular convex polygon, the best case occurs when each step in the gift$-$wrapping process can find the next hull point quickly.

The algorithm will still examine each point to determine the next hull vertex, but it will only do so n times, as each point is part of the hull.

Time Complexity in the Best Case:

Number of Iterations: Since each iteration of the algorithm identifies one vertex of the convex hull, and in the best case, there are exactly n points on the hull, the algorithm will make n iterations. Points Examination per Iteration: During each iteration, the algorithm evaluates all n points to find the next vertex on the convex hull. Therefore, in the best$-$case scenario where all n points

are on the convex hull, the time complexity is: O$($n$2).$ Explanation:

Number of Iterations $($Finding Hull Points$)$: The algorithm performs n iterations to find each point on the convex hull.

Work Done per Iteration $($Finding the Next Point$)$: In each iteration, the algorithm needs to evaluate all n points to determine the next point that forms the smallest angle with the current edge of the convex hull. Even in the best$-$case scenario, the gift$-$wrapping algorithm still performs O$($n$2)$ operations due to the need to check all n points in each of the n iterations. Thus, while the best case is e$$icient in terms of the number of iterations, the time complexity remains O$($n$2)$ because each iteration requires evaluating all n points.$$

Give a critique of this answer. Is it completely correct? $($It is OK to state so$,$ if it is true.$)$ Or does it have any inaccuracies $($minor or major$)?$ What grade out of $100$ would you score this answer to problem $3($b$)?($this last part is a judgement call $$ you will not be graded on your answer to the grade out of $100$ estimate$).$