Differentiating between BubbleSort and InsertionSort

Question

This is a homework I'm doing, but I couldn't find an answer, hopefully you guys can shine some light on this.

The problem is this:
You have two unknown sorting algorithms, one is Bubble Sort, the other Insertion Sort.
Identify which is which given that you can only differentiate them via differences in asymptotic time for different data sets. (Meaning no run time cross comparison between the two sorts, as our professor may add constant time costs making Insertion Sort slower).

Edit: To clarify, we can specify any array of length n, with a particular sequence if you wish, to see the time it took to sort.
However, you may not compare times between the two sorts. E.g. This sort is faster, therefore this is Insertion Sort.

My findings:

            Insertion Sort | Bubble Sort  
Best case:            O(n) | O(n)  
Worst case:         O(n^2) | O(n^2)  
Average case:       O(n^2) | O(n^2)  
Stable:                Yes | Yes  
In-place:              Yes | Yes  
Nearly Sorted:        O(n) | O(n)
Reversed:           O(n^2) | O(n^2)  
Few Unique: closer to O(n) | closer to O(n^2) // not sure

Given how similar the two sorts are, it's really hard to distinct them...

Answer 1

Unfortunately, your problem statement is contradictory, so the problem cannot be solved in its current form. Your question states that we can obtain the running time for finite values of $n$. However, it also states that we only learn the asymptotic running time. This is self-contradictory, because it is not possible to infer the asymptotic running time based on any finite number of measurements.

If I had to speculate, I'd guess that maybe what the instructor actually wants you to do is find an infinite sequence of inputs $a_1,a_2,a_3,\dots$ such that: (1) $a_i$ is an array of length $i$; (2) the asymptotic running time of Insertion Sort on these inputs differs by more than a constant factor from the asymptotic running time of Bubble Sort on these inputs. If that's correct, your next step should be to investigate various ways you might construct such a sequence. I emphasize that this is only speculation, and not something that can be definitively derived from the problem statement as provided.