is to analyse your friend's algorithm:

$($i$)$ Suppose the next turn is the ith turn. What is the time complexity of a step which does not find an ideal turn?

$($ii$)$ Suppose the next turn is the ith turn. What is the time complexity of a step which does find an ideal turn?

$($iii$)$ What is the maximum number of ideal turns there can be in terms of n and m$?$

$($iv$)$ Therefore, what is the time complexity of the entire algorithm?

This is an example of amortisation, where a single operation may be slow but the sequence of operations is guaranteed to be fast.

Hint: The time complexity should account for setting up array f$,$ as well as the time spent at each ideal turn and at each not ideal turn.