$2$ Complexity analysis

$2\mathrm{.}1$ Complexity of find$\_$pivot$\_$largest$\_$subscript

Example adopting the previous code in section $1\mathrm{.}1.$

Lines $2$ and $5$ take $O(1).$

Each time the block of lines $4-$to$-7$ is repeated:

Lines $4-$to$-5O(1)$

Each time it is repeated, line $7$ takes $O(1).$ Since, due to line $6,$ it is repeated $O(n)$ times, it overall takes $O(n).$

Overall, lines $4-$to$-7$ take $O(n).$

Overall, we have a complexity of:

$O(1)+O(n)(O(1)+O(n))=O(n^2)$

$2\mathrm{.}2$ Complexity of find$\_$pivot$\_$smallest$\_$change

Example adopting the previous code in section $.$

Lines $2$ and $5$ take $O(1).$

Each time the block of lines $4-$to$-7$ is repeated:

Lines $4-$to$-5O(1).$

Each time it is repeated, line $7$ takes $O(1).$ Since, due to line $6,$ it is repeated $O(n)$ times, it overall takes $O(n).$

Overall, lines $4-$to$-7$ take $O(n).$