In the following, we ask you to propose or reason about different modifications made to the

Dijkstra algorithm $($i$.$e$.$ the algorithm used by OSPF$).$

$($i$)$ Describe a $($simple$)$ modification to the Dijkstra algorithm so as to make it return the path$($s$)$ containing the smallest number of links.

$($ii$)$ One of your friend proposes you to modify the Dijkstra algorithm to find the longest paths instead of the shortest ones by considering the inverse of each link cost $($e$.$g$.1/2$

instead of $2).$ Explain with an example why this would not work. $(2$ Points$)$

$($iii$)$ After realizing his mistake, the same friend now proposes you to take the negative of each link cost $($e$.$g$.2$ instead of $2)$ to find the longest paths instead of the shortest ones. Explain with an example why this would not work.

$($iv$)$ Yet another friend of yours modified the Dijkstra algorithm so that every router selects and sends traffic along its second best path $($i$.$e$.$ the path with the second lowest sum of link costs$)$ instead of the shortest path. Explain with an example why this modification can lead to forwarding loops and therefore does not work $($again$!).$