$($a$)$Find $($by inspection$)$ shortest paths from all nodes to all other nodes.

$($b$)$Verify that every subpath of the optimal $1$ to $4$ path in part $($a$)$ is itself optimal.

$($c$)$Detailyouroptimalsolutionsofpart$($a$)$in functional notation n$[$k$][/]$ and xi$,$j $[$k$][/].$

$($d$)$Write functional equations for the short$-$ est path problems of part $($a$).$

$($e$)$Verify that the n$[$k$][/]$ of part $($c$)$ satisfy functional equations of part $($d$).$