Q$3(39$ pts$)$ Use vertex $1$ as the starting $($root$)$ vertex and run the Dijkstra's algorithm and its modified versions to determine each of the following trees, rooted at vertex $1.$ Show all the steps in detail for each of $($a$),($b$)$ and $($c$).$

$($a $-13$ pts$)$ Shortest path tree

$($b $-13$ pts$)$ Min$-$max bottleneck path tree

$($c $-13$ pts$)$ Max$-$min bottleneck path tree