Consider the following directed network, where $d_{i,j}$ denotes the distance between vertices i and $j$ :

$d_{1,2}=7,d_{1,3}=8,d_{1,4}=8,d_{1,7}=18$

$d_{2,5}=10,d_{2,6}=17,d_{2,7}=8$

$d_{3,5}=2,d_{3,6}=10,d_{3,7}=5,$

$d_{4,5}=9,d_{4,6}=9,d_{4,7}=6,$

$d_{5,7}=8,d_{5,8}=6,d_{5,9}=7,d_{5,10}=2,$

$d_{6,7}=9,d_{6,8}=4,d_{6,9}=6,d_{6,10}=8,$

$d_{7,8}=4,d_{7,10}=9,$

$d_{8,11}=5,d_{8,12}=9,d_{8,13}=12,$

$d_{9,11}=5,d_{9,12}=7,$

$d_{11,13}=2$

$d_{12,13}=3.$

Use backward recursion to find the shortest path between vertices $1$ and $13.$

For problem $1,$ find the shortest path between vertices $3$ and $11.$

For problem $1,$ find the shortest path between vertices $2$ and $13.$

For problem $1,$ find the longest path between vertices $1$ and $13.$

For problem $1,$ find the shortest path between vertices $1$ and $13$ which does not contain node $5.$

For problem $1,$ find the length of the myopic strategy to find the shortest path between nodes $1$ and $13.$

Problems involving minimizing the worst effect are frequently found in practice. In problem $1,$ suppose

that the cost of a path is the largest arc cost along the path. For example, the cost of the path

$1258$ is equal to $10.$ Find the minimum$-$cost path between vertices $1$ and $13.$

For problem $1,$ use forward recursion to find the shortest path between vertices $1$ and $13.($For each

node $i$ : $1)$ define $P_i$ to be the set of immediate predecessors of $\{$:$i,2)$ find $f_i,$ the minimum distance from

node $1$ to node $i($i$.$e$.,$ moving forward$)$; Find $f_{13}).$