Graduate based, proof. Description:

The state requires that \(s\neq t\) to be true.

By length, the question probably mean the total cost of the directed path.

(a) I think this is just a direct application of the characterization of shortest path conditions, \(d(i, j) = d(i, k) + d(k, j) \; \forall i, j \in N\). (b) I have a feeling that this is a similar proof compare to the Floyd Warshall algorithm as discussed before.