given a weighted, directed graph. Give an O(VE)-time algorithm to find, for each vertex v∈V, the value δ∗(v)=Min u∈V{δ(u,v)}?
What is the question?

A vertex v ∈ G.V can have shortest paths from other vertices u ∈ G.V - {v}. What is the minimum of all these paths to v? This is *(u, v). You  have to calculate this minimum value for all the vertices

Why the solution works?

To get to a vertex v you have 2 options:


Take minimum incoming edge from u to v i.e. w(u, v)
Get to u and then take the edge from u to v i.e. u.d + w(u, v)


You have to find the minimum of these 2 options. So min(w(u, v), u.d + w(u, v))

In either of the cases the v.pi will be u