Vertex-weighted shortest paths. [20 points] Let G = (V, E) be an undirected graph and let w: V → R ≥ 0 be a function that assigns non-negative weights to the vertices of G. We define the weight of a path P = v₁, ... ,vₖ, denoted by weight (P), to be the sum of the weights of the vertices in P; that is weight (P) = ᵏ∑ᵢ₌₁ w(vᵢ). Let s,t ∊ V. Design a polynomial-time algorithm that computes a path from s to t of minimum weight, if one exists. Show that your algorithm is correct and that it runs in polynomial time.