Let G = (V, E) be a weighted, directed graph with positive weight function w : E → {1, 2, . . . ,W} for some positive integer W, and assume that no two vertices have the same shortest-path weights from source vertex s. Now suppose that we define an unweighted, directed graph G′ = (V ∪ V′, E′) by replacing each edge (u, ν) ∈ E with w (u, ν) unit-weight edges in series. How many vertices does G′ have? Now suppose that we run a breadth-first search on G′. Show that the order in which the breadth-first search of G′ colors vertices in V black is the same as the order in which Dijkstra's algorithm extracts the vertices of V from the priority queue when it runs on G.