Shortest Path : Dijkstra ’ s and Bellman - Ford

(We will use this subroutine later on in the lecture for another algorithm, which is why we are defining it as a separate procedure). Informally, we think of d[v] as our current estimate for the shortest path from s to v. The algorithm begins by initializing each d[v] ← ∞, except for our source vertex s, which we initialize so that d[s] = 0 (trivially, the shortest path from s to s is length 0). We then build a min-heap H on the vertex set that is organized based on the these d[·] values. We proceed by repeatedly dequeuing vertex u with the minimum d[u] value, and then we use this value to update the values of d[v] for all vertices v that are adjacent to u.