CMSC 451 : Lecture 5 Graph Shortest Paths : Dijkstra and Bellman - Ford Tuesday , Sep 12 , 2017

Shortest Paths: Today we consider the problem of computing shortest paths in a directed graph. We are given a digraph G = (V,E) and a source vertex s ∈ V , and we want to compute the shortest path from s to every other vertex in G. This is called the single source shortest path problem. The algorithms we will present work for undirected graphs as well, by simply assuming that each undirected edge consists of two directed edges going in opposite directions. In general we assume that the graph is weighted, meaning that each edge (u, v) ∈ E has a numeric edge weight w(u, v). The cost of a path is the sum of edge weights along the path. Define the distance from any vertex u to any vertex v to be the minimum cost over all the paths from u to v. We denote this by δ(u, v). Thus, we are interested in computing δ(s, v) for all v ∈ V . We assume that every vertex has a trivial path of cost zero to itself, and hence δ(v, v) = 0, for all v ∈ V .