Clauset 1 Shortest Paths with Weights

Recall that BFS is a special case the general Search-Tree(G,s) algorithm in which we store the vertices that border our growing tree T in a first-in-first-out (FIFO) queue. In this way, the algorithm explores every vertex of distance ` from the source vertex s before we explore any vertex of distance ` + 1, for all `. Its output is a tree T that contains the union of “geodesic” or shortest paths from some source vertex s to all other vertices V − s. However, distances for BFS are only defined in terms of the number of edges, and thus the algorithm is only applicable to undirected and unweighted graphs (but, it is still correct for multigraphs and graphs with self-loops; do you see why?). But, not all graphs are so simple. In particular, many graphs of interest are weighted and this complicates the discovery of short paths.