Shortest Weighted Paths Parallel and Sequential Data

It is often necessary to associate weights or other values with the edges of a graph. Such a “weighted” or “edge-labeled” graph can be defined as a triple G = (E, V, w) where w : E → eVal is a function mapping edges or directed edges to their values, and eVal is the set (type) of possible values. For a weighted graph eVal would typically be the real numbers, but for edge-labeled graphs they could be any type. There are a few ways to represent a weighted or edge-labeled graph. The first representation translates directly from representing the function w. In particular we can use a table that maps each edge (a pair of vertex identifiers) to its value. This would have type eVal edgeTable That is, the keys of the table are edges and the values are of type eVal. For example, for a weighted graph we might have: W = {(0,1) 7→ 0.7, (1,2) 7→ −2.0, (0,2) 7→ 1.5} This representation would allow us to look up the weight of an edge e using: w(e) = find W e. Another way to associate values with edges is to use a structure similar to the adjacency set representation for unweighted graphs and associate a value with each out edge of a vertex. In particular, instead of associating with each vertex a set of out-neighbors, we can associate each vertex with a table that maps each out-neighbor to its value. It would have type: (eVal vertexTable) vertexTable . The graph above would then be represented as: G = {0 7→ {1 7→ 0.7, 2 7→ 1.5} , 1 7→ {2 7→ −2.0} , 2 7→ {}} . We will mostly be using this second representation. †Lecture notes by Umut A. Acar, Guy E Blelloch, Margaret Reid-Miller, and Kanat Tangwongsan.