Algorithms [Sedgewick, Robert]

It is obvious how to represent weighted graphs: in the adjacency matrix representation, the matrix can contain edge weights rather than boolean values, and in the adjacency structure representation, each list element (which represents an edge) can contain a weight.

We’ll start by assuming that all of the weights are positive. Some of the algorithms can be adapted to handle negative weights, but they become significantly more complicated. In other cases, negative weights change the nature of the problem in an essential way, and require far more sophisticated algorithms than those considered here. For an example of the type of difficulty that can arise, suppose that we have a situation where the sum of the weights of the edges around a cycle is negative: an infinitely short path could be generated by simply spinning around the cycle.

Minimum Spanning Tree

A minimum spanning tree of a weighted graph is a collection of edges that connects all the vertices such that the sum of the weights of the edges is at least as small as the sum of the weights of any other collection of edges that connects all the vertices. The minimum spanning tree need not be unique: for example, the following diagram shows three other minimum spanning trees for our sample graph.