Q: In the implementations presented for computing minimum spanning trees and shortest paths, weighted graphs are represented by storing the weights of edges in the graphs themselves. What is an alternative to this?
A: For graphs containing edges weighted by factors that do not change frequently, the approach used in this chapter works well. However, a more general way to think of an edge’s weight is as a function w (u, v), where u and v are the vertices that define the edge to which the weight function applies. To determine the weight of an edge, we simply call the function as needed. An advantage to this approach is that it lets us compute weights dynamically in applications where we expect weights to change frequently. On the other hand, a disadvantage is that if the weight function is complicated, it may be inefficient to compute over and over again.
Q: When solving the traveling-salesman problem, we saw that computing an optimal tour is intractable except when the tour contains very few points. Thus, an approximation algorithm based on the nearest-neighbor heuristic was used. What is another way to approximate a traveling-salesman tour? What is the running time of the approach? How close does the approach come to an optimal tour?
A: Another approach to solving the traveling-salesman problem using an approximation algorithm is to compute a minimum spanning tree, then traverse the tree using a preorder traversal (see Chapter 9). The running time of this approach ...