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Additional info for Algorithms and Complexity
What have we learned? 619n ) units if the input graph G has n vertices. This is a little improvement of the most simple-minded possible 41 algorithm that one might think of for this problem, which is to examine every single subset of the vertices of of G and ask if it is an independent set or not. That algorithm would take Θ(2n ) time units because there are 2n subsets of vertices to look at. 619n by being a little bit cagey about the algorithm. Can we do still better? There have in fact been a number of improvements of the basic maxset1 algorithm worked out.
82 (1976), 711-712. 43 Fig. 5(a) Fig. 5(b) of this construction, we show in Fig. 5(a) a map of a distant planet, and in Fig. 5(b) the graph that results from the construction that we have just described. By a ‘planar graph’ we mean a graph G that can be drawn in the plane in such a way that two edges never cross (except that two edges at the same vertex have that vertex in common). The graph that results from changing a map of countries into a graph as described above is always a planar graph.
There have in fact been a number of improvements of the basic maxset1 algorithm worked out. Of these the most successful is perhaps the one of Tarjan and Trojanowski that is cited in the bibliography at the end of this chapter. We are not going to work out all of those ideas here, but instead we will show what kind of improvements on the basic idea will help us to do better in the time estimate. We can obviously do better if we choose v ∗ in such a way as to be certain that it has at least two neighbors.
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