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Qksort} Once we have qksort, of course, Quicksort is no problem: we call qksort with left := 1 and right := n. The next item on the agenda is the little question of how to create a splitter in an array. Suppose we are working with a subarray x[lef t], x[lef t + 1], . . , x[right]. The first step is to choose one of the subarray elements (the element itself, not the position of the element) to be the splitter, and the second step is to make it happen. The choice of the splitter element in the Quicksort algorithm is done very simply: at random.

Two of these vertices will be connected by an edge of the graph G if the two countries that they correspond to have a common stretch of border (we keep saying ‘stretch of border’ to emphasize that if the two countries have just a single point in common they are allowed to have the same color). As an illustration * Every planar map is four colorable, Bull. Amer. Math. , 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.

Of course the mere fact that our proved time estimate is O(2E ) doesn’t necessarily mean that the algorithm can be that slow, because maybe our complexity analysis wasn’t as sharp as it might have been. However, consider the graph G(s, t) that consists of s disjoint edges and t isolated vertices, for a total of 2s + t vertices altogether. If we choose an edge of G(s, t) and delete it, we get G(s − 1, t + 2), whereas the graph G/{e} is G(s − 1, t + 1). Each of these two new graphs has s − 1 edges.

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