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We will illustrate the dramatically increased complexity with a recursive algorithm for the ‘maximum independent set problem,’ one which has received a great deal of attention in recent years. Suppose a graph G is given. By an independent set of vertices of G we mean a set of vertices no two of which are connected by an edge of G. In the graph in Fig. 1 the set {1, 2, 6} is an independent set and so is the set {1, 3}. The largest independent set of vertices in the graph shown there is the set {1, 2, 3, 6}.

N in some order. There are n! possible orders in which these elements might appear, so we are considering the action of Quicksort on just these n! inputs. Second, for each particular one of these inputs, the choices of the splitting elements will be made by choosing, at random, one of the entries of the array at each step of the recursion. We will also average over all such random choices of the splitting elements. 2 Quicksort is done first of all over all n! of the possible input orderings of the array elements, and second, for each such input ordering, we average also over all sequences of choices of the splitting elements.

By an independent set of vertices of G we mean a set of vertices no two of which are connected by an edge of G. In the graph in Fig. 1 the set {1, 2, 6} is an independent set and so is the set {1, 3}. The largest independent set of vertices in the graph shown there is the set {1, 2, 3, 6}. The problem of finding the size of the largest independent set in a given graph is computationally very difficult. All algorithms known to date require exponential amounts of time, in their worst cases, although no one has proved the nonexistence of fast (polynomial time) algorithms.

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Algorithms and Complexity by wilf


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