On the Average Number of Maxima in a set of Vectors and Applications

A maximal vector of a set is one which is not less than any other vector in all components. We derive a recurrence relation for computing the average number of maximal vectors in a set of n vectors in d-space under the assumption that all (n!)d relative orderings are equally probable. Solving the recurrence shows that the average number of maxima is 0((ln n)d-1) We use this result to construct an algorithm for finding all the maxima that has expected running time linear in n (for sets of vectors drawn under our assumptions.) For a given set of random points, the result in also used to derive an upper bound on the expected number of points from the set which are on the boundary of the convex hull of the set.