On the average number of maxima in a set of vectors and applications

ABSTRACT. A maximal vector of a set ~s one which is not less than any other vector m all components We derive a recurrence relation for computing the average number of maxunal vectors m a set of n vectors m d-space under the assumpUon that all (nl) a relative ordermgs are equally probable. Solving the recurrence shows that the average number of maxmaa is O((ln n) a-~) for fixed d We use this result to construct an algorithm for finding all the maxima that have expected running tmae hnear m n (for sets of vectors drawn under our assumptions) We then use the result to find an upper bound on the expected number of convex hull points m a random point set KE ~ WORDS AND eHRASES maxtma of a set of vectors, average number of maxtma, expected-tsme algorithms, analysts of algorithms, convex hulls, dynamtc programming CR CATEGORIES " 5 25, 5.39, 5.42 1

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.</p