Smoothed Analysis of the Expected Number of Maximal Points in Two Dimensions (original) (raw)

2018, arXiv (Cornell University)

The Maximal points in a set S are those that aren't dominated by any other point in S. Such points arise in multiple application settings in which they are called by a variety of different names, e.g., maxima, Pareto optimums, skylines. Because of their ubiquity, there is a large literature on the expected number of maxima in a set S of n points chosen IID from some distribution. Most such results assume that the underlying distribution is uniform over some spatial region and strongly use this uniformity in their analysis. This work was initially motivated by the question of how this expected number changes if the input distribution is perturbed by random noise. More specifically, let B p denote the uniform distribution from the 2-d unit L p ball, δB q denote the 2-d L q-ball, of radius δ and B p + δB q be the convolution of the two distributions, i.e., a point v ∈ B p is reported with an error chosen from δB q. The question is how the expected number of maxima change as a function of δ. Although the original motivation is for small δ the problem is well defined for any δ and our analysis treats the general case. More specifically, we study, as a function of n, δ, the expected number of maximal points when the n points in S are chosen IID from distributions of the type B p + δB q where p, q ∈ {1, 2, ∞} for δ > 0 and also of the type B ∞ + δB q where q ∈ [1, ∞) for δ > 0.