1 research outputs found
Smoothed Analysis of the Expected Number of Maximal Points in Two Dimensions
The {\em Maximal} points in a set S are those that aren't {\em 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 {\em 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 Ballp denote the uniform distribution from the 2-d unit
Lp ball, delta Ballq denote the 2-d Lq-ball, of radius delta and Ballpq be the
convolution of the two distributions, i.e., a point v in Ballp is reported with
an error chosen from delta Ballq. The question is how the expected number of
maxima change as a function of delta. Although the original motivation is for
small delta the problem is well defined for any delta and our analysis treats
the general case. More specifically, we study, as a function of n,\delta, the
expected number of maximal points when the n points in S are chosen IID from
distributions of the type Ballpq where p,q in {1,2,infty} for delta > 0 and
also of the type Ballp infty-q, where q in [1,infty) for delta > 0.Comment: 95 pages, 35 figure