Threshold Phenomena in $k$-Dominant Skylines of Random Samples

Skylines emerged as a useful notion in database queries for selecting representative groups in multivariate data samples for further decision making, multiobjective optimization, or data processing, and the $k$-dominant skylines were naturally introduced to resolve the abundance of skylines when the dimensionality grows or when the coordinates are negatively correlated. We prove in this paper that the expected number of $k$-dominant skylines is asymptotically zero for large samples when $1\leq k\leq d-1$ under two reasonable (continuous) probability assumptions of the input points, $d$ being the (finite) dimensionality, in contrast to the asymptotic unboundedness when $k=d$. In addition to such an asymptotic zero-infinity property, we also establish a sharp threshold phenomenon for the expected $(d-1)$-dominant skylines when the dimensionality is allowed to grow with $n$, the sample size. Several related issues, such as the dominant cycle structures, the numerical aspects, and the practical implications, ...