Threshold phenomena in k-dominant skylines of random samples

TL;DR

This paper investigates the behavior of k-dominant skylines in large random samples, revealing a threshold phenomenon and showing that their expected number tends to zero for certain parameters, contrasting with the case when k equals the dimension.

Contribution

It establishes the asymptotic zero expectation of k-dominant skylines for large samples when 1 ≤ k ≤ d-1 and identifies a sharp threshold for the expected number as dimension grows.

Findings

Expected number of k-dominant skylines tends to zero for large samples when 1 ≤ k ≤ d-1.

A sharp threshold phenomenon is identified for the expected (d-1)-dominant skylines as dimension increases.

Contrasts the behavior with the case when k equals the dimension, where the expected number is unbounded.

Abstract

Skylines emerged as a useful notion in database queries for selecting representative groups in multivariate data samples for further decision making, multi-objective 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 $1\le k\le d-1$ 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 $n$. Several related issues such as the dominant cycle structures and numerical aspects, are also briefly studied.