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

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Chen, Wei-Mei Hwang, Hsien-Kuei Tsai, Tsung-Hsi
Copyright Year	2013
Abstract	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, are also briefly studied.
Starting Page	405
Ending Page	441
Page Count	37
File Format	PDF
ISSN	00975397
DOI	10.1137/110856952
e-ISSN	10957111
Journal	SIAM Journal on Computing (SMJCAT)
Issue Number	2
Volume Number	42
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2013-03-06
Access Restriction	Subscribed
Subject Keyword	asymptotic approximations computational geometry Combinatorial probability skyline Zero-one laws Phase transitions random samples multiobjective optimization Analysis of algorithms and problem complexity maxima Pareto optimality average-case analysis of algorithms threshold phenomena dominance Database theory
Content Type	Text
Resource Type	Article
Subject	Mathematics Computer Science

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