Computing Maximal Layers Of Points in $E^{f(n)}$

TL;DR

This paper introduces a randomized algorithm for efficiently computing maximal layers of points in high-dimensional spaces where the dimension is a function of the number of points, achieving sub-quadratic runtime for fixed polynomial dimensions.

Contribution

The paper presents the first non-trivial polynomial-time algorithm for computing maximal layers in high-dimensional spaces with dimensions depending on the number of points, using a novel data structure.

Findings

Achieves expected runtime bounds for random and arbitrary point sets

Runtime remains polynomial when the dimension is a polynomial function of the number of points

Provides a new data structure for dominance queries in high-dimensional spaces

Abstract

In this paper we present a randomized algorithm for computing the collection of maximal layers for a point set in $E^{k}$ ($k = f(n)$). The input to our algorithm is a point set $P = \{p_1,...,p_n\}$ with $p_i \in E^{k}$. The proposed algorithm achieves a runtime of $O\left(kn^{2 - {1 \over \log{k}} + \log_k{\left(1 + {2 \over {k+1}}\right)}}\log{n}\right)$ when $P$ is a random order and a runtime of $O(k^2 n^{3/2 + (\log_{k}{(k-1)})/2}\log{n})$ for an arbitrary $P$. Both bounds hold in expectation. Additionally, the run time is bounded by $O(kn^2)$ in the worst case. This is the first non-trivial algorithm whose run-time remains polynomial whenever $f(n)$ is bounded by some polynomial in $n$ while remaining sub-quadratic in $n$ for constant $k$. The algorithm is implemented using a new data-structure for storing and answering dominance queries over the set of incomparable points.