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LATIN 2016: LATIN 2016: Theoretical Informatics pp 138-151

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

• Indranil Banerjee
• Dana Richards
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9644)

## 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 $$P = \{p_1,\ldots ,p_n\}$$ 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 (in expectation). The algorithm is implemented using a new data-structure for storing and answering dominance queries over the set of incomparable points.

## Keywords

Maximal layers Random order Complexity

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## Copyright information

© Springer-Verlag Berlin Heidelberg 2016

## Authors and Affiliations

1. 1.Department of Computer ScienceGeorge Mason UniversityFairfaxUSA