A Fast Algorithm for Three-Dimensional Layers of Maxima Problem

TL;DR

This paper presents a novel, faster algorithm for the three-dimensional layers-of-maxima problem, achieving sub-logarithmic factors in time complexity with optimal space usage.

Contribution

It introduces the first algorithms solving the problem in o(n log n) time in the word RAM model, improving over previous solutions.

Findings

Algorithm runs in O(n(log log n)^3) deterministic time

Expected runtime is O(n(log log n)^2)

Uses O(n) space, optimal in the pointer machine model

Abstract

We show that the three-dimensional layers-of-maxima problem can be solved in $o(n\log n)$ time in the word RAM model. Our algorithm runs in $O(n(\log \log n)^3)$ deterministic time or $O(n(\log\log n)^2)$ expected time and uses O(n) space. We also describe an algorithm that uses optimal O(n) space and solves the three-dimensional layers-of-maxima problem in $O(n\log n)$ time in the pointer machine model.