Algorithms for orthogonal partitioning into four parts

TL;DR

This paper presents a linear-time algorithm for orthogonal partitioning of points in the plane, based on the pancake theorem, and extends the approach to higher dimensions with polynomial-time solutions.

Contribution

It introduces an optimal linear-time algorithm for orthogonal partitioning into four parts and generalizes the problem to higher dimensions with polynomial-time algorithms.

Findings

Linear-time algorithm for 2D orthogonal partitioning

Optimality of the algorithm's complexity

Polynomial-time solutions for higher-dimensional cases

Abstract

The famous pancake theorem states that for every finite set $X$ in the plane, there exist two orthogonal lines that divide $X$ into four equal parts. We propose an algorithm whose running time is linear in the number of points in $X$ and prove that this complexity is optimal. We also consider generalizations of the pancake theorem and show that orthogonal hyperplanes can be found in polynomial time.