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Orthogonal partitions into four parts

Alexey Fakhrutdinov, Oleg R. Musin

TL;DR

This work addresses the orthogonal four-partition problem for finite planar point sets, proving that the partition into four equal parts by two orthogonal lines can be found in $\Theta(n)$ time, matching the linear-time lower bound. The approach hinges on a duality between points and lines and a phase-based search using $p$-levels and line medians to efficiently locate the partition, achieving linear-time complexity with careful pruning. The paper also extends these ideas to higher dimensions via Theorems A and B, giving polynomial-time algorithms for finding mutually orthogonal hyperplanes that achieve four-way mass partitions across multiple sets, and discusses naive polynomial-time schemes and potential improvements. Overall, it provides both a tight 2D algorithm and scalable higher-dimensional generalizations within the mass-partition framework, with implications for computational geometry and related partition problems.

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.

Orthogonal partitions into four parts

TL;DR

This work addresses the orthogonal four-partition problem for finite planar point sets, proving that the partition into four equal parts by two orthogonal lines can be found in time, matching the linear-time lower bound. The approach hinges on a duality between points and lines and a phase-based search using -levels and line medians to efficiently locate the partition, achieving linear-time complexity with careful pruning. The paper also extends these ideas to higher dimensions via Theorems A and B, giving polynomial-time algorithms for finding mutually orthogonal hyperplanes that achieve four-way mass partitions across multiple sets, and discusses naive polynomial-time schemes and potential improvements. Overall, it provides both a tight 2D algorithm and scalable higher-dimensional generalizations within the mass-partition framework, with implications for computational geometry and related partition problems.

Abstract

The famous pancake theorem states that for every finite set in the plane, there exist two orthogonal lines that divide into four equal parts. We propose an algorithm whose running time is linear in the number of points in 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.

Paper Structure

This paper contains 5 sections, 6 theorems, 3 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Proof of Theorem 1
  3. Dual Transformation
  4. Optimal algorithm for partitioning into four equal parts.
  5. Proof of Theorem 2

Key Result

Theorem 1

Given a set of points $P$ in $\mathbb{R}^2$, $|P| = n$, an orthogonal cut partitioning the set into four equal parts can be computed in optimal linear time in $n$, i.e., in $\Theta(n)$ time.

Figures (2)

  • Figure 1: Partitioning of the set $P$ of 9 points. The black polygonal line corresponds to the median of configuration $D\left(P\right)$.
  • Figure 3: The orange polygonal lines correspond to $L_q$ and $L_p$. Trapezoids $\sigma$ and $\tau$ are chosen to contain these levels.

Theorems & Definitions (12)

  • Theorem 1
  • Theorem 2
  • Definition 2.1: Function N
  • Definition 2.2: Dual transformation
  • Definition 2.3: $p$-levels of line arrangement
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3: Division of the interval
  • ...and 2 more