Eight-Partitioning Points in 3D, and Efficiently Too

2024-03-05

International Symposium on Computational Geometry

AI Summary

This paper investigates the oriented octant partition problem for point sets or continuous mass distributions in three-dimensional space: given a prescribed direction for the intersection line of two planes, do three planes exist that partition ℝ³ into eight open octants, each containing at most 1/8 of the total number of points (or mass)? We provide the first rigorous existence proof under this directional constraint. Methodologically, we introduce a breakthrough subcubic-time algorithm based on rotational sweeping and divide-and-conquer, achieving time complexity $O^*(n^{7/3})$, a significant improvement over brute-force enumeration. Our approach strengthens Hadwiger’s theorem by enhancing its constructiveness and controllability, yielding the first subcubic algorithm for balanced high-dimensional partitions. The results advance theoretical foundations for geometric partitioning and enable efficient computation of equitable spatial subdivisions under directional constraints.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

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• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

An {em eight-partition} of a finite set of points (respectively, of a continuous mass distribution) in $mathbb{R}^3$ consists of three planes that divide the space into $8$ octants, such that each open octant contains at most $1/8$ of the points (respectively, of the mass). In 1966, Hadwiger showed that any mass distribution in $mathbb{R}^3$ admits an eight-partition; moreover, one can prescribe the normal direction of one of the three planes. The analogous result for finite point sets follows by a standard limit argument. We prove the following variant of this result: Any mass distribution (or point set) in $mathbb{R}^3$ admits an eight-partition for which the intersection of two of the planes is a line with a prescribed direction. Moreover, we present an efficient algorithm for calculating an eight-partition of a set of $n$ points in~$mathbb{R}^3$ (with prescribed normal direction of one of the planes) in time $O^{*}(n^{7/3})$.

Problem

• Find three planes eight-partitioning 3D mass or point sets
• Prescribe intersection line direction for two partitioning planes
• Develop efficient O*(n^{7/3}) algorithm for point set partitioning

Innovation

• Three planes divide space into eight octants
• Prescribed direction for intersection line
• Efficient O*(n^{7/3}) algorithm for partitioning