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NumPSLA -- An experimental research tool for pseudoline arrangements and order types

Günter Rote

TL;DR

NumPSLA presents an experimental tool for enumerating pseudoline arrangements and abstract order types by incremental construction of x-monotone PSLAs, leveraging succ/pred data structures, local sequences, and a duality-based mapping to OAOTs/AOTs. The framework supports efficient enumeration, duplication elimination through lexicographic comparison of local sequences, and parallel execution, achieving speeds competitive with precomputed order-type databases while enabling realizability filtering for up to 11 points. The authors provide extensive experimental data on hull sizes, crossing numbers, and symmetries, and demonstrate scalability to 12–13 points with substantial compute resources. This work offers a practical platform for computer-assisted exploration in discrete geometry, enabling precise enumeration and statistical analysis of small configurations to validate conjectures and discover new structural patterns.

Abstract

We present a program for enumerating all pseudoline arrangements with a small number of pseudolines and abstract order types of small point sets. This program supports computer experiments with these structures, and it complements the order-type database of Aichholzer, Aurenhammer, and Krasser. This system makes it practical to explore the abstract order types for 12 points, and the pseudoline arrangements of 11 pseudolines.

NumPSLA -- An experimental research tool for pseudoline arrangements and order types

TL;DR

NumPSLA presents an experimental tool for enumerating pseudoline arrangements and abstract order types by incremental construction of x-monotone PSLAs, leveraging succ/pred data structures, local sequences, and a duality-based mapping to OAOTs/AOTs. The framework supports efficient enumeration, duplication elimination through lexicographic comparison of local sequences, and parallel execution, achieving speeds competitive with precomputed order-type databases while enabling realizability filtering for up to 11 points. The authors provide extensive experimental data on hull sizes, crossing numbers, and symmetries, and demonstrate scalability to 12–13 points with substantial compute resources. This work offers a practical platform for computer-assisted exploration in discrete geometry, enabling precise enumeration and statistical analysis of small configurations to validate conjectures and discover new structural patterns.

Abstract

We present a program for enumerating all pseudoline arrangements with a small number of pseudolines and abstract order types of small point sets. This program supports computer experiments with these structures, and it complements the order-type database of Aichholzer, Aurenhammer, and Krasser. This system makes it practical to explore the abstract order types for 12 points, and the pseudoline arrangements of 11 pseudolines.

Paper Structure

This paper contains 31 sections, 10 equations, 15 figures, 6 tables.

Table of Contents

  1. Introduction
  2. The order type of a point set.
  3. The order-type database.
  4. Big results from small sets.
  5. Line arrangements and pseudoline arrangements
  6. Pseudoline arrangements and abstract order types of points.
  7. Overview
  8. Enumeration of pseudoline arrangement s
  9. Representing a pseudoline arrangement
  10. Incremental generation of pseudoline arrangement s
  11. Duality between pseudoline arrangement s and abstract order types
  12. Parallelization
  13. Enumerating only the realizable AOTs
  14. Experiments and Extensions
  15. Acknowledgements.
  16. ...and 16 more sections

Figures (15)

  • Figure 1: Left: A pseudoline arrangement of 5 lines, extended by a line 0 "at infinity". Middle and right: wiring diagrams
  • Figure 2: The first three three levels of the enumeration tree and a few nodes of the fourth level. The last inserted pseudoline is highlighted in red. For some nodes, the Dewey decimal notation is indicated.
  • Figure 3: Left: Threading line 6 through a PSLA of 5 lines. Right: The dual DAG of this PSLA
  • Figure 4: Continuing line $n$ after entering a face.
  • Figure 5: Relation between different concepts. An arrow in one direction indicates a specialization.
  • ...and 10 more figures