polypart 0.2.0

PolyPart is a Python package for partitioning d-dimensional convex polytopes by affine hyperplane arrangements. The algorithm works by iteratively slicing the polytope with hyperplanes and organizing the resulting subdivisions into a decision tree, which enables:

• Counting the exact number of regions
• Efficient point classification into the corresponding region

All computations use exact rational arithmetic, which ensures robustness and eliminates floating-point errors. The implementation builds on pycddlib with GMP to provide efficient and reliable polyhedral computations.

🚀 Quick Start

Prerequisites

• Python >= 3.10.6
• numpy >= 1.24.4
• pycddlib >= 3.0.2
• gmpy2 >= 2.1.5

Installation

1. Install the package from PyPI

pip install polypart

2. Import the package in your Python code

import polypart

Basic Usage

from polypart.ftyping import Fraction
from polypart.geometry import Polytope, Hyperplane
from polypart.ppart import build_partition_tree
from polypart.io import save_tree

# 1. Create initial polytope object in H-representation
# unit square: 0 <= x <= 1, 0 <= y <= 1
A = [[-1, 0], [1, 0], [0, -1], [0, 1]]
b = [0, 1, 0, 1]
square = Polytope(A, b) # expressed as A x <= b
square.extreme()  # compute vertices (Optional)

h1 = Hyperplane.from_coefficients([1, 0, Fraction(1, 3)]) # x = 1/3
h2 = Hyperplane.from_coefficients([0, 1, Fraction(1, 3)]) # y = 1/3

tree, n_parts = build_partition_tree(square, [h1, h2])
save_tree(tree, "data/square_partitions.json")
print("Number of regions:", n_parts)

# Output: 'Number of regions: 4'

🎯 Examples

Complete Jupyter notebooks providing guided, reproducible demonstrations of how to use PolyPart are available in the examples folder:

• Partitioning_Unit_Square.ipynb

  Demonstrates the algorithm on a simple, visual case: partitioning the unit square with three hyperplanes. Includes a plot of the resulting regions and new random points classified into their respective regions using the partition tree.

• Moduli_Stability_Chambers.ipynb A higher-dimensional research application on moduli spaces of parabolic vector bundles.

Experiments

Extensive computational experiments have been conducted to evaluate the performance and scalability of the PolyPart package. The experiment configurations, execution scripts, and results analysis notebooks are available in the experiments folder.

📜 License

This project is licensed under the MIT License - see the LICENSE file for details.

👥 Authors

• Sergio Herreros Pérez, Institute for Research in Technology, ICAI, Comillas Pontifical University
• José Portela González, Department of Quantitative Methods, ICADE, Comillas Pontifical University and Institute for Research in Technology, IIT, Comillas Pontifical University
• David Alfaya Sánchez, Department of Applied Mathematics and Institute for Research in Technology, ICAI, Comillas Pontifical University
• Jaime Pizarroso Gonzalo, Department of Telematics and Computing and Institute for Research in Technology, ICAI, Comillas Pontifical University, and Santalucía Chair of Analytics for Education.

🙌 Acknowledgments

This research was supported by project CIAMOD (Applications of computational methods and artificial intelligence to the study of moduli spaces, project PP2023_9) funded by Convocatoria de Financiación de Proyectos de Investigación Propios 2023, Universidad Pontificia Comillas, and by grants PID2022-142024NB-I00 and RED2022-134463-T funded by MCIN/AEI/10.13039/501100011033.

Find more about the CIAMOD project in the project webpage and the IIT proyect webpage.

Special thanks to everyone who contributed to the project:

• David Alfaya Sánchez (PI), Department of Applied Mathematics and Institute for Research in Technology, ICAI, Comillas Pontifical University
• Javier Rodrigo Hitos, Department of Applied Mathematics, ICAI, Comillas Pontifical University
• Luis Ángel Calvo Pascual, Department of Quantitative Methods, ICADE, Comillas Pontifical University
• Anitha Srinivasan, Department of Quantitative Methods, ICADE, Comillas Pontifical University
• José Portela González, Department of Quantitative Methods, ICADE, IIT, Comillas Pontifical University
• Jaime Pizarroso Gonzalo, Department of Telematics and Computing and Institute for Research in Technology, ICAI, Comillas Pontifical University
• Tomás Luis Gómez de Quiroga, Institute of Mathematical Sciences, UAM-UCM-UC3M-CSIC
• Daniel Sánchez Sánchez, Student of the Degree in Mathematical Engineering and Artificial Intelligence, Institute for Research in Technology, ICAI, Comillas Pontifical University
• Alejandro Martínez de Guinea García, Student of the Degree in Mathematical Engineering and Artificial Intelligence, Institute for Research in Technology, ICAI, Comillas Pontifical University
• Sergio Herreros Pérez, Student of the Degree in Mathematical Engineering and Artificial Intelligence, Institute for Research in Technology, ICAI, Comillas Pontifical University

📚 References

• David Alfaya, Sergio Herreros, Jaime Pizarroso, José Portela, and Javier Rodrigo, "A Computational Analysis of Ismorphism Classes of Moduli Spaces of Parabolic Vector Bundles", in preparation, 2025.
• David Alfaya, Sergio Herreros, Jaime Pizarroso and José Portela, "PolyPart: Exact Partitioning of Convex Polytopes uisng Decision Trees", in preparation, 2025.