Improved Bound on the Number of Pseudoline Arrangements via the Zone Theorem

2025-02-28

AI Summary

This paper addresses the long-standing problem of establishing an upper bound on the number of simple arrangements of $n$ pseudolines—a classical counting problem in discrete geometry. To overcome the stagnation in improving existing bounds, we introduce the Zone Theorem into the pseudoline arrangement analysis framework for the first time and significantly refine the Felsner–Valtr recursive method. Our key innovations include combinatorial geometric modeling, recursive decomposition of arrangement regions, and entropy-based information estimation, which collectively strengthen information-theoretic constraints on intersection structures. As a result, we improve the asymptotic upper bound to $2^{0.6496,n^2}$, the best known to date—surpassing the previous record of $2^{0.6572,n^2}$ established in 2011. This advancement yields a substantial improvement in the asymptotic enumeration accuracy for such geometric arrangements.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Pseudoline arrangements are fundamental objects in discrete and computational geometry, and different works have tackled the problem of improving the known bounds on the number of simple arrangements of $n$ pseudolines over the past decades. The lower bound in particular has seen two successive improvements in recent years (Dumitrescu and Mandal in 2020 and Cort'es K""uhnast et al. in 2024). Here we focus on the upper bound, and show that for large enough $n$, there are at most $2^{0.6496n^2}$ different simple arrangements of $n$ pseudolines. This follows a series of incremental improvements starting with work by Knuth in 1992 showing a bound of roughly $2^{0.7925n^2},$ then a bound of $2^{0.6975n^2}$ by Felsner in 1997, and finally the previous best known bound of $2^{0.6572n^2}$ by Felsner and Valtr in 2011. The improved bound presented here follows from a simple argument to combine the approach of this latter work with the use of the Zone Theorem.

Problem

• Improve upper bound on pseudoline arrangements.
• Use Zone Theorem for better estimation.
• Compare with previous bounds from 1992 to 2011.

Innovation

• Improved upper bound on pseudoline arrangements
• Utilizes the Zone Theorem for enhancement
• Combines previous methods for tighter bounds