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

TL;DR

This paper advances the upper bound on the number of simple pseudoline arrangements by merging the Felsner–Valtr cutpath-encoding framework with the Zone Theorem. Through a refined combinatorial-optimization bound on the maximum number of cutpaths $\gamma_n$, they prove $\gamma_n \le n^6 2^{\alpha n}$ with $\alpha < 1.2992$, which implies the total count satisfies $B_n \le 2^{0.6496 n^2}$ for large $n$. Additionally, a constructive stacking method yields $\Omega(2.083^n)$ cutpaths, showing the approach is tight in parts and providing a concrete lower bound. They further discuss potential improvements via conjectured zone-complexity properties, which could push the exponent down to $0.6074 n^2$, signaling a feasible route for future tightening of the bound in this central problem of discrete geometry.

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 Cortés Kühnast 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.

Figures (5)

• Figure 1: An arbitrarily drawn pseudoline arrangement (left) and the same arrangement drawn as a wiring diagram (right). In both cases, the north-cell is marked by a star.
• Figure 2: A pseudoline arrangement and its associated directed acyclic dual graph.
• Figure 3: Arrangement of $13$ pseudolines obtained by stacking two copies of the same arrangement of $5$ pseudolines together with $3$ additional pseudolines. The dotted line represents a cutpath.
• Figure 4: The "odd-even sort arrangement" on $20$ pseudolines.
• Figure 5: The arrangement $A_3$, obtained by stacking two copies of $A_1$ and adding the missing intersections on the left, then stacking a third copy of $A_1$ on top and adding missing intersections on the right.

Theorems & Definitions (12)

• Lemma 1: knuth1992axioms
• Lemma 2: felsner2011coding
• Lemma 3
• Theorem 4: Zone Theorem bern1990horizonpinchasi2011zone
• Lemma 5
• Lemma 6
• Lemma 7
• Lemma 8
• Theorem 9
• Theorem 10