New Lower Bounds for the Number of Pseudoline Arrangements

TL;DR

This paper establishes a new, improved lower bound on the number of nonisomorphic pseudoline arrangements, demonstrating that their logarithmic count grows at least quadratically with the number of pseudolines, using elementary geometric methods.

Contribution

It provides the first significant improvement in the lower bound for the number of pseudoline arrangements since 2011, using elementary geometric constructions.

Findings

Proves that the logarithm of the number of arrangements grows at least as 0.2083 n^2.

Improves previous lower bounds from 0.1887 n^2 to 0.2083 n^2.

Uses elementary and geometric arguments to derive the bounds.

Abstract

Arrangements of lines and pseudolines are fundamental objects in discrete and computational geometry. They also appear in other areas of computer science, such as the study of sorting networks. Let $B_n$ be the number of nonisomorphic arrangements of $n$ pseudolines and let $b_n=\log_2{B_n}$. The problem of estimating $B_n$ was posed by Knuth in 1992. Knuth conjectured that $b_n \leq {n \choose 2} + o(n^2)$ and also derived the first upper and lower bounds: $b_n \leq 0.7924 (n^2 +n)$ and $b_n \geq n^2/6 -O(n)$. The upper bound underwent several improvements, $b_n \leq 0.6988\, n^2$ (Felsner, 1997), and $b_n \leq 0.6571\, n^2$ (Felsner and Valtr, 2011), for large $n$. Here we show that $b_n \geq cn^2 -O(n \log{n})$ for some constant $c>0.2083$. In particular, $b_n \geq 0.2083\, n^2$ for large $n$. This improves the previous best lower bound, $b_n \geq 0.1887\, n^2$, due to Felsner and Valtr (2011). Our arguments are elementary and geometric in nature. Further, our constructions are likely to spur new developments and improved lower bounds for related problems, such as in topological graph drawings.