Two Counting Problems in Geometric Triangulations and Pseudoline Arrangements

Abstract

The purpose of this dissertation is to study two problems in combinatorial geometry in regard to obtaining better bounds on the number of geometric objects of interest: (i) monotone paths in geometric triangulations and (ii) pseudoline arrangements. \medskip(i) A directed path in a graph is monotone in direction of $\mathbf{u}$ if every edge in the path has a positive inner product with $\mathbf{u}$. A path is monotone if it is monotone in some direction. Monotone paths are studied in optimization problems, specially in classical simplex algorithm in linear programming. We prove that the (maximum) number of monotone paths in a geometric triangulation of $n$ points in the plane is $O(1.7864^n)$. This improves an earlier upper bound of $O(1.8393^n)$; the current best lower bound is $\Omega(1.7003^n)$ (Dumitrescu~\etal, 2016). \medskip (ii) Arrangements of lines and pseudolines are fundamental objects in discrete and computational geometry. They also appear in other areas of computer science, for instance in 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, $b_n \leq 0.6974\, n^2$ (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. \medskip Developing efficient algorithms and computer search were key to verifying the validity of both results.