On the Rectangles Induced by Points

Abstract

$ \newcommand{\Re}{\mathbb{R}} \newcommand{\reals}{\mathbb{R}} \newcommand{\SetX}{\mathsf{X}} \newcommand{\rad}{r} \newcommand{\Mh}[1]{#1} \newcommand{\query}{q} \newcommand{\eps}{\varepsilon} \newcommand{\VorX}[1]{\mathcal{V} \pth{#1}} \newcommand{\Polygon}{\mathsf{P}} \newcommand{\IntRange}[1]{[ #1 ]} \newcommand{\Space}{\overline{\mathsf{m}}} \newcommand{\pth}[2][\!]{#1\left({#2}\right)} \newcommand{\polylog}{\mathrm{polylog}} \newcommand{\N}{\mathbb N} \newcommand{\Z}{\mathbb Z} \newcommand{\pt}{p} \newcommand{\distY}[2]{\left\| {#1} - {#2} \right\|} \newcommand{\ptq}{q} \newcommand{\pts}{s}$ A set $P$ of $n$ points in the plane, induces a set of Delaunay-type axis-parallel rectangles $\mathcal{R}$, potentially of quadratic size, where an axis-parallel rectangle is in $\mathcal{R}$, if it has two points of $P$ as corners, and no other point of $P$ in it. We study various algorithmic problems related to this set of rectangles, including how to compute it, in near linear time, and handle various algorithmic tasks on it, such as computing its union and depth. The set of rectangles $\mathcal{R}$ induces the rectangle influence graph $G = (P,\mathcal{R})$, which we also study. Potentially our most interesting result is showing that this graph can be described as the union of $O(n)$ bicliques, where the total weight of the bicliques is $O(n \log^2 n)$. Here, the weight of a bicliques is the cardinality of its vertices.