A fast algorithm for computing a planar support for non-piercing rectangles

TL;DR

It is shown that if for a family of axis-parallel rectangles, any point in the plane is contained in at most $k$ pairwise \emph{crossing} rectangles, then a support can be obtained as the union of $k$ planar graphs.

Abstract

For a hypergraph $\mathcal{H}=(X,\mathcal{E})$ a \emph{support} is a graph $G$ on $X$ such that for each $E\in\mathcal{E}$, the induced subgraph of $G$ on the elements in $E$ is connected. If $G$ is planar, we call it a planar support. A set of axis parallel rectangles $\mathcal{R}$ forms a non-piercing family if for any $R_1, R_2 \in \mathcal{R}$, $R_1 \setminus R_2$ is connected. Given a set $P$ of $n$ points in $\mathbb{R}^2$ and a set $\mathcal{R}$ of $m$ \emph{non-piercing} axis-aligned rectangles, we give an algorithm for computing a planar support for the hypergraph $(P,\mathcal{R})$ in $O(n\log^2 n + (n+m)\log m)$ time, where each $R\in\mathcal{R}$ defines a hyperedge consisting of all points of $P$ contained in~$R$. We use this result to show that if for a family of axis-parallel rectangles, any point in the plane is contained in at most $k$ pairwise \emph{crossing} rectangles (a pair of intersecting rectangles such that neither contains a corner of the other is called a crossing pair of rectangles), then we can obtain a support as the union of $k$ planar graphs.

Figures (4)

• Figure 1: The figure above shows $\textsc{Ub}(p),\textsc{Lb}(p)$, and the upper and lower piercing barriers $\textsc{Lpb}(R,p)$ and $\textsc{Upb}(R,p)$ of $\textsc{Piece}(R,p)$. The slab $\textsc{Slab}(R,p)$ containing $p$ defined by $\textsc{Upb}(R,p)$ and $\textsc{Lpb}(R,p)$ is shaded. The dark grey part shows the $\textsc{Subslab}(R,p)$.
• Figure 2: The figure shows the construction of the strips $\textsc{Strip}_0$, $\textsc{Strip}_1$ and $\textsc{Strip}_2$. The vertical line segment through $p_i$, $i\in\{1,2\}$ shows that $p_i$ is the rightmost point among the points in the strip $i$.
• Figure 3: The figure above shows the strip $\textsc{Strip}_i$, the slab $\textsc{Slab}(R,p)$ in grey, and the corridor $\textsc{Corridor}_i$ as the region shaded in red between $\pi^u_i$ and $\pi^{\ell}_i$.
• Figure 4: Top left: $R'$ cannot be empty if $p_i$ is above $R'$, Top right: If $R'$ does not contain a point of $P$ between $x(q)$ and $x_+(R)$, then $y(q)>y_+(\textsc{Strip}_i)$, Bottom: Since $qq_1$ and $qq_2$ are Delaunay, then $R_2$ pierces $v(qq_1)$.

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