129 research outputs found

### Unions of Onions: Preprocessing Imprecise Points for Fast Onion Decomposition

Let $\mathcal{D}$ be a set of $n$ pairwise disjoint unit disks in the plane.
We describe how to build a data structure for $\mathcal{D}$ so that for any
point set $P$ containing exactly one point from each disk, we can quickly find
the onion decomposition (convex layers) of $P$.
Our data structure can be built in $O(n \log n)$ time and has linear size.
Given $P$, we can find its onion decomposition in $O(n \log k)$ time, where $k$
is the number of layers. We also provide a matching lower bound. Our solution
is based on a recursive space decomposition, combined with a fast algorithm to
compute the union of two disjoint onionComment: 10 pages, 5 figures; a preliminary version appeared at WADS 201

### Minimum Cuts in Geometric Intersection Graphs

Let $\mathcal{D}$ be a set of $n$ disks in the plane. The disk graph
$G_\mathcal{D}$ for $\mathcal{D}$ is the undirected graph with vertex set
$\mathcal{D}$ in which two disks are joined by an edge if and only if they
intersect. The directed transmission graph $G^{\rightarrow}_\mathcal{D}$ for
$\mathcal{D}$ is the directed graph with vertex set $\mathcal{D}$ in which
there is an edge from a disk $D_1 \in \mathcal{D}$ to a disk $D_2 \in
\mathcal{D}$ if and only if $D_1$ contains the center of $D_2$.
Given $\mathcal{D}$ and two non-intersecting disks $s, t \in \mathcal{D}$, we
show that a minimum $s$-$t$ vertex cut in $G_\mathcal{D}$ or in
$G^{\rightarrow}_\mathcal{D}$ can be found in $O(n^{3/2}\text{polylog} n)$
expected time. To obtain our result, we combine an algorithm for the maximum
flow problem in general graphs with dynamic geometric data structures to
manipulate the disks.
As an application, we consider the barrier resilience problem in a
rectangular domain. In this problem, we have a vertical strip $S$ bounded by
two vertical lines, $L_\ell$ and $L_r$, and a collection $\mathcal{D}$ of
disks. Let $a$ be a point in $S$ above all disks of $\mathcal{D}$, and let $b$
a point in $S$ below all disks of $\mathcal{D}$. The task is to find a curve
from $a$ to $b$ that lies in $S$ and that intersects as few disks of
$\mathcal{D}$ as possible. Using our improved algorithm for minimum cuts in
disk graphs, we can solve the barrier resilience problem in
$O(n^{3/2}\text{polylog} n)$ expected time.Comment: 11 pages, 4 figure

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