626 research outputs found
On -Guarding Thin Orthogonal Polygons
Guarding a polygon with few guards is an old and well-studied problem in
computational geometry. Here we consider the following variant: We assume that
the polygon is orthogonal and thin in some sense, and we consider a point
to guard a point if and only if the minimum axis-aligned rectangle spanned
by and is inside the polygon. A simple proof shows that this problem is
NP-hard on orthogonal polygons with holes, even if the polygon is thin. If
there are no holes, then a thin polygon becomes a tree polygon in the sense
that the so-called dual graph of the polygon is a tree. It was known that
finding the minimum set of -guards is polynomial for tree polygons, but the
run-time was . We show here that with a different approach
the running time becomes linear, answering a question posed by Biedl et al.
(SoCG 2011). Furthermore, the approach is much more general, allowing to
specify subsets of points to guard and guards to use, and it generalizes to
polygons with holes or thickness , becoming fixed-parameter tractable in
.Comment: 18 page
Mobile vs. point guards
We study the problem of guarding orthogonal art galleries with horizontal
mobile guards (alternatively, vertical) and point guards, using "rectangular
vision". We prove a sharp bound on the minimum number of point guards required
to cover the gallery in terms of the minimum number of vertical mobile guards
and the minimum number of horizontal mobile guards required to cover the
gallery. Furthermore, we show that the latter two numbers can be calculated in
linear time.Comment: This version covers a previously missing case in both Phase 2 &
Engineering Art Galleries
The Art Gallery Problem is one of the most well-known problems in
Computational Geometry, with a rich history in the study of algorithms,
complexity, and variants. Recently there has been a surge in experimental work
on the problem. In this survey, we describe this work, show the chronology of
developments, and compare current algorithms, including two unpublished
versions, in an exhaustive experiment. Furthermore, we show what core
algorithmic ingredients have led to recent successes
Securing Pathways with Orthogonal Robots
The protection of pathways holds immense significance across various domains,
including urban planning, transportation, surveillance, and security. This
article introduces a groundbreaking approach to safeguarding pathways by
employing orthogonal robots. The study specifically addresses the challenge of
efficiently guarding orthogonal areas with the minimum number of orthogonal
robots. The primary focus is on orthogonal pathways, characterized by a
path-like dual graph of vertical decomposition. It is demonstrated that
determining the minimum number of orthogonal robots for pathways can be
achieved in linear time. However, it is essential to note that the general
problem of finding the minimum number of robots for simple polygons with
general visibility, even in the orthogonal case, is known to be NP-hard.
Emphasis is placed on the flexibility of placing robots anywhere within the
polygon, whether on the boundary or in the interior.Comment: 8 pages, 5 figure
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