434 research outputs found
Minimum Hidden Guarding of Histogram Polygons
A hidden guard set is a set of point guards in polygon that all
points of the polygon are visible from some guards in under the
constraint that no two guards may see each other. In this paper, we consider
the problem for finding minimum hidden guard sets in histogram polygons under
orthogonal visibility. Two points and are orthogonally visible if
the orthogonal bounding rectangle for and lies within . It is
known that the problem is NP-hard for simple polygon with general visibility
and it is true for simple orthogonal polygon. We proposed a linear time exact
algorithm for finding minimum hidden guard set in histogram polygons under
orthogonal visibility. In our algorithm, it is allowed that guards place
everywhere in the polygon
Some Results on Open Edge and Open Mobile Guarding of Polygons and Triangulations
This paper focuses on a variation of the Art Gallery problem that considers
open edge guards and open mobile guards. A mobile guard can be placed on edges
and diagonals of a polygon, and the "open" prefix means that the endpoints of
such edge or diagonal are not taken into account for visibility purposes. This
paper studies the number of guards that are sufficient and sometimes necessary
to guard some classes of simple polygons for both open edge and open mobile
guards. This problem is also considered for planar triangulation graphs using
open edge guards.Comment: 13 page
Smoothed Analysis of the Art Gallery Problem
In the Art Gallery Problem we are given a polygon on
vertices and a number . We want to find a guard set of size , such
that each point in is seen by a guard in . Formally, a guard sees a
point if the line segment is fully contained inside the polygon
. The history and practical findings indicate that irrational coordinates
are a "very rare" phenomenon. We give a theoretical explanation. Next to worst
case analysis, Smoothed Analysis gained popularity to explain the practical
performance of algorithms, even if they perform badly in the worst case. The
idea is to study the expected performance on small perturbations of the worst
input. The performance is measured in terms of the magnitude of the
perturbation and the input size. We consider four different models of
perturbation. We show that the expected number of bits to describe optimal
guard positions per guard is logarithmic in the input and the magnitude of the
perturbation. This shows from a theoretical perspective that rational guards
with small bit-complexity are typical. Note that describing the guard position
is the bottleneck to show NP-membership. The significance of our results is
that algebraic methods are not needed to solve the Art Gallery Problem in
typical instances. This is the first time an -complete
problem was analyzed by Smoothed Analysis.Comment: 24 pages, 12 Figure
Distance domination, guarding and vertex cover for maximal outerplanar graph
This paper discusses a distance guarding concept on triangulation graphs,
which can be associated with distance domination and distance vertex cover. We
show how these subjects are interconnected and provide tight bounds for any
n-vertex maximal outerplanar graph: the 2d-guarding number, g_{2d}(n) = n/5;
the 2d-distance domination number, gamma_{2d}(n) = n/5; and the 2d-distance
vertex cover number, beta_{2d}(n) = n/4
Approximability of Guarding Weak Visibility Polygons
The art gallery problem enquires about the least number of guards that are
sufficient to ensure that an art gallery, represented by a polygon , is
fully guarded. In 1998, the problems of finding the minimum number of point
guards, vertex guards, and edge guards required to guard were shown to be
APX-hard by Eidenbenz, Widmayer and Stamm. In 1987, Ghosh presented
approximation algorithms for vertex guards and edge guards that achieved a
ratio of , which was improved upto by King and Kirkpatrick in 2011. It has been conjectured that
constant-factor approximation algorithms exist for these problems. We settle
the conjecture for the special class of polygons that are weakly visible from
an edge and contain no holes by presenting a 6-approximation algorithm for
finding the minimum number of vertex guards that runs in
time. On the other hand, for weak visibility polygons with holes, we present a
reduction from the Set Cover problem to show that there cannot exist a
polynomial time algorithm for the vertex guard problem with an approximation
ratio better than for any , unless NP=P.
We also show that, for the special class of polygons without holes that are
orthogonal as well as weakly visible from an edge, the approximation ratio can
be improved to 3. Finally, we consider the Point Guard problem and show that it
is NP-hard in the case of polygons weakly visible from an edge.Comment: 23 pages, 21 figures, 30 citation
Guard Placement For Wireless Localization
Motivated by secure wireless networking, we consider the problem of placing
fixed localizers that enable mobile communication devices to prove they belong
to a secure region that is defined by the interior of a polygon. Each localizer
views an infinite wedge of the plane, and a device can prove membership in the
secure region if it is inside the wedges for a set of localizers whose common
intersection contains no points outside the polygon. This model leads to a
broad class of new art gallery type problems, for which we provide upper and
lower bounds
Constant Approximation Algorithms for Guarding Simple Polygons using Vertex Guards
The art gallery problem enquires about the least number of guards sufficient
to ensure that an art gallery, represented by a simple polygon , is fully
guarded. Most standard versions of this problem are known to be NP-hard. In
1987, Ghosh provided a deterministic -approximation
algorithm for the case of vertex guards and edge guards in simple polygons. In
the same paper, Ghosh also conjectured the existence of constant ratio
approximation algorithms for these problems. We present here three
polynomial-time algorithms with a constant approximation ratio for guarding an
-sided simple polygon using vertex guards. (i) The first algorithm, that
has an approximation ratio of 18, guards all vertices of in
time. (ii) The second algorithm, that has the same
approximation ratio of 18, guards the entire boundary of in
time. (iii) The third algorithm, that has an approximation
ratio of 27, guards all interior and boundary points of in
time. Further, these algorithms can be modified to obtain
similar approximation ratios while using edge guards. The significance of our
results lies in the fact that these results settle the conjecture by Ghosh
regarding the existence of constant-factor approximation algorithms for this
problem, which has been open since 1987 despite several attempts by
researchers. Our approximation algorithms exploit several deep visibility
structures of simple polygons which are interesting in their own right.Comment: 39 pages, 31 figure
A Leapfrog Strategy for Pursuit-Evasion in a Polygonal Environment
We study pursuit-evasion in a polygonal environment with polygonal obstacles.
In this turn based game, an evader is chased by pursuers . The players have full information about the environment and the
location of the other players. The pursuers are allowed to coordinate their
actions. On the pursuer turn, each can move to any point at distance at
most 1 from his current location. On the evader turn, he moves similarly. The
pursuers win if some pursuer becomes co-located with the evader in finite time.
The evader wins if he can evade capture forever.
It is known that one pursuer can capture the evader in any simply-connected
polygonal environment, and that three pursuers are always sufficient in any
polygonal environment (possibly with polygonal obstacles). We contribute two
new results to this field. First, we fully characterize when an environment
with a single obstacles is one-pursuer-win or two-pursuer-win. Second, we give
sufficient (but not necessary) conditions for an environment to have a winning
strategy for two pursuers. Such environments can be swept by a \emph{leapfrog
strategy} in which the two cops alternately guard/increase the currently
controlled area. The running time of this algorithm is where is the number of vertices, is the number of obstacles
and is the diameter of .
More concretely, for an environment with vertices, we describe an
algorithm that (1) determines whether the obstacles are
well-separated, and if so, (2) constructs the required partition for a leapfrog
strategy.Comment: 25 pages, 10 figure
Combinatorics of Beacon-based Routing in Three Dimensions
A beacon is a point-like object which can be enabled to exert a magnetic pull
on other point-like objects in space. Those objects then move towards the
beacon in a greedy fashion until they are either stuck at an obstacle or reach
the beacon's location. Beacons placed inside polyhedra can be used to route
point-like objects from one location to another. A second use case is to cover
a polyhedron such that every point-like object at an arbitrary location in the
polyhedron can reach at least one of the beacons once the latter is activated.
The notion of beacon-based routing and guarding was introduced by Biro et al.
[FWCG'11] in 2011 and covered in detail by Biro in his PhD thesis [SUNY-SB'13],
which focuses on the two-dimensional case.
We extend Biro's result to three dimensions by considering beacon routing in
polyhedra. We show that beacons are always
sufficient and sometimes necessary to route between any pair of points in a
given polyhedron , where is the number of tetrahedra in a tetrahedral
decomposition of . This is one of the first results that show that beacon
routing is also possible in three dimensions.Comment: v1 published in "LATIN 2018: Theoretical Informatics"; v2 to be
published in "Computational Geometry - Theory and Application
Vertex Guarding for Dynamic Orthogonal Art Galleries
We devise an algorithm for surveying a dynamic orthogonal polygonal domain by
placing one guard at each vertex in a subset of its vertices, i.e., whenever an
orthogonal polygonal domain {\cal P'} is modified to result in another
orthogonal polygonal domain {\cal P}, our algorithm updates the set of vertex
guards surveying {\cal P'} so that the updated guard set surveys {\cal P}. Our
algorithm modifies the guard placement in O(k \lg{(n+n')}) amortized time while
ensuring the updated orthogonal polygonal domain with h holes and n vertices is
guarded using at most \lfloor (n+2h)/4 \rfloor vertex guards. For the special
case of the initial orthogonal polygon being hole-free and each update
resulting in a hole-free orthogonal polygon, our guard update algorithm takes
O(k\lg{(n+n')}) worst-case time. Here, n' and n are the number of vertices of
the orthogonal polygon before and after the update, respectively; and, k is the
sum of |n - n'| and the number of updates to a few structures maintained by our
algorithm. Further, by giving a construction, we show it suffices for the
algorithm to consider only the case in which the parity of the number of reflex
vertices of both {\cal P'} and {\cal P} are equal.Comment: accepted to IJCG
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