125 research outputs found
The VC-Dimension of Limited Visibility Terrains
Visibility problems are fundamental to computational geometry, and many versions of geometric set cover where coverage is based on visibility have been considered. In most settings, points can see "infinitely far" so long as visibility is not "blocked" by some obstacle. In many applications, this may be an unreasonable assumption. In this paper, we consider a new model of visibility where no point can see any other point beyond a sight radius ?. In particular, we consider this visibility model in the context of terrains. We show that the VC-dimension of limited visibility terrains is exactly 7. We give lower bound construction that shatters a set of 7 points, and we prove that shattering 8 points is not possible
Computational Geometry Column 42
A compendium of thirty previously published open problems in computational
geometry is presented.Comment: 7 pages; 72 reference
An approximation algorithm for the art gallery problem
Given a simple polygon P on n vertices, two points x, y in P are said to be visible to each other if the line segment between x and y is contained in P. The Point Guard Art Gallery problem asks for a minimum-size set S such that every point in P is visible from a point in S. The set S is referred to as guards. Assuming integer coordinates and a specific general position on the vertices of P, we present the first O(log OPT)-approximation algorithm for the point guard problem. This algorithm combines ideas in papers of Efrat and Har-Peled and Deshpande et al. We also point out a mistake in the latter
An Approximation Algorithm for the Art Gallery Problem
Given a simple polygon on vertices, two points in
are said to be visible to each other if the line segment between
and is contained in . The Point Guard Art Gallery problem
asks for a minimum set such that every point in is visible
from a point in . The set is referred to as guards. Assuming integer
coordinates and a specific general position assumption, we present the first
-approximation algorithm for the point guard problem for
simple polygons. This algorithm combines ideas of a paper of Efrat and
Har-Peled [Inf. Process. Lett. 2006] and Deshpande et. al. [WADS 2007]. We also
point out a mistake in the latter.Comment: 25 pages, 4 pages proof ideas, many figure
matching, interpolation, and approximation ; a survey
In this survey we consider geometric techniques which have been used to
measure the similarity or distance between shapes, as well as to approximate
shapes, or interpolate between shapes. Shape is a modality which plays a key
role in many disciplines, ranging from computer vision to molecular biology.
We focus on algorithmic techniques based on computational geometry that have
been developed for shape matching, simplification, and morphing
Algorithms for Geometric Facility Location: Centers in a Polygon and Dispersion on a Line
We study three geometric facility location problems in this thesis.
First, we consider the dispersion problem in one dimension. We are given an ordered list
of (possibly overlapping) intervals on a line. We wish to choose exactly one point from
each interval such that their left to right ordering on the line matches the input order.
The aim is to choose the points so that the distance between the closest pair of points is
maximized, i.e., they must be socially distanced while respecting the order. We give a new
linear-time algorithm for this problem that produces a lexicographically optimal solution.
We also consider some generalizations of this problem.
For the next two problems, the domain of interest is a simple polygon with n vertices.
The second problem concerns the visibility center. The convention is to think of a polygon
as the top view of a building (or art gallery) where the polygon boundary represents opaque
walls. Two points in the domain are visible to each other if the line segment joining them
does not intersect the polygon exterior. The distance to visibility from a source point to a
target point is the minimum geodesic distance from the source to a point in the polygon
visible to the target. The question is: Where should a single guard be located within the
polygon to minimize the maximum distance to visibility? For m point sites in the polygon,
we give an O((m + n) log (m + n)) time algorithm to determine their visibility center.
Finally, we address the problem of locating the geodesic edge center of a simple polygon—a
point in the polygon that minimizes the maximum geodesic distance to any edge. For a
triangle, this point coincides with its incenter. The geodesic edge center is a generalization
of the well-studied geodesic center (a point that minimizes the maximum distance to any
vertex). Center problems are closely related to farthest Voronoi diagrams, which are well-
studied for point sites in the plane, and less well-studied for line segment sites in the plane.
When the domain is a polygon rather than the whole plane, only the case of point sites has
been addressed—surprisingly, more general sites (with line segments being the simplest
example) have been largely ignored. En route to our solution, we revisit, correct, and
generalize (sometimes in a non-trivial manner) existing algorithms and structures tailored
to work specifically for point sites. We give an optimal linear-time algorithm for finding
the geodesic edge center of a simple polygon
Visibility Domains and Complexity
Two problems in discrete and computational geometry are considered that are related to questions about the combinatorial complexity of arrangements of visibility domains and about the hardness of path planning under cost measures defined using visibility domains. The first problem is to estimate the VC-dimension of visibility domains. The VC-dimension is a fundamental parameter of every range space that is typically used to derive upper bounds on the size of hitting sets. Better bounds on the VC-dimension directly translate into better bounds on the size of hitting sets. Estimating the VC-dimension of visibility domains has proven to be a hard problem. In this thesis, new tools to tackle this problem are developed. Encircling arguments are combined with decomposition techniques of a new kind. The main ingredient of the novel approach is the idea of relativization that makes it possible to replace in the analysis of intersections the complicated visibility domains by simpler geometric ranges. The main result here is the new upper bound of 14 on the VC-dimension of visibility polygons in simple polygons that improves significantly upon the previously known best upper bound of 23. For the VC-dimension of perimeter visibility domains, the new techniques yield an upper bound of 7 that leaves only a very small gap to the best known lower bound of 5. The second problem considered is to compute the barrier resilience of visibility domains. In barrier resilience problems, one is given a set of barriers and two points s and t in R^d. The task is to find the minimum number of barriers one has to remove such that there is a way between s and t that does not cross a barrier. In the field of sensor networks, the barriers are interpreted as sensor ranges and the barrier resilience of a network is a measure for its vulnerability. In this thesis the very natural special case where the barriers are visibility domains is investigated. It can also be formulated in terms of finding a so-called minimum witness path. For visibility domains in simple polygons it is shown that one can find an optimal path efficiently. For polygons with holes an approximation hardness result is shown that is stronger than previous hardness results in geometric settings. Two different three-dimensional settings are considered and their respective relations to the Minimum Neighborhood Path problem and the Minimum Color Path problem in graphs are demonstrated. For one of the three-dimensional problems a 2-approximation algorithm is designed. For the general problem of finding minimum witness paths among polyhedral obstacles it turns out that it is not approximable in a strong sense
Twin-width VIII: delineation and win-wins
We introduce the notion of delineation. A graph class is said
delineated if for every hereditary closure of a subclass of
, it holds that has bounded twin-width if and only if
is monadically dependent. An effective strengthening of
delineation for a class implies that tractable FO model checking
on is perfectly understood: On hereditary closures of
subclasses of , FO model checking is fixed-parameter tractable
(FPT) exactly when has bounded twin-width. Ordered graphs
[BGOdMSTT, STOC '22] and permutation graphs [BKTW, JACM '22] are effectively
delineated, while subcubic graphs are not. On the one hand, we prove that
interval graphs, and even, rooted directed path graphs are delineated. On the
other hand, we show that segment graphs, directed path graphs, and visibility
graphs of simple polygons are not delineated. In an effort to draw the
delineation frontier between interval graphs (that are delineated) and
axis-parallel two-lengthed segment graphs (that are not), we investigate the
twin-width of restricted segment intersection classes. It was known that
(triangle-free) pure axis-parallel unit segment graphs have unbounded
twin-width [BGKTW, SODA '21]. We show that -free segment graphs, and
axis-parallel -free unit segment graphs have bounded twin-width, where
is the half-graph or ladder of height . In contrast, axis-parallel
-free two-lengthed segment graphs have unbounded twin-width. Our new
results, combined with the known FPT algorithm for FO model checking on graphs
given with -sequences, lead to win-win arguments. For instance, we derive
FPT algorithms for -Ladder on visibility graphs of 1.5D terrains, and
-Independent Set on visibility graphs of simple polygons.Comment: 51 pages, 19 figure
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