2,237 research outputs found
Rectilinear Link Diameter and Radius in a Rectilinear Polygonal Domain
We study the computation of the diameter and radius under the rectilinear
link distance within a rectilinear polygonal domain of vertices and
holes. We introduce a \emph{graph of oriented distances} to encode the distance
between pairs of points of the domain. This helps us transform the problem so
that we can search through the candidates more efficiently. Our algorithm
computes both the diameter and the radius in time, where denotes the matrix
multiplication exponent and is the number of
edges of the graph of oriented distances. We also provide a faster algorithm
for computing the diameter that runs in time
Clustering with Neighborhoods
In the standard planar -center clustering problem, one is given a set
of points in the plane, and the goal is to select center points, so as
to minimize the maximum distance over points in to their nearest center.
Here we initiate the systematic study of the clustering with neighborhoods
problem, which generalizes the -center problem to allow the covered objects
to be a set of general disjoint convex objects rather than just a
point set . For this problem we first show that there is a PTAS for
approximating the number of centers. Specifically, if is the optimal
radius for centers, then in time we can produce a
set of centers with radius . If instead one
considers the standard goal of approximating the optimal clustering radius,
while keeping as a hard constraint, we show that the radius cannot be
approximated within any factor in polynomial time unless , even
when is a set of line segments. When is a set of
unit disks we show the problem is hard to approximate within a factor of
. This hardness result
complements our main result, where we show that when the objects are disks, of
possibly differing radii, there is a approximation
algorithm. Additionally, for unit disks we give an time -approximation to the optimal
radius, that is, an FPTAS for constant whose running time depends only
linearly on . Finally, we show that the one dimensional version of the
problem, even when intersections are allowed, can be solved exactly in time
Discretization of Planar Geometric Cover Problems
We consider discretization of the 'geometric cover problem' in the plane:
Given a set of points in the plane and a compact planar object ,
find a minimum cardinality collection of planar translates of such that
the union of the translates in the collection contains all the points in .
We show that the geometric cover problem can be converted to a form of the
geometric set cover, which has a given finite-size collection of translates
rather than the infinite continuous solution space of the former. We propose a
reduced finite solution space that consists of distinct canonical translates
and present polynomial algorithms to find the reduce solution space for disks,
convex/non-convex polygons (including holes), and planar objects consisting of
finite Jordan curves.Comment: 16 pages, 5 figure
An Algorithmic Study of Manufacturing Paperclips and Other Folded Structures
We study algorithmic aspects of bending wires and sheet metal into a
specified structure. Problems of this type are closely related to the question
of deciding whether a simple non-self-intersecting wire structure (a
carpenter's ruler) can be straightened, a problem that was open for several
years and has only recently been solved in the affirmative.
If we impose some of the constraints that are imposed by the manufacturing
process, we obtain quite different results. In particular, we study the variant
of the carpenter's ruler problem in which there is a restriction that only one
joint can be modified at a time. For a linkage that does not self-intersect or
self-touch, the recent results of Connelly et al. and Streinu imply that it can
always be straightened, modifying one joint at a time. However, we show that
for a linkage with even a single vertex degeneracy, it becomes NP-hard to
decide if it can be straightened while altering only one joint at a time. If we
add the restriction that each joint can be altered at most once, we show that
the problem is NP-complete even without vertex degeneracies.
In the special case, arising in wire forming manufacturing, that each joint
can be altered at most once, and must be done sequentially from one or both
ends of the linkage, we give an efficient algorithm to determine if a linkage
can be straightened.Comment: 28 pages, 14 figures, Latex, to appear in Computational Geometry -
Theory and Application
Fast approximation of centrality and distances in hyperbolic graphs
We show that the eccentricities (and thus the centrality indices) of all
vertices of a -hyperbolic graph can be computed in linear
time with an additive one-sided error of at most , i.e., after a
linear time preprocessing, for every vertex of one can compute in
time an estimate of its eccentricity such that
for a small constant . We
prove that every -hyperbolic graph has a shortest path tree,
constructible in linear time, such that for every vertex of ,
. These results are based on an
interesting monotonicity property of the eccentricity function of hyperbolic
graphs: the closer a vertex is to the center of , the smaller its
eccentricity is. We also show that the distance matrix of with an additive
one-sided error of at most can be computed in
time, where is a small constant. Recent empirical studies show that
many real-world graphs (including Internet application networks, web networks,
collaboration networks, social networks, biological networks, and others) have
small hyperbolicity. So, we analyze the performance of our algorithms for
approximating centrality and distance matrix on a number of real-world
networks. Our experimental results show that the obtained estimates are even
better than the theoretical bounds.Comment: arXiv admin note: text overlap with arXiv:1506.01799 by other author
On the Line-Separable Unit-Disk Coverage and Related Problems
Given a set of points and a set of disks in the plane, the
disk coverage problem asks for a smallest subset of disks that together cover
all points of . The problem is NP-hard. In this paper, we consider a
line-separable unit-disk version of the problem where all disks have the same
radius and their centers are separated from the points of by a line .
We present an time
algorithm for the problem. This improves the previously best result of time. Our techniques also solve the line-constrained version of the
problem, where centers of all disks of are located on a line while
points of can be anywhere in the plane. Our algorithm runs in time, which improves the previously best result of
time. In addition, our results lead to an algorithm of
time for a half-plane coverage problem (given
half-planes and points, find a smallest subset of half-planes covering all
points); this improves the previously best algorithm of time.
Further, if all half-planes are lower ones, our algorithm runs in
time while the previously best algorithm takes
time.Comment: To appear in ISAAC 202
Discriminating Codes in Geometric Setups
We study geometric variations of the discriminating code problem. In the
\emph{discrete version} of the problem, a finite set of points and a finite
set of objects are given in . The objective is to choose a
subset of minimum cardinality such that for each point , the subset covering satisfies , and each pair , , we have . In the \emph{continuous version} of the problem, the solution set
can be chosen freely among a (potentially infinite) class of allowed geometric
objects. In the 1-dimensional case (), the points in are placed on a
horizontal line , and the objects in are finite-length line segments
aligned with (called intervals). We show that the discrete version of this
problem is NP-complete. This is somewhat surprising as the continuous version
is known to be polynomial-time solvable. Still, for the 1-dimensional discrete
version, we design a polynomial-time -approximation algorithm. We also
design a PTAS for both discrete and continuous versions in one dimension, for
the restriction where the intervals are all required to have the same length.
We then study the 2-dimensional case () for axis-parallel unit square
objects. We show that both continuous and discrete versions are NP-complete,
and design polynomial-time approximation algorithms that produce -approximate and -approximate solutions respectively,
using rounding of suitably defined integer linear programming problems. We show
that the identifying code problem for axis-parallel unit square intersection
graphs (in ) can be solved in the same manner as for the discrete version
of the discriminating code problem for unit square objects
Probabilistic embeddings of the Fr\'echet distance
The Fr\'echet distance is a popular distance measure for curves which
naturally lends itself to fundamental computational tasks, such as clustering,
nearest-neighbor searching, and spherical range searching in the corresponding
metric space. However, its inherent complexity poses considerable computational
challenges in practice. To address this problem we study distortion of the
probabilistic embedding that results from projecting the curves to a randomly
chosen line. Such an embedding could be used in combination with, e.g.
locality-sensitive hashing. We show that in the worst case and under reasonable
assumptions, the discrete Fr\'echet distance between two polygonal curves of
complexity in , where , degrades
by a factor linear in with constant probability. We show upper and lower
bounds on the distortion. We also evaluate our findings empirically on a
benchmark data set. The preliminary experimental results stand in stark
contrast with our lower bounds. They indicate that highly distorted projections
happen very rarely in practice, and only for strongly conditioned input curves.
Keywords: Fr\'echet distance, metric embeddings, random projectionsComment: 27 pages, 11 figure
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