4,411 research outputs found
Polylogarithmic Approximation for Generalized Minimum Manhattan Networks
Given a set of terminals, which are points in -dimensional Euclidean
space, the minimum Manhattan network problem (MMN) asks for a minimum-length
rectilinear network that connects each pair of terminals by a Manhattan path,
that is, a path consisting of axis-parallel segments whose total length equals
the pair's Manhattan distance. Even for , the problem is NP-hard, but
constant-factor approximations are known. For , the problem is
APX-hard; it is known to admit, for any \eps > 0, an
O(n^\eps)-approximation.
In the generalized minimum Manhattan network problem (GMMN), we are given a
set of terminal pairs, and the goal is to find a minimum-length
rectilinear network such that each pair in is connected by a Manhattan
path. GMMN is a generalization of both MMN and the well-known rectilinear
Steiner arborescence problem (RSA). So far, only special cases of GMMN have
been considered.
We present an -approximation algorithm for GMMN (and, hence,
MMN) in dimensions and an -approximation algorithm for 2D.
We show that an existing -approximation algorithm for RSA in 2D
generalizes easily to dimensions.Comment: 14 pages, 5 figures; added appendix and figure
Halving Balls in Deterministic Linear Time
Let \D be a set of pairwise disjoint unit balls in and the
set of their center points. A hyperplane \Hy is an \emph{-separator} for
\D if each closed halfspace bounded by \Hy contains at least points
from . This generalizes the notion of halving hyperplanes, which correspond
to -separators. The analogous notion for point sets has been well studied.
Separators have various applications, for instance, in divide-and-conquer
schemes. In such a scheme any ball that is intersected by the separating
hyperplane may still interact with both sides of the partition. Therefore it is
desirable that the separating hyperplane intersects a small number of balls
only. We present three deterministic algorithms to bisect or approximately
bisect a given set of disjoint unit balls by a hyperplane: Firstly, we present
a simple linear-time algorithm to construct an -separator for balls
in , for any , that intersects at most
balls, for some constant that depends on and . The number of
intersected balls is best possible up to the constant . Secondly, we present
a near-linear time algorithm to construct an -separator in
that intersects balls. Finally, we give a linear-time algorithm to
construct a halving line in that intersects
disks.
Our results improve the runtime of a disk sliding algorithm by Bereg,
Dumitrescu and Pach. In addition, our results improve and derandomize an
algorithm to construct a space decomposition used by L{\"o}ffler and Mulzer to
construct an onion (convex layer) decomposition for imprecise points (any point
resides at an unknown location within a given disk)
Minimum d-dimensional arrangement with fixed points
In the Minimum -Dimensional Arrangement Problem (d-dimAP) we are given a
graph with edge weights, and the goal is to find a 1-1 map of the vertices into
(for some fixed dimension ) minimizing the total
weighted stretch of the edges. This problem arises in VLSI placement and chip
design.
Motivated by these applications, we consider a generalization of d-dimAP,
where the positions of some of the vertices (pins) is fixed and specified as
part of the input. We are asked to extend this partial map to a map of all the
vertices, again minimizing the weighted stretch of edges. This generalization,
which we refer to as d-dimAP+, arises naturally in these application domains
(since it can capture blocked-off parts of the board, or the requirement of
power-carrying pins to be in certain locations, etc.). Perhaps surprisingly,
very little is known about this problem from an approximation viewpoint.
For dimension , we obtain an -approximation
algorithm, based on a strengthening of the spreading-metric LP for 2-dimAP. The
integrality gap for this LP is shown to be . We also show that
it is NP-hard to approximate 2-dimAP+ within a factor better than
\Omega(k^{1/4-\eps}). We also consider a (conceptually harder, but
practically even more interesting) variant of 2-dimAP+, where the target space
is the grid , instead of
the entire integer lattice . For this problem, we obtain a -approximation using the same LP relaxation. We complement
this upper bound by showing an integrality gap of , and an
\Omega(k^{1/2-\eps})-inapproximability result.
Our results naturally extend to the case of arbitrary fixed target dimension
An Optimal Algorithm for Higher-Order Voronoi Diagrams in the Plane: The Usefulness of Nondeterminism
We present the first optimal randomized algorithm for constructing the
order- Voronoi diagram of points in two dimensions. The expected running
time is , which improves the previous, two-decades-old result
of Ramos (SoCG'99) by a factor. To obtain our result, we (i)
use a recent decision-tree technique of Chan and Zheng (SODA'22) in combination
with Ramos's cutting construction, to reduce the problem to verifying an
order- Voronoi diagram, and (ii) solve the verification problem by a new
divide-and-conquer algorithm using planar-graph separators.
We also describe a deterministic algorithm for constructing the -level of
lines in two dimensions in time, and constructing
the -level of planes in three dimensions in
time. These time bounds (ignoring the term) match the current best
upper bounds on the combinatorial complexity of the -level. Previously, the
same time bound in two dimensions was obtained by Chan (1999) but with
randomization.Comment: To appear in SODA 2024. 16 pages, 1 figur
Higher-order Voronoi diagrams of polygonal objects
Higher-order Voronoi diagrams are fundamental geometric structures which encode the k-nearest neighbor information. Thus, they aid in computations that require proximity information beyond the nearest neighbor. They are related to various favorite structures in computational geometry and are a fascinating combinatorial problem to study. While higher-order Voronoi diagrams of points have been studied a lot, they have not been considered for other types of sites. Points lack dimensionality which makes them unable to represent various real-life instances. Points are the simplest kind of geometric object and therefore higher- order Voronoi diagrams of points can be considered as the corner case of all higher-order Voronoi diagrams. The goal of this dissertation is to move away from the corner and bring the higher-order Voronoi diagram to more general geometric instances. We focus on certain polygonal objects as they provide flexibility and are able to represent real-life instances. Before this dissertation, higher-order Voronoi diagrams of polygonal objects had been studied only for the nearest neighbor and farthest Voronoi diagrams. In this dissertation we investigate structural and combinatorial properties and discover that the dimensionality of geometric objects manifests itself in numerous ways which do not exist in the case of points. We prove that the structural complexity of the order-k Voronoi diagram of non-crossing line segments is O(k(n-k)), as in the case of points. We study disjoint line segments, intersecting line segments, line segments forming a planar straight-line graph and extend the results to the Lp metric, 1<=p<=infty. We also establish the connection between two mathematical abstractions: abstract Voronoi diagrams and the Clarkson-Shor framework. We design several construction algorithms that cover the case of non-point sites. While computational geometry provides several approaches to study the structural complexity that give tight realizable bounds, developing an effective construction algorithm is still a challenging problem even for points. Most of the construction algorithms are designed to work with points as they utilize their simplicity and relations with data-structures that work specifically for points. We extend the iterative and the sweepline approaches that are quite efficient in constructing all order-i Voronoi diagrams, for i<=k and we also give three randomized construction algorithms for abstract higher-order Voronoi diagrams that deal specifically with the construction of the order-k Voronoi diagrams
An Elimination Method for Solving Bivariate Polynomial Systems: Eliminating the Usual Drawbacks
We present an exact and complete algorithm to isolate the real solutions of a
zero-dimensional bivariate polynomial system. The proposed algorithm
constitutes an elimination method which improves upon existing approaches in a
number of points. First, the amount of purely symbolic operations is
significantly reduced, that is, only resultant computation and square-free
factorization is still needed. Second, our algorithm neither assumes generic
position of the input system nor demands for any change of the coordinate
system. The latter is due to a novel inclusion predicate to certify that a
certain region is isolating for a solution. Our implementation exploits
graphics hardware to expedite the resultant computation. Furthermore, we
integrate a number of filtering techniques to improve the overall performance.
Efficiency of the proposed method is proven by a comparison of our
implementation with two state-of-the-art implementations, that is, LPG and
Maple's isolate. For a series of challenging benchmark instances, experiments
show that our implementation outperforms both contestants.Comment: 16 pages with appendix, 1 figure, submitted to ALENEX 201
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