4,394 research outputs found
Minimizing the stabbing number of matchings, trees, and triangulations
The (axis-parallel) stabbing number of a given set of line segments is the
maximum number of segments that can be intersected by any one (axis-parallel)
line. This paper deals with finding perfect matchings, spanning trees, or
triangulations of minimum stabbing number for a given set of points. The
complexity of these problems has been a long-standing open question; in fact,
it is one of the original 30 outstanding open problems in computational
geometry on the list by Demaine, Mitchell, and O'Rourke. The answer we provide
is negative for a number of minimum stabbing problems by showing them NP-hard
by means of a general proof technique. It implies non-trivial lower bounds on
the approximability. On the positive side we propose a cut-based integer
programming formulation for minimizing the stabbing number of matchings and
spanning trees. We obtain lower bounds (in polynomial time) from the
corresponding linear programming relaxations, and show that an optimal
fractional solution always contains an edge of at least constant weight. This
result constitutes a crucial step towards a constant-factor approximation via
an iterated rounding scheme. In computational experiments we demonstrate that
our approach allows for actually solving problems with up to several hundred
points optimally or near-optimally.Comment: 25 pages, 12 figures, Latex. To appear in "Discrete and Computational
Geometry". Previous version (extended abstract) appears in SODA 2004, pp.
430-43
On the complexity of range searching among curves
Modern tracking technology has made the collection of large numbers of
densely sampled trajectories of moving objects widely available. We consider a
fundamental problem encountered when analysing such data: Given polygonal
curves in , preprocess into a data structure that answers
queries with a query curve and radius for the curves of that
have \Frechet distance at most to .
We initiate a comprehensive analysis of the space/query-time trade-off for
this data structuring problem. Our lower bounds imply that any data structure
in the pointer model model that achieves query time, where is
the output size, has to use roughly space in
the worst case, even if queries are mere points (for the discrete \Frechet
distance) or line segments (for the continuous \Frechet distance). More
importantly, we show that more complex queries and input curves lead to
additional logarithmic factors in the lower bound. Roughly speaking, the number
of logarithmic factors added is linear in the number of edges added to the
query and input curve complexity. This means that the space/query time
trade-off worsens by an exponential factor of input and query complexity. This
behaviour addresses an open question in the range searching literature: whether
it is possible to avoid the additional logarithmic factors in the space and
query time of a multilevel partition tree. We answer this question negatively.
On the positive side, we show we can build data structures for the \Frechet
distance by using semialgebraic range searching. Our solution for the discrete
\Frechet distance is in line with the lower bound, as the number of levels in
the data structure is , where denotes the maximal number of vertices
of a curve. For the continuous \Frechet distance, the number of levels
increases to
Sign rank versus VC dimension
This work studies the maximum possible sign rank of sign
matrices with a given VC dimension . For , this maximum is {three}. For
, this maximum is . For , similar but
slightly less accurate statements hold. {The lower bounds improve over previous
ones by Ben-David et al., and the upper bounds are novel.}
The lower bounds are obtained by probabilistic constructions, using a theorem
of Warren in real algebraic topology. The upper bounds are obtained using a
result of Welzl about spanning trees with low stabbing number, and using the
moment curve.
The upper bound technique is also used to: (i) provide estimates on the
number of classes of a given VC dimension, and the number of maximum classes of
a given VC dimension -- answering a question of Frankl from '89, and (ii)
design an efficient algorithm that provides an multiplicative
approximation for the sign rank.
We also observe a general connection between sign rank and spectral gaps
which is based on Forster's argument. Consider the adjacency
matrix of a regular graph with a second eigenvalue of absolute value
and . We show that the sign rank of the signed
version of this matrix is at least . We use this connection to
prove the existence of a maximum class with VC
dimension and sign rank . This answers a question
of Ben-David et al.~regarding the sign rank of large VC classes. We also
describe limitations of this approach, in the spirit of the Alon-Boppana
theorem.
We further describe connections to communication complexity, geometry,
learning theory, and combinatorics.Comment: 33 pages. This is a revised version of the paper "Sign rank versus VC
dimension". Additional results in this version: (i) Estimates on the number
of maximum VC classes (answering a question of Frankl from '89). (ii)
Estimates on the sign rank of large VC classes (answering a question of
Ben-David et al. from '03). (iii) A discussion on the computational
complexity of computing the sign-ran
Ramsey-type theorems for lines in 3-space
We prove geometric Ramsey-type statements on collections of lines in 3-space.
These statements give guarantees on the size of a clique or an independent set
in (hyper)graphs induced by incidence relations between lines, points, and
reguli in 3-space. Among other things, we prove that: (1) The intersection
graph of n lines in R^3 has a clique or independent set of size Omega(n^{1/3}).
(2) Every set of n lines in R^3 has a subset of n^{1/2} lines that are all
stabbed by one line, or a subset of Omega((n/log n)^{1/5}) such that no
6-subset is stabbed by one line. (3) Every set of n lines in general position
in R^3 has a subset of Omega(n^{2/3}) lines that all lie on a regulus, or a
subset of Omega(n^{1/3}) lines such that no 4-subset is contained in a regulus.
The proofs of these statements all follow from geometric incidence bounds --
such as the Guth-Katz bound on point-line incidences in R^3 -- combined with
Tur\'an-type results on independent sets in sparse graphs and hypergraphs.
Although similar Ramsey-type statements can be proved using existing generic
algebraic frameworks, the lower bounds we get are much larger than what can be
obtained with these methods. The proofs directly yield polynomial-time
algorithms for finding subsets of the claimed size.Comment: 18 pages including appendi
Witness (Delaunay) Graphs
Proximity graphs are used in several areas in which a neighborliness
relationship for input data sets is a useful tool in their analysis, and have
also received substantial attention from the graph drawing community, as they
are a natural way of implicitly representing graphs. However, as a tool for
graph representation, proximity graphs have some limitations that may be
overcome with suitable generalizations. We introduce a generalization, witness
graphs, that encompasses both the goal of more power and flexibility for graph
drawing issues and a wider spectrum for neighborhood analysis. We study in
detail two concrete examples, both related to Delaunay graphs, and consider as
well some problems on stabbing geometric objects and point set discrimination,
that can be naturally described in terms of witness graphs.Comment: 27 pages. JCCGG 200
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
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