311 research outputs found
Polychromatic Coloring for Half-Planes
We prove that for every integer , every finite set of points in the plane
can be -colored so that every half-plane that contains at least
points, also contains at least one point from every color class. We also show
that the bound is best possible. This improves the best previously known
lower and upper bounds of and respectively. We also show
that every finite set of half-planes can be colored so that if a point
belongs to a subset of at least of the half-planes then
contains a half-plane from every color class. This improves the best previously
known upper bound of . Another corollary of our first result is a new
proof of the existence of small size \eps-nets for points in the plane with
respect to half-planes.Comment: 11 pages, 5 figure
Coloring Hypergraphs Induced by Dynamic Point Sets and Bottomless Rectangles
We consider a coloring problem on dynamic, one-dimensional point sets: points
appearing and disappearing on a line at given times. We wish to color them with
k colors so that at any time, any sequence of p(k) consecutive points, for some
function p, contains at least one point of each color.
We prove that no such function p(k) exists in general. However, in the
restricted case in which points appear gradually, but never disappear, we give
a coloring algorithm guaranteeing the property at any time with p(k)=3k-2. This
can be interpreted as coloring point sets in R^2 with k colors such that any
bottomless rectangle containing at least 3k-2 points contains at least one
point of each color. Here a bottomless rectangle is an axis-aligned rectangle
whose bottom edge is below the lowest point of the set. For this problem, we
also prove a lower bound p(k)>ck, where c>1.67. Hence for every k there exists
a point set, every k-coloring of which is such that there exists a bottomless
rectangle containing ck points and missing at least one of the k colors.
Chen et al. (2009) proved that no such function exists in the case of
general axis-aligned rectangles. Our result also complements recent results
from Keszegh and Palvolgyi on cover-decomposability of octants (2011, 2012).Comment: A preliminary version was presented by a subset of the authors to the
European Workshop on Computational Geometry, held in Assisi (Italy) on March
19-21, 201
The Szemeredi-Trotter Theorem in the Complex Plane
It is shown that points and lines in the complex Euclidean plane
determine point-line incidences. This
bound is the best possible, and it generalizes the celebrated theorem by
Szemer\'edi and Trotter about point-line incidences in the real Euclidean plane
.Comment: 24 pages, 5 figures, to appear in Combinatoric
Intrusion Detection Systems for Community Wireless Mesh Networks
Wireless mesh networks are being increasingly used to provide affordable network connectivity to communities where wired deployment strategies are either not possible or are prohibitively expensive. Unfortunately, computer networks (including mesh networks) are frequently being exploited by increasingly profit-driven and insidious attackers, which can affect their utility for legitimate use. In response to this, a number of countermeasures have been developed, including intrusion detection systems that aim to detect anomalous behaviour caused by attacks. We present a set of socio-technical challenges associated with developing an intrusion detection system for a community wireless mesh network. The attack space on a mesh network is particularly large; we motivate the need for and describe the challenges of adopting an asset-driven approach to managing this space. Finally, we present an initial design of a modular architecture for intrusion detection, highlighting how it addresses the identified challenges
Unsplittable coverings in the plane
A system of sets forms an {\em -fold covering} of a set if every point
of belongs to at least of its members. A -fold covering is called a
{\em covering}. The problem of splitting multiple coverings into several
coverings was motivated by classical density estimates for {\em sphere
packings} as well as by the {\em planar sensor cover problem}. It has been the
prevailing conjecture for 35 years (settled in many special cases) that for
every plane convex body , there exists a constant such that every
-fold covering of the plane with translates of splits into
coverings. In the present paper, it is proved that this conjecture is false for
the unit disk. The proof can be generalized to construct, for every , an
unsplittable -fold covering of the plane with translates of any open convex
body which has a smooth boundary with everywhere {\em positive curvature}.
Somewhat surprisingly, {\em unbounded} open convex sets do not misbehave,
they satisfy the conjecture: every -fold covering of any region of the plane
by translates of such a set splits into two coverings. To establish this
result, we prove a general coloring theorem for hypergraphs of a special type:
{\em shift-chains}. We also show that there is a constant such that, for
any positive integer , every -fold covering of a region with unit disks
splits into two coverings, provided that every point is covered by {\em at
most} sets
Drawing Planar Graphs with a Prescribed Inner Face
Given a plane graph (i.e., a planar graph with a fixed planar embedding)
and a simple cycle in whose vertices are mapped to a convex polygon, we
consider the question whether this drawing can be extended to a planar
straight-line drawing of . We characterize when this is possible in terms of
simple necessary conditions, which we prove to be sufficient. This also leads
to a linear-time testing algorithm. If a drawing extension exists, it can be
computed in the same running time
Untangling polygons and graphs
Untangling is a process in which some vertices of a planar graph are moved to
obtain a straight-line plane drawing. The aim is to move as few vertices as
possible. We present an algorithm that untangles the cycle graph C_n while
keeping at least \Omega(n^{2/3}) vertices fixed. For any graph G, we also
present an upper bound on the number of fixed vertices in the worst case. The
bound is a function of the number of vertices, maximum degree and diameter of
G. One of its consequences is the upper bound O((n log n)^{2/3}) for all
3-vertex-connected planar graphs.Comment: 11 pages, 3 figure
Subsampling in Smoothed Range Spaces
We consider smoothed versions of geometric range spaces, so an element of the
ground set (e.g. a point) can be contained in a range with a non-binary value
in . Similar notions have been considered for kernels; we extend them to
more general types of ranges. We then consider approximations of these range
spaces through -nets and -samples (aka
-approximations). We characterize when size bounds for
-samples on kernels can be extended to these more general
smoothed range spaces. We also describe new generalizations for -nets to these range spaces and show when results from binary range spaces can
carry over to these smoothed ones.Comment: This is the full version of the paper which appeared in ALT 2015. 16
pages, 3 figures. In Algorithmic Learning Theory, pp. 224-238. Springer
International Publishing, 201
Role of TSH and excess Heart Age in Predicting Atrial Fibrillation Recurrence Post-Ablation
Background: The association between atrial fibrillation (AF) and thyroid disease as defined by thyroid stimulating hormone (TSH) is established in literature. However, the relationship between TSH and recurrence of AF post ablation has not been established.
Methods: We studied 207 patients (60.54±9.39yrs, 35.7% female) with persistent or paroxysmal AF who underwent either Cryo or RFA ablation between April 2011 and Jan 2015 at our center. Patients were stratified into hypothyroid (TSH \u3e \u3e4.5 U/mL), euthyroid (TSH 0.5-4.5 U/mL) and hyperthyroid (TSH \u3c 0.5 U/mL) based on pre procedure testing. Heart age was computed based on Framingham risk factors. Excess heart age was defined as the difference between actual age and heart age. Logistic regression and cox-proportional hazards model were implemented using R statistical software (v3.2.0).
Results: There was a statistically significant lower rate of AF recurrence among male patients (OR 2.92, p=0.003). In univariate analysis, there was no statistically significant relationship between TSH and incidence of AF recurrence (OR 1.05, p=0.74). Cox proportional hazards models did not show an association between recurrence and TSH states (HR 0.85, p=0.74 for hypothyroid and HR 0.75, p=0.56 for hyperthyroid).
Conclusions: This exploratory showed that TSH may not play a role in AF recurrence. While there is a tendency towards an association between TSH and AF recurrence, this was not statistically significant. We hypothesize that overt hyperthyroidism prior to ablation will not increase chance of recurrence. This was true after adjustment for Framingham risk factors. The limitation of this study was the small sample size of the patients with TSH in the hyperthyroid range. Further analysis using larger dataset is indicated
A Bichromatic Incidence Bound and an Application
We prove a new, tight upper bound on the number of incidences between points
and hyperplanes in Euclidean d-space. Given n points, of which k are colored
red, there are O_d(m^{2/3}k^{2/3}n^{(d-2)/3} + kn^{d-2} + m) incidences between
the k red points and m hyperplanes spanned by all n points provided that m =
\Omega(n^{d-2}). For the monochromatic case k = n, this was proved by Agarwal
and Aronov.
We use this incidence bound to prove that a set of n points, no more than n-k
of which lie on any plane or two lines, spans \Omega(nk^2) planes. We also
provide an infinite family of counterexamples to a conjecture of Purdy's on the
number of hyperplanes spanned by a set of points in dimensions higher than 3,
and present new conjectures not subject to the counterexample.Comment: 12 page
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