220 research outputs found
Locked and Unlocked Polygonal Chains in 3D
In this paper, we study movements of simple polygonal chains in 3D. We say
that an open, simple polygonal chain can be straightened if it can be
continuously reconfigured to a straight sequence of segments in such a manner
that both the length of each link and the simplicity of the chain are
maintained throughout the movement. The analogous concept for closed chains is
convexification: reconfiguration to a planar convex polygon. Chains that cannot
be straightened or convexified are called locked. While there are open chains
in 3D that are locked, we show that if an open chain has a simple orthogonal
projection onto some plane, it can be straightened. For closed chains, we show
that there are unknotted but locked closed chains, and we provide an algorithm
for convexifying a planar simple polygon in 3D with a polynomial number of
moves.Comment: To appear in Proc. 10th ACM-SIAM Sympos. Discrete Algorithms, Jan.
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Polymorphisms of the glucose transporter (GLUT1) gene are associated with diabetic nephropathy
Polymorphisms of the glucose transporter (GLUT1) gene are associated with diabetic nephropathy.BackgroundDiabetic nephropathy (DN) is a major cause of morbidity and mortality in patients with type 1 diabetes mellitus. Recent studies suggest that genetic factors, including polymorphisms in the flanking region of the aldose reductase gene (5′ALR2), play an important role in the pathogenesis of nephropathy. Glucose transporter (GLUT1) activity has been implicated in renal hypertrophy and extracellular matrix formation in mesangial cells. The aim was to investigate the frequency of a polymorphism within the GLUT1 gene in 186 Caucasoid patients with type 1 diabetes and 104 normal controls.MethodsAmplimers flanking the Xba-I polymorphic site in the second intron were employed to amplify DNA from subjects. The amplified DNA was restricted with endonuclease Xba-I, separated by gel electrophoresis, and visualized. In the absence of an Xba-I site, a fragment of 1.1 kilobase was seen, whereas fragments of 0.9 and 0.2 were generated if the Xba-I site was present.ResultsThere was a highly significant increase in the frequency of the 1.1 allele in those patients with nephropathy (N = 70) compared with those with no proteinuria or retinopathy after 20 years of diabetes (uncomplicated N = 44, 61.4 vs. 40.9%, respectively, P < 0.001). The 1.1/1.1 genotype was also significantly increased in the nephropathy group compared with the uncomplicated group of patients (37.1 vs. 13.6%, respectively, P < 0.01). The frequency of the 1.1/1.1 genotype was similar in 30 patients with retinopathy but not nephropathy when compared with the uncomplicated group of patients (13.6 vs. 16.7%). Furthermore, only 8 out of 49 patients with DN had the Z+2 5′ALR2 DN “protective” allele and the 0.9 GLUT1 allele in contrast to 21 out of 39 uncomplicated patients (P < 0.0002).ConclusionThese results suggest that the GLUT1 gene together with the aldose reductase gene are associated with susceptibility to DN in patients with type 1 diabetes
Locked and Unlocked Polygonal Chains in Three Dimensions
This paper studies movements of polygonal chains in three dimensions whose links are not allowed to cross or change length. Our main result is an algorithmic proof that any simple closed chain that initially takes the form of a planar polygon can be made convex in three dimensions. Other results include an algorithm for straightening open chains having a simple orthogonal projection onto some plane, and an algorithm for making convex any open chain initially configured on the surface of a polytope. All our algorithms require only O (n) basic moves.
Local Guarantees in Graph Cuts and Clustering
Correlation Clustering is an elegant model that captures fundamental graph
cut problems such as Min Cut, Multiway Cut, and Multicut, extensively
studied in combinatorial optimization. Here, we are given a graph with edges
labeled or and the goal is to produce a clustering that agrees with the
labels as much as possible: edges within clusters and edges across
clusters. The classical approach towards Correlation Clustering (and other
graph cut problems) is to optimize a global objective. We depart from this and
study local objectives: minimizing the maximum number of disagreements for
edges incident on a single node, and the analogous max min agreements
objective. This naturally gives rise to a family of basic min-max graph cut
problems. A prototypical representative is Min Max Cut: find an cut
minimizing the largest number of cut edges incident on any node. We present the
following results: an -approximation for the problem of
minimizing the maximum total weight of disagreement edges incident on any node
(thus providing the first known approximation for the above family of min-max
graph cut problems), a remarkably simple -approximation for minimizing
local disagreements in complete graphs (improving upon the previous best known
approximation of ), and a -approximation for
maximizing the minimum total weight of agreement edges incident on any node,
hence improving upon the -approximation that follows from
the study of approximate pure Nash equilibria in cut and party affiliation
games
A PTAS for planar group Steiner tree via spanner bootstrapping and prize collecting
We present the first polynomial-time approximation scheme (PTAS), i.e., (1 + ϵ)-approximation algorithm for any constant ϵ > 0, for the planar group Steiner tree problem (in which each group lies on a boundary of a face). This result improves on the best previous approximation factor of O(logn(loglogn)O(1)). We achieve this result via a novel and powerful technique called spanner bootstrapping, which allows one to bootstrap from a superconstant approximation factor (even superpolynomial in the input size) all the way down to a PTAS. This is in contrast with the popular existing approach for planar PTASs of constructing lightweight spanners in one iteration, which notably requires a constant-factor approximate solution to start from. Spanner bootstrapping removes one of the main barriers for designing PTASs for problems which have no known constant-factor approximation (even on planar graphs), and thus can be used to obtain PTASs for several difficult-to-approximate problems. Our second major contribution required for the planar group Steiner tree PTAS is a spanner construction, which reduces the graph to have total weight within a factor of the optimal solution while approximately preserving the optimal solution. This is particularly challenging because group Steiner tree requires deciding which terminal in each group to connect by the tree, making it much harder than recent previous approaches to construct spanners for planar TSP by Klein [SIAM J. Computing 2008], subset TSP by Klein [STOC 2006], Steiner tree by Borradaile, Klein, and Mathieu [ACM Trans. Algorithms 2009], and Steiner forest by Bateni, Hajiaghayi, and Marx [J. ACM 2011] (and its improvement to an efficient PTAS by Eisenstat, Klein, and Mathieu [SODA 2012]. The main conceptual contribution here is realizing that selecting which terminals may be relevant is essentially a complicated prize-collecting process: we have to carefully weigh the cost and benefits of reaching or avoiding certain terminals in the spanner. Via a sequence of involved prize-collecting procedures, we can construct a spanner that reaches a set of terminals that is sufficient for an almost-optimal solution. Our PTAS for planar group Steiner tree implies the first PTAS for geometric Euclidean group Steiner tree with obstacles, as well as a (2 + ϵ)-approximation algorithm for group TSP with obstacles, improving over the best previous constant-factor approximation algorithms. By contrast, we show that planar group Steiner forest, a slight generalization of planar group Steiner tree, is APX-hard on planar graphs of treewidth 3, even if the groups are pairwise disjoint and every group is a vertex or an edge
On Profit-Maximizing Pricing for the Highway and Tollbooth Problems
In the \emph{tollbooth problem}, we are given a tree \bT=(V,E) with
edges, and a set of customers, each of whom is interested in purchasing a
path on the tree. Each customer has a fixed budget, and the objective is to
price the edges of \bT such that the total revenue made by selling the paths
to the customers that can afford them is maximized. An important special case
of this problem, known as the \emph{highway problem}, is when \bT is
restricted to be a line.
For the tollbooth problem, we present a randomized -approximation,
improving on the current best -approximation. We also study a
special case of the tollbooth problem, when all the paths that customers are
interested in purchasing go towards a fixed root of \bT. In this case, we
present an algorithm that returns a -approximation, for any
, and runs in quasi-polynomial time. On the other hand, we rule
out the existence of an FPTAS by showing that even for the line case, the
problem is strongly NP-hard. Finally, we show that in the \emph{coupon model},
when we allow some items to be priced below zero to improve the overall profit,
the problem becomes even APX-hard
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Dynamic ham-sandwich cuts in the plane
We design efficient data structures for dynamically maintaining a ham-sandwich cut of two point sets in the plane subject to insertions and deletions of points in either set. A ham-sandwich cut is a line that simultaneously bisects the cardinality of both point sets. For general point sets, our first data structure supports each operation in O(n1/3+ε) amortized time and O(n4/3+ε) space. Our second data structure performs faster when each point set decomposes into a small number k of subsets in convex position: it supports insertions and deletions in O(logn) time and ham-sandwich queries in O(klog4n) time. In addition, if each point set has convex peeling depth k , then we can maintain the decomposition automatically using O(klogn) time per insertion and deletion. Alternatively, we can view each convex point set as a convex polygon, and we show how to find a ham-sandwich cut that bisects the total areas or total perimeters of these polygons in O(klog4n) time plus the O((kb)polylog(kb)) time required to approximate the root of a polynomial of degree O(k) up to b bits of precision. We also show how to maintain a partition of the plane by two lines into four regions each containing a quarter of the total point count, area, or perimeter in polylogarithmic time.Engineering and Applied Science
Deterministic Sampling and Range Counting in Geometric Data Streams
We present memory-efficient deterministic algorithms for constructing
epsilon-nets and epsilon-approximations of streams of geometric data. Unlike
probabilistic approaches, these deterministic samples provide guaranteed bounds
on their approximation factors. We show how our deterministic samples can be
used to answer approximate online iceberg geometric queries on data streams. We
use these techniques to approximate several robust statistics of geometric data
streams, including Tukey depth, simplicial depth, regression depth, the
Thiel-Sen estimator, and the least median of squares. Our algorithms use only a
polylogarithmic amount of memory, provided the desired approximation factors
are inverse-polylogarithmic. We also include a lower bound for non-iceberg
geometric queries.Comment: 12 pages, 1 figur
Space-optimal Heavy Hitters with Strong Error Bounds
The problem of finding heavy hitters and approximating the frequencies of items is at the heart of many problems in data stream analysis. It has been observed that several proposed solutions to this problem can outperform their worst-case guarantees on real data. This leads to the question of whether some stronger bounds can be guaranteed. We answer this in the positive by showing that a class of "counter-based algorithms" (including the popular and very space-efficient FREQUENT and SPACESAVING algorithms) provide much stronger approximation guarantees than previously known. Specifically, we show that errors in the approximation of individual elements do not depend on the frequencies of the most frequent elements, but only on the frequency of the remaining "tail." This shows that counter-based methods are the most space-efficient (in fact, space-optimal) algorithms having this strong error bound.
This tail guarantee allows these algorithms to solve the "sparse recovery" problem. Here, the goal is to recover a faithful representation of the vector of frequencies, f. We prove that using space O(k), the algorithms construct an approximation f* to the frequency vector f so that the L1 error ||f -- f*||[subscript 1] is close to the best possible error min[subscript f2] ||f2 -- f||[subscript 1], where f2 ranges over all vectors with at most k non-zero entries. This improves the previously best known space bound of about O(k log n) for streams without element deletions (where n is the size of the domain from which stream elements are drawn). Other consequences of the tail guarantees are results for skewed (Zipfian) data, and guarantees for accuracy of merging multiple summarized streams.David & Lucile Packard Foundation (Fellowship)Center for Massive Data Algorithmics (MADALGO)National Science Foundation (U.S.). (Grant number CCF-0728645
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