1,771 research outputs found
Structures from Distances in Two and Three Dimensions using Stochastic Proximity Embedding
The point placement problem is to determine the locations of a set of distinct points uniquely (up to translation and reflection) by making the fewest possible pairwise distance queries of an adversary. Deterministic and randomized algorithms are available if distances are known exactly. In this thesis, we discuss a 1-round algorithm for approximate point placement in the plane in an adversarial model. The distance query graph presented to the adversary is chordal. The remaining distances are uniquely determined using the Stochastic Proximity Embedding (SPE) method due to Agrafiotis, and the layout of the points is also generated from SPE. We have also computed the distances uniquely using a distance matrix completion algorithm for chordal graphs, based on a result by Bakonyi and Johnson. The layout of the points is determined using the traditional Young- Householder approach. We compared the layout of both the method and discussed briefly inside. The modified version of SPE is proposed to overcome the highest translation embedding that the method faces when dealing with higher learning rates. We also discuss the computation of molecular structures in three-dimensional space, with only a subset of the pairwise atomic distances available. The subset of distances is obtained using the Philips model for creating artificial backbone chain of molecular structures. We have proposed the Degree of Freedom Approach to solve this problem and carried out our implementation using SPE and the Distance matrix completion Approac
The point placement problem in an inexact model and its applications
In the recent years, due to the advancement in computational tools and techniques to analyze the biological data, biologists have been actively engaged in conducting different experiments to study the arrangements of nucleotide sequence in a chromosome. This masters thesis focuses on the area of the computational methods for the genomic map problem. Though the probe location problem under consideration is known to be NP complete, it is possible to obtain approximate solutions. The distance geometry approach for achieving efficient and better results is shown here. This also solves the point placement problem when the available distance bounds on some probe pairs, correspond to adversarial responses to distance queries between some pairs of points. DGPL program has also been implemented to construct a probe map. Finally some chosen results from the experiments and their significance have been discussed. The screenshots of the working of DGPL algorithm have been attached for better understanding
Mixed Map Labeling
Point feature map labeling is a geometric problem, in which a set of input
points must be labeled with a set of disjoint rectangles (the bounding boxes of
the label texts). Typically, labeling models either use internal labels, which
must touch their feature point, or external (boundary) labels, which are placed
on one of the four sides of the input points' bounding box and which are
connected to their feature points by crossing-free leader lines. In this paper
we study polynomial-time algorithms for maximizing the number of internal
labels in a mixed labeling model that combines internal and external labels.
The model requires that all leaders are parallel to a given orientation , whose value influences the geometric properties and hence the
running times of our algorithms.Comment: Full version for the paper accepted at CIAC 201
Optimal Data Collection For Informative Rankings Expose Well-Connected Graphs
Given a graph where vertices represent alternatives and arcs represent
pairwise comparison data, the statistical ranking problem is to find a
potential function, defined on the vertices, such that the gradient of the
potential function agrees with the pairwise comparisons. Our goal in this paper
is to develop a method for collecting data for which the least squares
estimator for the ranking problem has maximal Fisher information. Our approach,
based on experimental design, is to view data collection as a bi-level
optimization problem where the inner problem is the ranking problem and the
outer problem is to identify data which maximizes the informativeness of the
ranking. Under certain assumptions, the data collection problem decouples,
reducing to a problem of finding multigraphs with large algebraic connectivity.
This reduction of the data collection problem to graph-theoretic questions is
one of the primary contributions of this work. As an application, we study the
Yahoo! Movie user rating dataset and demonstrate that the addition of a small
number of well-chosen pairwise comparisons can significantly increase the
Fisher informativeness of the ranking. As another application, we study the
2011-12 NCAA football schedule and propose schedules with the same number of
games which are significantly more informative. Using spectral clustering
methods to identify highly-connected communities within the division, we argue
that the NCAA could improve its notoriously poor rankings by simply scheduling
more out-of-conference games.Comment: 31 pages, 10 figures, 3 table
External polygon containment problems
AbstractGiven a convex polygonal object P with k vertices and an environment consisting of polygonal obstacles with a total of n corners, we seek a placement for the largest copy of P that does not intersect any of the obstacles, allowing translation, rotation and scaling. We employ the parametric search technique of Megiddo (1983), and the fixed size polygon placement algorithms developed by Leven and Sharir (1987), to obtain an algorithm that runs in time O(k2nλ6(kn)log3(kn)loglog(kn)). We also present several other efficient algorithms for restricted variants of the extremal polygon containment problem, using the same ideas. These variants include: placement of the largest homothetic copies of one or two convex polygons in another convex polygon and placement of the largest similar copy of a triangle in a convex polygon
Optimally fast incremental Manhattan plane embedding and planar tight span construction
We describe a data structure, a rectangular complex, that can be used to
represent hyperconvex metric spaces that have the same topology (although not
necessarily the same distance function) as subsets of the plane. We show how to
use this data structure to construct the tight span of a metric space given as
an n x n distance matrix, when the tight span is homeomorphic to a subset of
the plane, in time O(n^2), and to add a single point to a planar tight span in
time O(n). As an application of this construction, we show how to test whether
a given finite metric space embeds isometrically into the Manhattan plane in
time O(n^2), and add a single point to the space and re-test whether it has
such an embedding in time O(n).Comment: 39 pages, 15 figure
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
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