44,495 research outputs found
On Geometric Range Searching, Approximate Counting and Depth Problems
In this thesis we deal with problems connected to range searching,
which is one of the central areas of computational geometry.
The dominant problems in this area are
halfspace range searching, simplex range searching and orthogonal range searching and
research into these problems has spanned decades.
For many range searching problems, the best possible
data structures cannot offer fast (i.e., polylogarithmic) query
times if we limit ourselves to near linear storage.
Even worse, it is conjectured (and proved in some cases)
that only very small improvements to these might be possible.
This inefficiency has encouraged many researchers to seek alternatives through approximations.
In this thesis we continue this line of research and focus on
relative approximation of range counting problems.
One important problem where it is possible to achieve significant speedup
through approximation is halfspace range counting in 3D.
Here we continue the previous research done
and obtain the first optimal data structure for approximate halfspace range counting in 3D.
Our data structure has the slight advantage of being Las Vegas (the result is always correct) in contrast
to the previous methods that were Monte Carlo (the correctness holds with high probability).
Another series of problems where approximation can provide us with
substantial speedup comes from robust statistics.
We recognize three problems here:
approximate Tukey depth, regression depth and simplicial depth queries.
In 2D, we obtain an optimal data structure capable of approximating
the regression depth of a query hyperplane.
We also offer a linear space data structure which can answer approximate
Tukey depth queries efficiently in 3D.
These data structures are obtained by applying our ideas for the
approximate halfspace counting problem.
Approximating the simplicial depth turns out to be much more
difficult, however.
Computing the simplicial depth of a given point is more computationally
challenging than most other definitions of data depth.
In 2D we obtain the first data structure which uses near linear space
and can answer approximate simplicial depth queries in polylogarithmic time.
As applications of this result, we provide two non-trivial methods to
approximate the simplicial depth of a given point in higher dimension.
Along the way, we establish a tight combinatorial relationship between
the Tukey depth of any given point and its simplicial depth.
Another problem investigated in this thesis is the dominance reporting problem,
an important special case of orthogonal range reporting.
In three dimensions, we solve this
problem in the pointer machine model and the external memory model
by offering the first optimal data structures in these models of computation.
Also, in the RAM model and for points from
an integer grid we reduce the space complexity of the fastest
known data structure to optimal.
Using known techniques in the literature, we can use our
results to obtain solutions for the orthogonal range searching problem as well.
The query complexity offered by our orthogonal range reporting data structures
match the most efficient query complexities
known in the literature but our space bounds are lower than the previous methods in the external
memory model and RAM model where the input is a subset of an integer grid.
The results also yield improved orthogonal range searching in
higher dimensions (which shows the significance
of the dominance reporting problem).
Intersection searching is a generalization of range searching where
we deal with more complicated geometric objects instead of points.
We investigate the rectilinear disjoint polygon counting problem
which is a specialized intersection counting problem.
We provide a linear-size data structure capable of counting
the number of disjoint rectilinear polygons
intersecting any rectilinear polygon of constant size.
The query time (as well as some other properties of our data structure) resembles
the classical simplex range searching data structures
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
Recommended from our members
Assessment of mechanical properties and microstructure characterizing techniques in their ability to quantify amount of cold work in 316l alloy
Stress corrosion cracking (SCC) behavior is a matter of concern for structural materials, namely, stainless steels and nickel alloys, in nuclear power plants. High levels of cold work (CW) have shown to both reduce crack initiation times and increase crack growth rates. Cold working has numerous effects on a material, including changes in microstructure, mechanical properties, and residual stress state, yet it is typically reported as a simple percent change in geometry. There is need to develop a strategy for quantitative assessment of cold-work level in order to better understand stress corrosion cracking test data. Five assessment techniques, commonly performed alongside stress corrosion cracking testing (optical microscopy (OM), electron backscatter diffraction (EBSD), X-ray diffraction (XRD), tensile testing, and hardness testing) are evaluated with respect to their ability to quantify the level of CW in a component. The test material is stainless steel 316L that has been cold-rolled to three conditions: 0%, 20%, and 30% CW. Measurement results for each assessment method include correlation with CW condition and repeatability data. Measured values showed significant spatial variation, illustrating that CW level is not uniform throughout a component. Mechanical properties (tensile testing, hardness) were found to correlate most linearly with the amount of imparted CW
Dynamic Range Majority Data Structures
Given a set of coloured points on the real line, we study the problem of
answering range -majority (or "heavy hitter") queries on . More
specifically, for a query range , we want to return each colour that is
assigned to more than an -fraction of the points contained in . We
present a new data structure for answering range -majority queries on a
dynamic set of points, where . Our data structure uses O(n)
space, supports queries in time, and updates in amortized time. If the coordinates of the points are integers,
then the query time can be improved to . For constant values of , this improved query
time matches an existing lower bound, for any data structure with
polylogarithmic update time. We also generalize our data structure to handle
sets of points in d-dimensions, for , as well as dynamic arrays, in
which each entry is a colour.Comment: 16 pages, Preliminary version appeared in ISAAC 201
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