162 research outputs found
Restructuring Expression Dags for Efficient Parallelization
In the field of robust geometric computation it is often necessary to make exact decisions based on inexact floating-point arithmetic. One common approach is to store the computation history in an arithmetic expression dag and to re-evaluate the expression with increasing precision until an exact decision can be made. We show that exact-decisions number types based on expression dags can be evaluated faster in practice through parallelization on multiple cores. We compare the impact of several restructuring methods for the expression dag on its running time in a parallel environment
A face cover perspective to embeddings of planar graphs
It was conjectured by Gupta et al. [Combinatorica04] that every planar graph
can be embedded into with constant distortion. However, given an
-vertex weighted planar graph, the best upper bound on the distortion is
only , by Rao [SoCG99]. In this paper we study the case where
there is a set of terminals, and the goal is to embed only the terminals
into with low distortion. In a seminal paper, Okamura and Seymour
[J.Comb.Theory81] showed that if all the terminals lie on a single face, they
can be embedded isometrically into . The more general case, where the
set of terminals can be covered by faces, was studied by Lee and
Sidiropoulos [STOC09] and Chekuri et al. [J.Comb.Theory13]. The state of the
art is an upper bound of by Krauthgamer, Lee and Rika
[SODA19]. Our contribution is a further improvement on the upper bound to
. Since every planar graph has at most faces, any
further improvement on this result, will be a major breakthrough, directly
improving upon Rao's long standing upper bound. Moreover, it is well known that
the flow-cut gap equals to the distortion of the best embedding into .
Therefore, our result provides a polynomial time -approximation to the sparsest cut problem on planar graphs, for the
case where all the demand pairs can be covered by faces
Metric Embedding via Shortest Path Decompositions
We study the problem of embedding shortest-path metrics of weighted graphs
into spaces. We introduce a new embedding technique based on low-depth
decompositions of a graph via shortest paths. The notion of Shortest Path
Decomposition depth is inductively defined: A (weighed) path graph has shortest
path decomposition (SPD) depth . General graph has an SPD of depth if it
contains a shortest path whose deletion leads to a graph, each of whose
components has SPD depth at most . In this paper we give an
-distortion embedding for graphs of SPD
depth at most . This result is asymptotically tight for any fixed ,
while for it is tight up to second order terms.
As a corollary of this result, we show that graphs having pathwidth embed
into with distortion . For
, this improves over the best previous bound of Lee and Sidiropoulos that
was exponential in ; moreover, for other values of it gives the first
embeddings whose distortion is independent of the graph size . Furthermore,
we use the fact that planar graphs have SPD depth to give a new
proof that any planar graph embeds into with distortion . Our approach also gives new results for graphs with bounded treewidth,
and for graphs excluding a fixed minor
Air Force Institute of Technology Research Report 2009
This report summarizes the research activities of the Air Force Institute of Technology’s Graduate School of Engineering and Management. It describes research interests and faculty expertise; lists student theses/dissertations; identifies research sponsors and contributions; and outlines the procedures for contacting the school. Included in the report are: faculty publications, conference presentations, consultations, and funded research projects. Research was conducted in the areas of Aeronautical and Astronautical Engineering, Electrical Engineering and Electro-Optics, Computer Engineering and Computer Science, Systems and Engineering Management, Operational Sciences, Mathematics, Statistics and Engineering Physics
Air Force Institute of Technology Research Report 2007
This report summarizes the research activities of the Air Force Institute of Technology’s Graduate School of Engineering and Management. It describes research interests and faculty expertise; lists student theses/dissertations; identifies research sponsors and contributions; and outlines the procedures for contacting the school. Included in the report are: faculty publications, conference presentations, consultations, and funded research projects. Research was conducted in the areas of Aeronautical and Astronautical Engineering, Electrical Engineering and Electro-Optics, Computer Engineering and Computer Science, Systems and Engineering Management, Operational Sciences, Mathematics, Statistics and Engineering Physics
High-Dimensional Geometric Streaming in Polynomial Space
Many existing algorithms for streaming geometric data analysis have been
plagued by exponential dependencies in the space complexity, which are
undesirable for processing high-dimensional data sets. In particular, once
, there are no known non-trivial streaming algorithms for problems
such as maintaining convex hulls and L\"owner-John ellipsoids of points,
despite a long line of work in streaming computational geometry since [AHV04].
We simultaneously improve these results to bits of
space by trading off with a factor distortion. We
achieve these results in a unified manner, by designing the first streaming
algorithm for maintaining a coreset for subspace embeddings with
space and distortion. Our
algorithm also gives similar guarantees in the \emph{online coreset} model.
Along the way, we sharpen results for online numerical linear algebra by
replacing a log condition number dependence with a dependence,
answering a question of [BDM+20]. Our techniques provide a novel connection
between leverage scores, a fundamental object in numerical linear algebra, and
computational geometry.
For subspace embeddings, we give nearly optimal trade-offs between
space and distortion for one-pass streaming algorithms. For instance, we give a
deterministic coreset using space and
distortion for , whereas previous deterministic algorithms incurred a
factor in the space or the distortion [CDW18].
Our techniques have implications in the offline setting, where we give
optimal trade-offs between the space complexity and distortion of subspace
sketch data structures. To do this, we give an elementary proof of a "change of
density" theorem of [LT80] and make it algorithmic.Comment: Abstract shortened to meet arXiv limits; v2 fix statements concerning
online condition numbe
- …