1,163 research outputs found
Fully Dynamic Maximum Independent Sets of Disks in Polylogarithmic Update Time
A fundamental question in computational geometry is for a dynamic collection
of geometric objects in Euclidean space, whether it is possible to maintain a
maximum independent set in polylogarithmic update time. Already, for a set of
intervals, it is known that no dynamic algorithm can maintain an exact maximum
independent set with sublinear update time. Therefore, the typical objective is
to explore the trade-off between update time and solution size. Substantial
efforts have been made in recent years to understand this question for various
families of geometric objects, such as intervals, hypercubes, hyperrectangles,
and fat objects.
We present the first fully dynamic approximation algorithm for disks of
arbitrary radii in the plane that maintains a constant-factor approximate
maximum independent set in polylogarithmic update time. First, we show that for
a fully dynamic set of unit disks in the plane, a -approximate maximum
independent set can be maintained with worst-case update time ,
and optimal output-sensitive reporting. Moreover, this result generalizes to
fat objects of comparable sizes in any fixed dimension , where the
approximation ratio depends on the dimension and the fatness parameter. Our
main result is that for a fully dynamic set of disks of arbitrary radii in the
plane, an -approximate maximum independent set can be maintained in
polylogarithmic expected amortized update time.Comment: Abstract is shortened to meet Arxiv's requirement on the number of
character
A Bicriteria Approximation for the Reordering Buffer Problem
In the reordering buffer problem (RBP), a server is asked to process a
sequence of requests lying in a metric space. To process a request the server
must move to the corresponding point in the metric. The requests can be
processed slightly out of order; in particular, the server has a buffer of
capacity k which can store up to k requests as it reads in the sequence. The
goal is to reorder the requests in such a manner that the buffer constraint is
satisfied and the total travel cost of the server is minimized. The RBP arises
in many applications that require scheduling with a limited buffer capacity,
such as scheduling a disk arm in storage systems, switching colors in paint
shops of a car manufacturing plant, and rendering 3D images in computer
graphics.
We study the offline version of RBP and develop bicriteria approximations.
When the underlying metric is a tree, we obtain a solution of cost no more than
9OPT using a buffer of capacity 4k + 1 where OPT is the cost of an optimal
solution with buffer capacity k. Constant factor approximations were known
previously only for the uniform metric (Avigdor-Elgrabli et al., 2012). Via
randomized tree embeddings, this implies an O(log n) approximation to cost and
O(1) approximation to buffer size for general metrics. Previously the best
known algorithm for arbitrary metrics by Englert et al. (2007) provided an
O(log^2 k log n) approximation without violating the buffer constraint.Comment: 13 page
Minimum Cuts in Geometric Intersection Graphs
Let be a set of disks in the plane. The disk graph
for is the undirected graph with vertex set
in which two disks are joined by an edge if and only if they
intersect. The directed transmission graph for
is the directed graph with vertex set in which
there is an edge from a disk to a disk if and only if contains the center of .
Given and two non-intersecting disks , we
show that a minimum - vertex cut in or in
can be found in
expected time. To obtain our result, we combine an algorithm for the maximum
flow problem in general graphs with dynamic geometric data structures to
manipulate the disks.
As an application, we consider the barrier resilience problem in a
rectangular domain. In this problem, we have a vertical strip bounded by
two vertical lines, and , and a collection of
disks. Let be a point in above all disks of , and let
a point in below all disks of . The task is to find a curve
from to that lies in and that intersects as few disks of
as possible. Using our improved algorithm for minimum cuts in
disk graphs, we can solve the barrier resilience problem in
expected time.Comment: 11 pages, 4 figure
Parallelization of cycle-based logic simulation
Verification of digital circuits by Cycle-based simulation can be performed in parallel. The parallel implementation requires two phases: the compilation phase, that sets up the data needed for the
execution of the simulation, and the simulation phase, that consists in executing the parallel simulation of the considered circuit for a certain number of cycles. During the early phase of design, compilation phase has to be repeated each time a bug is found. Thus, if the time of the compilation phase is too high, the advantages stemming from the parallel approach may be lost. In this work we propose an
effective version of the compilation phase and compute the corresponding execution time. We also analyze the percentage of execution time required by the different steps of the compilation phase for
a set of literature benchmarks. Further, we implemented the simulation phase exploiting the GPU architecture, and we computed the execution times for a set of benchmarks obtaining values comparable
with literature ones. Finally, we implemented the sequential version of the Cycle-based simulation in such a way that the execution time is optimized. We used the sequential values to compute the speedup
of the parallel version for the considered set of benchmarks
Parallel and Distributed Data Series Processing on Modern and Emerging Hardware
This paper summarizes state-of-the-art results on data series processing with
the emphasis on parallel and distributed data series indexes that exploit the
computational power of modern computing platforms. The paper comprises a
summary of the tutorial the author delivered at the 15th International
Conference on Management of Digital EcoSystems (MEDES'23).Comment: This paper will appear in the Proceedings of the 15th International
Conference on Management of Digital EcoSystems (MEDES'23
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