5,267 research outputs found
Upward Three-Dimensional Grid Drawings of Graphs
A \emph{three-dimensional grid drawing} of a graph is a placement of the
vertices at distinct points with integer coordinates, such that the straight
line segments representing the edges do not cross. Our aim is to produce
three-dimensional grid drawings with small bounding box volume. We prove that
every -vertex graph with bounded degeneracy has a three-dimensional grid
drawing with volume. This is the broadest class of graphs admiting
such drawings. A three-dimensional grid drawing of a directed graph is
\emph{upward} if every arc points up in the z-direction. We prove that every
directed acyclic graph has an upward three-dimensional grid drawing with
volume, which is tight for the complete dag. The previous best upper
bound was . Our main result is that every -colourable directed
acyclic graph ( constant) has an upward three-dimensional grid drawing with
volume. This result matches the bound in the undirected case, and
improves the best known bound from for many classes of directed
acyclic graphs, including planar, series parallel, and outerplanar
Measuring and Understanding Throughput of Network Topologies
High throughput is of particular interest in data center and HPC networks.
Although myriad network topologies have been proposed, a broad head-to-head
comparison across topologies and across traffic patterns is absent, and the
right way to compare worst-case throughput performance is a subtle problem.
In this paper, we develop a framework to benchmark the throughput of network
topologies, using a two-pronged approach. First, we study performance on a
variety of synthetic and experimentally-measured traffic matrices (TMs).
Second, we show how to measure worst-case throughput by generating a
near-worst-case TM for any given topology. We apply the framework to study the
performance of these TMs in a wide range of network topologies, revealing
insights into the performance of topologies with scaling, robustness of
performance across TMs, and the effect of scattered workload placement. Our
evaluation code is freely available
Almost-Tight Distributed Minimum Cut Algorithms
We study the problem of computing the minimum cut in a weighted distributed
message-passing networks (the CONGEST model). Let be the minimum cut,
be the number of nodes in the network, and be the network diameter. Our
algorithm can compute exactly in time. To the best of our knowledge, this is the first paper that
explicitly studies computing the exact minimum cut in the distributed setting.
Previously, non-trivial sublinear time algorithms for this problem are known
only for unweighted graphs when due to Pritchard and
Thurimella's -time and -time algorithms for
computing -edge-connected and -edge-connected components.
By using the edge sampling technique of Karger's, we can convert this
algorithm into a -approximation -time algorithm for any . This improves
over the previous -approximation -time algorithm and
-approximation -time algorithm of Ghaffari and Kuhn. Due to the lower
bound of by Das Sarma et al. which holds for any
approximation algorithm, this running time is tight up to a factor.
To get the stated running time, we developed an approximation algorithm which
combines the ideas of Thorup's algorithm and Matula's contraction algorithm. It
saves an factor as compared to applying Thorup's tree
packing theorem directly. Then, we combine Kutten and Peleg's tree partitioning
algorithm and Karger's dynamic programming to achieve an efficient distributed
algorithm that finds the minimum cut when we are given a spanning tree that
crosses the minimum cut exactly once
An efficient multi-core implementation of a novel HSS-structured multifrontal solver using randomized sampling
We present a sparse linear system solver that is based on a multifrontal
variant of Gaussian elimination, and exploits low-rank approximation of the
resulting dense frontal matrices. We use hierarchically semiseparable (HSS)
matrices, which have low-rank off-diagonal blocks, to approximate the frontal
matrices. For HSS matrix construction, a randomized sampling algorithm is used
together with interpolative decompositions. The combination of the randomized
compression with a fast ULV HSS factorization leads to a solver with lower
computational complexity than the standard multifrontal method for many
applications, resulting in speedups up to 7 fold for problems in our test
suite. The implementation targets many-core systems by using task parallelism
with dynamic runtime scheduling. Numerical experiments show performance
improvements over state-of-the-art sparse direct solvers. The implementation
achieves high performance and good scalability on a range of modern shared
memory parallel systems, including the Intel Xeon Phi (MIC). The code is part
of a software package called STRUMPACK -- STRUctured Matrices PACKage, which
also has a distributed memory component for dense rank-structured matrices
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