7,622 research outputs found
Theoretically Efficient Parallel Graph Algorithms Can Be Fast and Scalable
There has been significant recent interest in parallel graph processing due
to the need to quickly analyze the large graphs available today. Many graph
codes have been designed for distributed memory or external memory. However,
today even the largest publicly-available real-world graph (the Hyperlink Web
graph with over 3.5 billion vertices and 128 billion edges) can fit in the
memory of a single commodity multicore server. Nevertheless, most experimental
work in the literature report results on much smaller graphs, and the ones for
the Hyperlink graph use distributed or external memory. Therefore, it is
natural to ask whether we can efficiently solve a broad class of graph problems
on this graph in memory.
This paper shows that theoretically-efficient parallel graph algorithms can
scale to the largest publicly-available graphs using a single machine with a
terabyte of RAM, processing them in minutes. We give implementations of
theoretically-efficient parallel algorithms for 20 important graph problems. We
also present the optimizations and techniques that we used in our
implementations, which were crucial in enabling us to process these large
graphs quickly. We show that the running times of our implementations
outperform existing state-of-the-art implementations on the largest real-world
graphs. For many of the problems that we consider, this is the first time they
have been solved on graphs at this scale. We have made the implementations
developed in this work publicly-available as the Graph-Based Benchmark Suite
(GBBS).Comment: This is the full version of the paper appearing in the ACM Symposium
on Parallelism in Algorithms and Architectures (SPAA), 201
The Walk Distances in Graphs
The walk distances in graphs are defined as the result of appropriate
transformations of the proximity measures, where
is the weighted adjacency matrix of a graph and is a sufficiently small
positive parameter. The walk distances are graph-geodetic; moreover, they
converge to the shortest path distance and to the so-called long walk distance
as the parameter approaches its limiting values. We also show that the
logarithmic forest distances which are known to generalize the resistance
distance and the shortest path distance are a subclass of walk distances. On
the other hand, the long walk distance is equal to the resistance distance in a
transformed graph.Comment: Accepted for publication in Discrete Applied Mathematics. 26 pages, 3
figure
Applications of Graphical Condensation for Enumerating Matchings and Tilings
A technique called graphical condensation is used to prove various
combinatorial identities among numbers of (perfect) matchings of planar
bipartite graphs and tilings of regions. Graphical condensation involves
superimposing matchings of a graph onto matchings of a smaller subgraph, and
then re-partitioning the united matching (actually a multigraph) into matchings
of two other subgraphs, in one of two possible ways. This technique can be used
to enumerate perfect matchings of a wide variety of bipartite planar graphs.
Applications include domino tilings of Aztec diamonds and rectangles, diabolo
tilings of fortresses, plane partitions, and transpose complement plane
partitions.Comment: 25 pages; 21 figures Corrected typos; Updated references; Some text
revised, but content essentially the sam
Better Tradeoffs for Exact Distance Oracles in Planar Graphs
We present an -space distance oracle for directed planar graphs
that answers distance queries in time. Our oracle both
significantly simplifies and significantly improves the recent oracle of
Cohen-Addad, Dahlgaard and Wulff-Nilsen [FOCS 2017], which uses
-space and answers queries in time. We achieve this by
designing an elegant and efficient point location data structure for Voronoi
diagrams on planar graphs.
We further show a smooth tradeoff between space and query-time. For any , we show an oracle of size that answers queries in time. This new tradeoff is currently the best (up to
polylogarithmic factors) for the entire range of and improves by polynomial
factors over all the previously known tradeoffs for the range
Structure of Triadic Relations in Multiplex Networks
Recent advances in the study of networked systems have highlighted that our
interconnected world is composed of networks that are coupled to each other
through different "layers" that each represent one of many possible subsystems
or types of interactions. Nevertheless, it is traditional to aggregate
multilayer networks into a single weighted network in order to take advantage
of existing tools. This is admittedly convenient, but it is also extremely
problematic, as important information can be lost as a result. It is therefore
important to develop multilayer generalizations of network concepts. In this
paper, we analyze triadic relations and generalize the idea of transitivity to
multiplex networks. By focusing on triadic relations, which yield the simplest
type of transitivity, we generalize the concept and computation of clustering
coefficients to multiplex networks. We show how the layered structure of such
networks introduces a new degree of freedom that has a fundamental effect on
transitivity. We compute multiplex clustering coefficients for several real
multiplex networks and illustrate why one must take great care when
generalizing standard network concepts to multiplex networks. We also derive
analytical expressions for our clustering coefficients for ensemble averages of
networks in a family of random multiplex networks. Our analysis illustrates
that social networks have a strong tendency to promote redundancy by closing
triads at every layer and that they thereby have a different type of multiplex
transitivity from transportation networks, which do not exhibit such a
tendency. These insights are invisible if one only studies aggregated networks.Comment: Main text + Supplementary Material included in a single file.
Published in New Journal of Physic
Heavy cycles in k-connected weighted graphs with large weighted degree sums
2008-2009 > Academic research: refereed > Publication in refereed journalAccepted ManuscriptPublishe
A Parallel Solver for Graph Laplacians
Problems from graph drawing, spectral clustering, network flow and graph
partitioning can all be expressed in terms of graph Laplacian matrices. There
are a variety of practical approaches to solving these problems in serial.
However, as problem sizes increase and single core speeds stagnate, parallelism
is essential to solve such problems quickly. We present an unsmoothed
aggregation multigrid method for solving graph Laplacians in a distributed
memory setting. We introduce new parallel aggregation and low degree
elimination algorithms targeted specifically at irregular degree graphs. These
algorithms are expressed in terms of sparse matrix-vector products using
generalized sum and product operations. This formulation is amenable to linear
algebra using arbitrary distributions and allows us to operate on a 2D sparse
matrix distribution, which is necessary for parallel scalability. Our solver
outperforms the natural parallel extension of the current state of the art in
an algorithmic comparison. We demonstrate scalability to 576 processes and
graphs with up to 1.7 billion edges.Comment: PASC '18, Code: https://github.com/ligmg/ligm
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