571 research outputs found
Meerkat: A framework for Dynamic Graph Algorithms on GPUs
Graph algorithms are challenging to implement due to their varying topology
and irregular access patterns. Real-world graphs are dynamic in nature and
routinely undergo edge and vertex additions, as well as, deletions. Typical
examples of dynamic graphs are social networks, collaboration networks, and
road networks. Applying static algorithms repeatedly on dynamic graphs is
inefficient. Unfortunately, we know little about how to efficiently process
dynamic graphs on massively parallel architectures such as GPUs. Existing
approaches to represent and process dynamic graphs are either not general or
inefficient. In this work, we propose a library-based framework for dynamic
graph algorithms that proposes a GPU-tailored graph representation and exploits
the warp-cooperative execution model. The library, named Meerkat, builds upon a
recently proposed dynamic graph representation on GPUs. This representation
exploits a hashtable-based mechanism to store a vertex's neighborhood. Meerkat
also enables fast iteration through a group of vertices, such as the whole set
of vertices or the neighbors of a vertex. Based on the efficient iterative
patterns encoded in Meerkat, we implement dynamic versions of the popular graph
algorithms such as breadth-first search, single-source shortest paths, triangle
counting, weakly connected components, and PageRank. Compared to the
state-of-the-art dynamic graph analytics framework Hornet, Meerkat is
, , and faster, for query, insert, and
delete operations, respectively. Using a variety of real-world graphs, we
observe that Meerkat significantly improves the efficiency of the underlying
dynamic graph algorithm. Meerkat performs for BFS,
for SSSP, for PageRank, and for WCC, better than
Hornet on average
The structure of unicellular maps, and a connection between maps of positive genus and planar labelled trees
A unicellular map is a map which has only one face. We give a bijection
between a dominant subset of rooted unicellular maps of fixed genus and a set
of rooted plane trees with distinguished vertices. The bijection applies as
well to the case of labelled unicellular maps, which are related to all rooted
maps by Marcus and Schaeffer's bijection.
This gives an immediate derivation of the asymptotic number of unicellular
maps of given genus, and a simple bijective proof of a formula of Lehman and
Walsh on the number of triangulations with one vertex. From the labelled case,
we deduce an expression of the asymptotic number of maps of genus g with n
edges involving the ISE random measure, and an explicit characterization of the
limiting profile and radius of random bipartite quadrangulations of genus g in
terms of the ISE.Comment: 27pages, 6 figures, to appear in PTRF. Version 2 includes corrections
from referee report in sections 6-
Counting Perfect Matchings and the Switch Chain
We examine the problem of exactly or approximately counting all perfect matchings in hereditary classes of nonbipartite graphs. In particular, we consider the switch Markov chain of Diaconis, Graham, and Holmes. We determine the largest hereditary class for which the chain is ergodic, and define a large new hereditary class of graphs for which it is rapidly mixing. We go on to show that the chain has exponential mixing time for a slightly larger class. We also examine the question of ergodicity of the switch chain in an arbitrary graph. Finally, we give exact counting algorithms for three classes
Unavoidable Immersions and Intertwines of Graphs
The topological minor and the minor relations are well-studied binary relations on the class of graphs. A natural weakening of the topological minor relation is an immersion. An immersion of a graph H into a graph G is a map that injects the vertex set of H into the vertex set of G such that edges between vertices of H are represented by pairwise-edge-disjoint paths of G. In this dissertation, we present two results: the first giving a set of unavoidable immersions of large 3-edge-connected graphs and the second on immersion intertwines of infinite graphs. These results, along with the methods used to prove them, are analogues of results on the graph minor relation. A conjecture for the unavoidable immersions of large 3-edge-connected graphs is also stated with a partial proof
A look at cycles containing specified elements of a graph
AbstractThis article is intended as a brief survey of problems and results dealing with cycles containing specified elements of a graph. It is hoped that this will help researchers in the area to identify problems and areas of concentration
A counterexample to the weak density of smooth maps between manifolds in Sobolev spaces
The present paper presents a counterexample to the sequentially weak density
of smooth maps between two manifolds and in the Sobolev space , in the case is an integer. It has been shown that, if
is not an integer and the -th homotopy group of is not
trivial, denoting the largest integer less then , then smooth maps are
not sequentially weakly dense in for the strong convergence.
On the other, in the case is an integer, examples have been
provided where smooth maps are actually sequentially weakly dense in with . This is the case for instance for , the standard ball in , and the
standard sphere of dimension , for which . The main
result of this paper shows however that such a property does not holds for
arbitrary manifolds and integers .Our counterexample deals with the case
, and , for which the homotopy group
is related to the Hopf fibration.Comment: 68 page
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