507 research outputs found
Performance Guarantees for Distributed Reachability Queries
In the real world a graph is often fragmented and distributed across
different sites. This highlights the need for evaluating queries on distributed
graphs. This paper proposes distributed evaluation algorithms for three classes
of queries: reachability for determining whether one node can reach another,
bounded reachability for deciding whether there exists a path of a bounded
length between a pair of nodes, and regular reachability for checking whether
there exists a path connecting two nodes such that the node labels on the path
form a string in a given regular expression. We develop these algorithms based
on partial evaluation, to explore parallel computation. When evaluating a query
Q on a distributed graph G, we show that these algorithms possess the following
performance guarantees, no matter how G is fragmented and distributed: (1) each
site is visited only once; (2) the total network traffic is determined by the
size of Q and the fragmentation of G, independent of the size of G; and (3) the
response time is decided by the largest fragment of G rather than the entire G.
In addition, we show that these algorithms can be readily implemented in the
MapReduce framework. Using synthetic and real-life data, we experimentally
verify that these algorithms are scalable on large graphs, regardless of how
the graphs are distributed.Comment: VLDB201
Graph Homomorphism Revisited for Graph Matching
In a variety of emerging applications one needs to decide whether a graph
G matches
another
G
p
,
i.e.
, whether
G
has a topological structure similar to that of
G
p
. The traditional notions of graph homomorphism and isomorphism often fall short of capturing the structural similarity in these applications. This paper studies revisions of these notions, providing a full treatment from complexity to algorithms. (1) We propose
p-homomorphism (p
-hom) and 1-1
p
-hom, which extend graph homomorphism and subgraph isomorphism, respectively, by mapping
edges
from one graph to
paths
in another, and by measuring
the similarity of nodes
. (2) We introduce metrics to measure graph similarity, and several optimization problems for
p
-hom and 1-1
p
-hom. (3) We show that the decision problems for
p
-hom and 1-1
p
-hom are NP-complete even for DAGs, and that the optimization problems are approximation-hard. (4) Nevertheless, we provide approximation algorithms with
provable guarantees
on match quality. We experimentally verify the effectiveness of the revised notions and the efficiency of our algorithms in Web site matching, using real-life and synthetic data.
</jats:p
Propagating functional dependencies with conditions
The dependency propagation problem is to determine, given a view defined on data sources and a set of dependencies on the sources, whether another dependency is guaranteed to hold on the view. This paper investigates dependency propagation for recently proposed conditional functional dependencies (CFDs). The need for this study is evident in data integration, exchange and cleaning since dependencies on data sources often only hold
conditionally
on the view. We investigate dependency propagation for views defined in various fragments of relational algebra, CFDs as view dependencies, and for source dependencies given as either CFDs or traditional functional dependencies (FDs). (a) We establish lower and upper bounds,
all matching
, ranging from PTIME to undecidable. These not only provide the
first
results for CFD propagation, but also extend the classical work of FD propagation by giving new complexity bounds in the presence of finite domains. (b) We provide the first algorithm for computing a minimal cover of
all
CFDs propagated via SPC views; the algorithm has the same complexity as one of the most efficient algorithms for computing a cover of FDs propagated via a projection view, despite the increased expressive power of CFDs and SPC views. (c) We experimentally verify that the algorithm is efficient.
</jats:p
On instability of a generic compressible two-fluid model in
We are concerned with the instability of a generic compressible two-fluid
model in the whole space , where the capillary pressure
is taken into account. For the case that the
capillary pressure is a strictly decreasing function near the equilibrium,
namely, , Evje-Wang-Wen established global stability of the constant
equilibrium state for the three-dimensional Cauchy problem under some smallness
assumptions. Recently, Wu-Yao-Zhang proved global stability of the constant
equilibrium state for the case (corresponding to ). In this
work, we investigate the instability of the constant equilibrium state for the
case that the capillary pressure is a strictly increasing function near the
equilibrium, namely, . First, by employing Hodge decomposition
technique and making detailed analysis of the Green's function for the
corresponding linearized system, we construct solutions of the linearized
problem that grow exponentially in time in the Sobolev space , thus
leading to a global instability result for the linearized problem. Moreover,
with the help of the global linear instability result and a local existence
theorem of classical solutions to the original nonlinear system, we can then
show the instability of the nonlinear problem in the sense of Hadamard by
making a delicate analysis on the properties of the semigroup. Therefore, our
result shows that for the case , the constant equilibrium state of the
two-fluid model is linearly globally unstable and nonlinearly locally unstable
in the sense of Hadamard, which is in contrast to the cases and
(corresponding to ) where the constant equilibrium state of
the two--fluid model is nonlinearly globally stable.Comment: 17. arXiv admin note: substantial text overlap with arXiv:2204.10706,
arXiv:2108.06974, arXiv:2010.1150
Global Stability and Non-Vanishing Vacuum States of 3D Compressible Navier-Stokes Equations
We investigate the global stability and non-vanishing vacuum states of large
solutions to the compressible Navier-Stokes equations on the torus
, and the main novelty of this work is three-fold: First, under
the assumption that the density verifies , it is shown that the solutions converge to
equilibrium state exponentially in -norm. Second, by employing some new
thoughts, we also show that the density converges to its equilibrium state
exponentially in -norm if additionally the initial density
satisfies
. Finally, we prove
that the vacuum states will not vanish for any time provided that the vacuum
states are present initially. This phenomenon is totally new and somewhat
surprising, and particularly is in contrast to the previous work of [H. L. Li
et al., Commun. Math. Phys., 281 (2008), 401-444], where the authors showed
that the vacuum states must vanish within finite time for the 1D compressible
Navier-Stokes equations with density-dependent viscosity
with .Comment: 17 page
Extending graph homomorphism and simulation for real life graph matching
Among the vital problems in a variety of emerging applications is the graph matching problem, which is to determine whether two graphs are similar, and if so, find all the valid matches in one graph for the other, based on specified metrics. Traditional graph matching approaches are mostly based on graph homomorphism and isomorphism, falling short of capturing both structural and semantic similarity in real life applications. Moreover, it is preferable while difficult to find all matches with high accuracy over complex graphs. Worse still, the graph structures in real life applications constantly bear modifications. In response to these challenges, this thesis presents a series of approaches for ef?ciently solving graph matching problems, over both static and dynamic real life graphs.
Firstly, the thesis extends graph homomorphism and subgraph isomorphism, respectively, by mapping edges from one graph to paths in another, and by measuring the semantic similarity of nodes. The graph similarity is then measured by the metrics based on these extensions. Several optimization problems for graph matching based on the new metrics are studied, with approximation algorithms having provable guarantees on match quality developed.
Secondly, although being extended in the above work, graph matching is defined in terms of functions, which cannot capture more meaningful matches and is usually hard to compute. In response to this, the thesis proposes a class of graph patterns, in which an edge denotes the connectivity in a data graph within a predefined number of hops. In addition, the thesis defines graph pattern matching based on a notion of bounded simulation relation, an extension of graph simulation. With this revision, graph pattern matching is in cubic-time by providing such an algorithm, rather than intractable.
Thirdly, real life graphs often bear multiple edge types. In response to this, the thesis further extends and generalizes the proposed revisions of graph simulation to a more powerful case: a novel set of reachability queries and graph pattern queries, constrained by a subclass of regular path expressions. Several fundamental problems of the queries are studied: containment, equivalence and minimization. The enriched reachability query does not increase the complexity of the above problems, shown by the corresponding algorithms. Moreover, graph pattern queries can be evaluated in cubic time, where two such algorithms are proposed.
Finally, real life graphs are frequently updated with small changes. The thesis investigates incremental algorithms for graph pattern matching defined in terms of graph simulation, bounded simulation and subgraph isomorphism. Besides studying the results on the complexity bounds, the thesis provides the experimental study verifying that these incremental algorithms significantly outperform their batch counterparts in response to small changes, using real-life data and synthetic data
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