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 R3\mathbb R^3

We are concerned with the instability of a generic compressible two-fluid model in the whole space $\mathbb{R}^3$, where the capillary pressure $f(\alpha^-\rho^-)=P^+-P^-\neq 0$ is taken into account. For the case that the capillary pressure is a strictly decreasing function near the equilibrium, namely, $f'(1)<0$, 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 $P^+=P^-$ (corresponding to $f'(1)=0$). 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, $f'(1)>0$. 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 $H^k$, 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 $f'(1)>0$, 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 $f'(1)<0$ and $P^+=P^-$ (corresponding to $f'(1)=0$) 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 $\mathbb{T}^3$, and the main novelty of this work is three-fold: First, under the assumption that the density $\rho({\bf{x}}, t)$ verifies $\sup_{t\geq 0}\|\rho(t)\|_{L^\infty}\leq M$, it is shown that the solutions converge to equilibrium state exponentially in $L^2$-norm. Second, by employing some new thoughts, we also show that the density converges to its equilibrium state exponentially in $L^\infty$-norm if additionally the initial density $\rho_0({\bf{x}})$ satisfies $\inf_{{\bf{x}}\in\mathbb{T}^3}\rho_0({\bf{x}})\geq c_0>0$. 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 $\mu(\rho)=\rho^\alpha$ with $\alpha>1/2$.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