Yinghui Wu

Graph pattern matching: from intractable to polynomial time

Graph pattern matching is typically defined in terms of subgraph isomorphism, which makes it an np-complete problem. Moreover, it requires bijective functions, which are often too restrictive to characterize patterns in emerging applications. We propose a class of graph patterns, in which an edge denotes the connectivity in a data graph within a predefined number of hops. In addition, we define matching based on a notion of bounded simulation, an extension of graph simulation. We show that with this revision, graph pattern matching can be performed in cubic-time, by providing such an algorithm. We also develop algorithms for incrementally finding matches when data graphs are updated, with performance guarantees for dag patterns. We experimentally verify that these algorithms scale well, and that the revised notion of graph pattern matching allows us to identify communities commonly found in real-world networks.

NeMa: fast graph search with label similarity

It is increasingly common to find real-life data represented as networks of labeled, heterogeneous entities. To query these networks, one often needs to identify the matches of a given query graph in a (typically large) network modeled as a target graph. Due to noise and the lack of fixed schema in the target graph, the query graph can substantially differ from its matches in the target graph in both structure and node labels, thus bringing challenges to the graph querying tasks. In this paper, we propose NeMa (Network Match), a neighborhood-based subgraph matching technique for querying real-life networks. (1) To measure the quality of the match, we propose a novel subgraph matching cost metric that aggregates the costs of matching individual nodes, and unifies both structure and node label similarities. (2) Based on the metric, we formulate the minimum cost subgraph matching problem. Given a query graph and a target graph, the problem is to identify the (top-k) matches of the query graph with minimum costs in the target graph. We show that the problem is NP-hard, and also hard to approximate. (3) We propose a heuristic algorithm for solving the problem based on an inference model. In addition, we propose optimization techniques to improve the efficiency of our method. (4) We empirically verify that NeMa is both effective and efficient compared to the keyword search and various state-of-the-art graph querying techniques.

Incremental graph pattern matching

Graph pattern matching has become a routine process in emerging applications such as social networks. In practice a data graph is typically large, and is frequently updated with small changes. It is often prohibitively expensive to recompute matches from scratch via batch algorithms when the graph is updated. With this comes the need for incremental algorithms that compute changes to the matches in response to updates, to minimize unnecessary recomputation. This paper investigates incremental algorithms for graph pattern matching defined in terms of graph simulation, bounded simulation and subgraph isomorphism. (1) For simulation, we provide incremental algorithms for unit updates and certain graph patterns. These algorithms are optimal: in linear time in the size of the changes in the input and output, which characterizes the cost that is inherent to the problem itself. For general patterns we show that the incremental matching problem is unbounded, i.e., its cost is not determined by the size of the changes alone. (2) For bounded simulation, we show that the problem is unbounded even for unit updates and path patterns. (3) For subgraph isomorphism, we show that the problem is intractable and unbounded for unit updates and path patterns. (4) For multiple updates, we develop an incremental algorithm for each of simulation, bounded simulation and subgraph isomorphism. We experimentally verify that these incremental algorithms significantly outperform their batch counterparts in response to small changes, using real-life data and synthetic data.

Adding regular expressions to graph reachability and pattern queries

It is increasingly common to find graphs in which edges bear different types, indicating a variety of relationships. For such graphs we propose a class of reachability queries and a class of graph patterns, in which an edge is specified with a regular expression of a certain form, expressing the connectivity in a data graph via edges of various types. In addition, we define graph pattern matching based on a revised notion of graph simulation. On graphs in emerging applications such as social networks, we show that these queries are capable of finding more sensible information than their traditional counterparts. Better still, their increased expressive power does not come with extra complexity. Indeed, (1) we investigate their containment and minimization problems, and show that these fundamental problems are in quadratic time for reachability queries and are in cubic time for pattern queries. (2) We develop an algorithm for answering reachability queries, in quadratic time as for their traditional counterpart. (3) We provide two cubic-time algorithms for evaluating graph pattern queries based on extended graph simulation, as opposed to the NP-completeness of graph pattern matching via subgraph isomorphism. (4) The effectiveness, efficiency and scalability of these algorithms are experimentally verified using real-life data and synthetic data.

Graph homomorphism revisited for graph matching

In a variety of emerging applications one needs to decide whether a graph G matches another Gp, i.e., whether G has a topological structure similar to that of Gp. 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.

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.

Diversified top-k graph pattern matching

Graph pattern matching has been widely used in e.g., social data analysis. A number of matching algorithms have been developed that, given a graph pattern Q and a graph G, compute the set M(Q,G) of matches of Q in G. However, these algorithms often return an excessive number of matches, and are expensive on large real-life social graphs. Moreover, in practice many social queries are to find matches of a specific pattern node, rather than the entire M(Q,G). This paper studies top-k graph pattern matching. (1) We revise graph pattern matching defined in terms of simulation, by supporting a designated output node uo. Given G and Q, it is to find those nodes in M(Q,G) that match uo, instead of the large set M(Q,G). (2) We study two classes of functions for ranking the matches: relevance functions δr() based on, e.g., social impact, and distance functions δd() to cover diverse elements. (3) We develop two algorithms for computing top-k matches of uo based on δr(), with the early termination property, i.e., they find top-k matches without computing the entire M(Q,G). (4) We also study diversified top-k matching, a bi-criteria optimization problem based on both δr() and δd(). We show that its decision problem is NP-complete. Nonetheless, we provide an approximation algorithm with performance guarantees and a heuristic one with the early termination property. (5) Using real-life and synthetic data, we experimentally verify that our (diversified) top-k matching algorithms are effective, and outperform traditional matching algorithms in efficiency.

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.

Distributed graph simulation: impossibility and possibility

This paper studies fundamental problems for distributed graph simulation. Given a pattern query Q and a graph G that is fragmented and distributed, a graph simulation algorithm A is to compute the matches Q(G) of Q in G. We say that A is parallel scalable in (a) response time if its parallel computational cost is determined by the largest fragment Fm of G and the size |Q| of query Q, and (b) data shipment if its total amount of data shipped is determined by |Q| and the number of fragments of G, independent of the size of graph G. (1) We prove an impossibility theorem: there exists no distributed graph simulation algorithm that is parallel scalable in either response time or data shipment. (2) However, we show that distributed graph simulation is partition bounded, i.e., its response time depends only on |Q|, |Fm| and the number |Vf| of nodes in G with edges across different fragments; and its data shipment depends on |Q| and the number |Ef| of crossing edges only. We provide the first algorithms with these performance guarantees. (3) We also identify special cases of patterns and graphs when parallel scalability is possible. (4) We experimentally verify the scalability and efficiency of our algorithms.

Querying big graphs within bounded resources

This paper studies the problem of querying graphs within bounded resources. Given a query Q, a graph G and a small ratio α, it aims to answer Q in G by accessing only a fraction GQ of G of size |GQ| ≤ α |G|. The need for this is evident when G is big while our available resources are limited, as indicated by α. We propose resource-bounded query answering via a dynamic scheme that reduces big G to GQ. We investigate when we can find the exact answers Q(G) from GQ, and if GQ cannot accommodate enough information, how accurate the approximate answers Q(GQ) are. To verify the effectiveness of the approach, we study two types of queries. One consists of pattern queries that have data locality, such as subgraph isomorphism and strong simulation. The other is the class of reachability queries, without data locality. We show that it is hard to get resource-bounded algorithms with 100% accuracy: NP-hard for pattern queries, and non-existing for reachability when α ≠ 1. Despite these, we develop resource-bounded algorithms for answering these queries. Using real-life and synthetic data, we experimentally evaluate the performance of the algorithms. We find that they scale well for both types of queries, and our approximate answers are accurate, even 100% for small α.