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.

This paper investigates three problems identified in [1] for annotation propagation, namely, the view side-effect, source side-effect, and annotation placement problems. Given annotations entered for a tuple or an attribute in a view, these problems ask what tuples or attributes in the source have to be annotated to produce the view annotations. As observed in [1], these problems are fundamental not only for data provenance but also for the management of view updates. For an annotation attached to a single existing tuple in a view, it has been shown that these problems are often intractable even for views defined in terms of simple SPJU queries [1]. We revisit these problems by considering several dichotomies: 1) views defined in various subclasses of SPJU, versus SPJU views under a practical key preserving condition; 2) annotations attached to existing tuples in a view versus annotations on tuples to be inserted into the view; and 3) a single-tuple annotation versus a group of annotations. We provide a complete picture of intractability and tractability for the three problems in all these settings. We show that key preserving views often simplify the propagation analysis. Indeed, some problems become tractable for certain key preserving views, as opposed to the intractability of their counterparts that are not key preserving. However, group annotations often make the analysis harder. In addition, the problems have quite diverse complexity when annotations are attached to existing tuples in a view and when they are entered for tuples to be inserted into the view.Index Terms-Annotation, view updates, view maintenance, SPJU queries.

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