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Graph data model and graph databases are very popular in various areas such as bioinformatics, semantic web, and social networks. One specific problem in the area is a path querying with constraints formulated in terms of formal grammars. The query in this approach is written as grammar, and paths querying is graph parsing with respect to given grammar. There are several solutions to it, but how to provide structural representation of query result which is practical for answer processing and debugging is still an open problem. In this paper we propose a graph parsing technique which allows one to build such representation with respect to given grammar in polynomial time and space for arbitrary context-free grammar and graph. Proposed algorithm is based on generalized LL parsing algorithm, while previous solutions are based mostly on CYK or Earley algorithms, which reduces time complexity in some cases

Graph data models are widely used in many areas, for example, bioinformatics, graph databases. In these areas, it is often required to process queries for large graphs. Some of the most common graph queries are navigational queries. The result of query evaluation is a set of implicit relations between nodes of the graph, i.e. paths in the graph. A natural way to specify these relations is by specifying paths using formal grammars over the alphabet of edge labels. An answer to a context-free path query in this approach is usually a set of triples (A, m, n) such that there is a path from the node m to the node n, whose labeling is derived from a non-terminal A of the given context-free grammar. This type of queries is evaluated using the relational query semantics. Another example of path query semantics is the single-path query semantics which requires presenting a single path from the node m to the node n, whose labeling is derived from a non-terminal A for all triples (A, m, n) evaluated using the relational query semantics. There is a number of algorithms for query evaluation which use these semantics but all of them perform poorly on large graphs. One of the most common technique for efficient big data processing is the use of a graphics processing unit (GPU) to perform computations, but these algorithms do not allow to use this technique efficiently. In this paper, we show how the contextfree path query evaluation using these query semantics can be reduced to the calculation of the matrix transitive closure. Also, we propose an algorithm for context-free path query evaluation which uses relational query semantics and is based on matrix operations that make it possible to speed up computations by using a GPU.

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