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Invited Talk

**Published in:** LIPIcs, Volume 155, 23rd International Conference on Database Theory (ICDT 2020)

As graph databases grow in popularity, decades of work in graph query languages and models are materialising in industry standards and in the construction of new graph database systems. However, this surge in graph systems has in turn opened up a series of new, interesting research problems related to graph databases.
Our first set of problems has to do with more efficient ways of computing the answers of graph queries, specifically graph patterns, path queries, and combinations between them. Traditionally, researchers in graph databases have pointed out that relational systems are ill-equipped to process these types of queries, and if one looks at the performance of native graph database systems, there is clearly a lot of room for improvement. The talk focuses on two possible directions for improving the state of the art in graph query processing. The first is implementing worst-case optimal algorithms for processing graph patterns that traduce in relational queries with several joins. Some advances are already in development (see e.g. Nguyen, Dung, et al. "Join processing for graph patterns: An old dog with new tricks." GRADES'15. or Hogan, Aidan, et al. "A Worst-Case Optimal Join Algorithm for SPARQL." ISWC’19.), but we are still far from a full fledged solution: most algorithms require complex data structures, or need further support in terms of heuristics to select an order in which joins are processed. Second, we need to understand what is the best way of evaluating path queries (that is, finding all pairs of nodes connected by a path), in such a way that these results can be further integrated with other query results in a graph system pipeline. We already have complexity results regarding path computation and enumeration for different semantics of path queries (see e.g. Martens, Wim, and Tina Trautner. "Evaluation and enumeration problems for regular path queries." ICDT'18. or Bagan, Guillaume, Angela Bonifati, and Benoit Groz. "A trichotomy for regular simple path queries on graphs." PODS'13.), but still very little is known in terms of optimal processing of path queries when inside a tractable fragment.
Our second set of problems is related to graph analytics, one of the current selling points of graph databases. Systems should be able to run more complex analytical queries involving tasks such as more complex path finding, centrality or clustering. It is also important to be able to run these algorithms not over native graphs, but perhaps over a certain set of nodes or edges previously selected by a graph query, and one may also want to pose further queries over the result of the analytics task. Finally, all of this should be done in an efficient way, specially in the prospect that graph databases may contain a huge amount of nodes. In this talk I will discuss possible approaches to perform these operations, covering aspects from the design of languages for graph analytics to efficient ways of processing them, and also comparing the expressive power of graph analytics solutions with other forms of graph computation.

Juan L. Reutter. Current Challenges in Graph Databases (Invited Talk). In 23rd International Conference on Database Theory (ICDT 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 155, p. 3:1, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{reutter:LIPIcs.ICDT.2020.3, author = {Reutter, Juan L.}, title = {{Current Challenges in Graph Databases}}, booktitle = {23rd International Conference on Database Theory (ICDT 2020)}, pages = {3:1--3:1}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-139-9}, ISSN = {1868-8969}, year = {2020}, volume = {155}, editor = {Lutz, Carsten and Jung, Jean Christoph}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICDT.2020.3}, URN = {urn:nbn:de:0030-drops-119272}, doi = {10.4230/LIPIcs.ICDT.2020.3}, annote = {Keywords: Graph databases, Join algorithms, path queries, graph analytics} }

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**Published in:** LIPIcs, Volume 155, 23rd International Conference on Database Theory (ICDT 2020)

Worst-case optimal join algorithms have gained a lot of attention in the database literature. We now count with several algorithms that are optimal in the worst case, and many of them have been implemented and validated in practice. However, the implementation of these algorithms often requires an enhanced indexing structure: to achieve optimality we either need to build completely new indexes, or we must populate the database with several instantiations of indexes such as B+-trees. Either way, this means spending an extra amount of storage space that may be non-negligible.
We show that optimal algorithms can be obtained directly from a representation that regards the relations as point sets in variable-dimensional grids, without the need of extra storage. Our representation is a compact quadtree for the static indexes, and a dynamic quadtree sharing subtrees (which we dub a qdag) for intermediate results. We develop a compositional algorithm to process full join queries under this representation, and show that the running time of this algorithm is worst-case optimal in data complexity. Remarkably, we can extend our framework to evaluate more expressive queries from relational algebra by introducing a lazy version of qdags (lqdags). Once again, we can show that the running time of our algorithms is worst-case optimal.

Gonzalo Navarro, Juan L. Reutter, and Javiel Rojas-Ledesma. Optimal Joins Using Compact Data Structures. In 23rd International Conference on Database Theory (ICDT 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 155, pp. 21:1-21:21, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{navarro_et_al:LIPIcs.ICDT.2020.21, author = {Navarro, Gonzalo and Reutter, Juan L. and Rojas-Ledesma, Javiel}, title = {{Optimal Joins Using Compact Data Structures}}, booktitle = {23rd International Conference on Database Theory (ICDT 2020)}, pages = {21:1--21:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-139-9}, ISSN = {1868-8969}, year = {2020}, volume = {155}, editor = {Lutz, Carsten and Jung, Jean Christoph}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICDT.2020.21}, URN = {urn:nbn:de:0030-drops-119453}, doi = {10.4230/LIPIcs.ICDT.2020.21}, annote = {Keywords: Join algorithms, Compact data structures, Quadtrees, AGM bound} }

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**Published in:** LIPIcs, Volume 31, 18th International Conference on Database Theory (ICDT 2015)

Graph databases are currently one of the most popular paradigms for storing data. One of the key conceptual differences between graph and relational databases is the focus on navigational queries that ask whether some nodes are connected by paths satisfying certain restrictions. This focus has driven the definition of several different query languages and the subsequent study of their fundamental properties.
We define the graph query language of Regular Queries, which is a natural extension of unions of conjunctive 2-way regular path queries (UC2RPQs) and unions of conjunctive nested 2-way regular path queries (UCN2RPQs). Regular queries allow expressing complex regular patterns between nodes. We formalize regular queries as nonrecursive Datalog programs with transitive closure rules. This language has been previously considered, but its algorithmic properties are not well understood.
Our main contribution is to show elementary tight bounds for the containment problem for regular queries. Specifically, we show that this problem is 2EXPSPACE-complete. For all extensions of regular queries known to date, the containment problem turns out to be non-elementary. Together with the fact that evaluating regular queries is not harder than evaluating UCN2RPQs, our results show that regular queries achieve a good balance between expressiveness and complexity, and constitute a well-behaved class that deserves further investigation.

Juan L. Reutter, Miguel Romero, and Moshe Y. Vardi. Regular Queries on Graph Databases. In 18th International Conference on Database Theory (ICDT 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 31, pp. 177-194, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2015)

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@InProceedings{reutter_et_al:LIPIcs.ICDT.2015.177, author = {Reutter, Juan L. and Romero, Miguel and Vardi, Moshe Y.}, title = {{Regular Queries on Graph Databases}}, booktitle = {18th International Conference on Database Theory (ICDT 2015)}, pages = {177--194}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-79-8}, ISSN = {1868-8969}, year = {2015}, volume = {31}, editor = {Arenas, Marcelo and Ugarte, Mart{\'\i}n}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICDT.2015.177}, URN = {urn:nbn:de:0030-drops-49842}, doi = {10.4230/LIPIcs.ICDT.2015.177}, annote = {Keywords: graph databases, conjunctive regular path queries, regular queries, containment.} }

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**Published in:** LIPIcs, Volume 31, 18th International Conference on Database Theory (ICDT 2015)

SPARQL has become the most popular language for querying RDF datasets, the standard data model for representing information in the Web. This query language has received a good deal of attention in the last few years: two versions of W3C standards have been issued, several SPARQL query engines have been deployed, and important theoretical foundations have been laid. However, many fundamental aspects of SPARQL queries are not yet fully understood. To this end, it is crucial to understand the correspondence between SPARQL and well-developed frameworks like relational algebra or first order logic. But one of the main obstacles on the way to such understanding is the fact that the well-studied fragments of SPARQL do not produce RDF as output.
In this paper we embark on the study of SPARQL CONSTRUCT queries, that is, queries which output RDF graphs. This class of queries takes rightful place in the standards and implementations, but contrary to SELECT queries, it has not yet attracted a worth-while theoretical research. Under this framework we are able to establish a strong connection between SPARQL and well-known logical and database formalisms. In particular, the fragment which does not allow for blank nodes in output templates corresponds to first order queries, its well-designed sub-fragment corresponds to positive first order queries, and the general language can be re-stated as a data exchange setting. These correspondences allow us to conclude that the general language is not composable, but the aforementioned blank-free fragments are. Finally, we enrich SPARQL with a recursion operator and establish fundamental properties of this extension.

Egor V. Kostylev, Juan L. Reutter, and Martín Ugarte. CONSTRUCT Queries in SPARQL. In 18th International Conference on Database Theory (ICDT 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 31, pp. 212-229, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{kostylev_et_al:LIPIcs.ICDT.2015.212, author = {Kostylev, Egor V. and Reutter, Juan L. and Ugarte, Mart{\'\i}n}, title = {{CONSTRUCT Queries in SPARQL}}, booktitle = {18th International Conference on Database Theory (ICDT 2015)}, pages = {212--229}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-79-8}, ISSN = {1868-8969}, year = {2015}, volume = {31}, editor = {Arenas, Marcelo and Ugarte, Mart{\'\i}n}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICDT.2015.212}, URN = {urn:nbn:de:0030-drops-49866}, doi = {10.4230/LIPIcs.ICDT.2015.212}, annote = {Keywords: RDF, SPARQL, Query Languages} }

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**Published in:** LIPIcs, Volume 13, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011)

We study regular expressions that use variables, or parameters, which are interpreted as alphabet letters. We consider two classes of languages denoted by such expressions: under the possibility semantics, a word belongs to the language if it is denoted by some regular expression obtained by replacing variables with letters; under the certainty semantics, the word must be denoted by every such expression. Such languages are regular, and we show that they naturally arise in several applications such as querying graph databases and program analysis. As the main contribution of the paper, we provide a complete characterization of the complexity of the main computational problems related to such languages: nonemptiness, universality, containment, membership, as well as the problem of constructing NFAs capturing such languages. We also look at the extension when domains of variables could be arbitrary regular languages, and show that under the certainty semantics, languages remain regular and the complexity of the main computational problems does not change.

Pablo Barceló, Leonid Libkin, and Juan L. Reutter. Parameterized Regular Expressions and Their Languages. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011). Leibniz International Proceedings in Informatics (LIPIcs), Volume 13, pp. 351-362, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2011)

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@InProceedings{barcelo_et_al:LIPIcs.FSTTCS.2011.351, author = {Barcel\'{o}, Pablo and Libkin, Leonid and Reutter, Juan L.}, title = {{Parameterized Regular Expressions and Their Languages}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011)}, pages = {351--362}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-34-7}, ISSN = {1868-8969}, year = {2011}, volume = {13}, editor = {Chakraborty, Supratik and Kumar, Amit}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2011.351}, URN = {urn:nbn:de:0030-drops-33334}, doi = {10.4230/LIPIcs.FSTTCS.2011.351}, annote = {Keywords: Regular expressions, complexity, decision problems, regular languages} }

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**Published in:** LIPIcs, Volume 255, 26th International Conference on Database Theory (ICDT 2023)

Conjunctive regular path queries (CRPQs) are one of the core classes of queries over graph databases. They are join intensive, inheriting their structure from the relational setting, but they also allow arbitrary length paths to connect points that are to be joined. However, despite their popularity, little is known about what are the best algorithms for processing CRPQs. We focus on worst-case optimal algorithms, which are algorithms that run in time bounded by the worst-case output size of queries, and have been recently deployed for simpler graph queries with very promising results. We show that the famous bound on the number of query results by Atserias, Grohe and Marx can be extended to CRPQs, but to obtain tight bounds one needs to work with slightly stronger cardinality profiles. We also discuss what algorithms follow from our analysis. If one pays the cost for fully materializing graph queries, then the techniques developed for conjunctive queries can be reused. If, on the other hand, one imposes constraint on the working memory of algorithms, then worst-case optimal algorithms must be adapted with care: the order of variables in which queries are processed can have striking implications on the running time of queries.

Tamara Cucumides, Juan Reutter, and Domagoj Vrgoč. Size Bounds and Algorithms for Conjunctive Regular Path Queries. In 26th International Conference on Database Theory (ICDT 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 255, pp. 13:1-13:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{cucumides_et_al:LIPIcs.ICDT.2023.13, author = {Cucumides, Tamara and Reutter, Juan and Vrgo\v{c}, Domagoj}, title = {{Size Bounds and Algorithms for Conjunctive Regular Path Queries}}, booktitle = {26th International Conference on Database Theory (ICDT 2023)}, pages = {13:1--13:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-270-9}, ISSN = {1868-8969}, year = {2023}, volume = {255}, editor = {Geerts, Floris and Vandevoort, Brecht}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICDT.2023.13}, URN = {urn:nbn:de:0030-drops-177552}, doi = {10.4230/LIPIcs.ICDT.2023.13}, annote = {Keywords: graph databases, regular path queries, worst-case optimal algorithms} }

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**Published in:** LIPIcs, Volume 154, 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)

In the consensus protocols used in most cryptocurrencies, participants called miners must find valid blocks of transactions and append them to a shared tree-like data structure. Ideally, the rules of the protocol should ensure that miners maximize their gains if they follow a default strategy, which consists on appending blocks only to the longest branch of the tree, called the blockchain. Our goal is to understand under which circumstances are miners encouraged to follow the default strategy. Unfortunately, most of the existing models work with simplified payoff functions, without considering the possibility that rewards decrease over time because of the game rules (like in Bitcoin), nor integrating the fact that a miner naturally prefers to be paid earlier than later (the economic concept of discount). In order to integrate these factors, we consider a more general model where issues such as economic discount and decreasing rewards can be set as parameters of an infinite stochastic game. In this model, we study the limit situation in which a miner does not receive a full reward for a block if it stops being in the blockchain. We show that if rewards are not decreasing, then miners do not have incentives to create new branches, no matter how high their computational power is. On the other hand, when working with decreasing rewards similar to those in Bitcoin, we show that miners have an incentive to create such branches. Nevertheless, this incentive only occurs when a miner controls a proportion of the computational power which is close to half of the computational power of the entire network.

Marcelo Arenas, Juan Reutter, Etienne Toussaint, Martín Ugarte, Francisco Vial, and Domagoj Vrgoč. Cryptocurrency Mining Games with Economic Discount and Decreasing Rewards. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 54:1-54:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{arenas_et_al:LIPIcs.STACS.2020.54, author = {Arenas, Marcelo and Reutter, Juan and Toussaint, Etienne and Ugarte, Mart{\'\i}n and Vial, Francisco and Vrgo\v{c}, Domagoj}, title = {{Cryptocurrency Mining Games with Economic Discount and Decreasing Rewards}}, booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)}, pages = {54:1--54:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-140-5}, ISSN = {1868-8969}, year = {2020}, volume = {154}, editor = {Paul, Christophe and Bl\"{a}ser, Markus}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.54}, URN = {urn:nbn:de:0030-drops-119150}, doi = {10.4230/LIPIcs.STACS.2020.54}, annote = {Keywords: cryptocurrency, game theory, cryptomining, economic discount, decreasing rewards} }

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