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**Published in:** LIPIcs, Volume 281, 37th International Symposium on Distributed Computing (DISC 2023)

Over the past decade, a long line of research has investigated the distributed complexity landscape of locally checkable labeling (LCL) problems on bounded-degree graphs, culminating in an almost-complete classification on general graphs and a complete classification on trees. The latter states that, on bounded-degree trees, any LCL problem has deterministic worst-case time complexity O(1), Θ(log^* n), Θ(log n), or Θ(n^{1/k}) for some positive integer k, and all of those complexity classes are nonempty. Moreover, randomness helps only for (some) problems with deterministic worst-case complexity Θ(log n), and if randomness helps (asymptotically), then it helps exponentially.
In this work, we study how many distributed rounds are needed on average per node in order to solve an LCL problem on trees. We obtain a partial classification of the deterministic node-averaged complexity landscape for LCL problems. As our main result, we show that every problem with worst-case round complexity O(log n) has deterministic node-averaged complexity O(log^* n). We further establish bounds on the node-averaged complexity of problems with worst-case complexity Θ(n^{1/k}): we show that all these problems have node-averaged complexity Ω̃(n^{1 / (2^k - 1)}), and that this lower bound is tight for some problems.

Alkida Balliu, Sebastian Brandt, Fabian Kuhn, Dennis Olivetti, and Gustav Schmid. On the Node-Averaged Complexity of Locally Checkable Problems on Trees. In 37th International Symposium on Distributed Computing (DISC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 281, pp. 7:1-7:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{balliu_et_al:LIPIcs.DISC.2023.7, author = {Balliu, Alkida and Brandt, Sebastian and Kuhn, Fabian and Olivetti, Dennis and Schmid, Gustav}, title = {{On the Node-Averaged Complexity of Locally Checkable Problems on Trees}}, booktitle = {37th International Symposium on Distributed Computing (DISC 2023)}, pages = {7:1--7:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-301-0}, ISSN = {1868-8969}, year = {2023}, volume = {281}, editor = {Oshman, Rotem}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2023.7}, URN = {urn:nbn:de:0030-drops-191330}, doi = {10.4230/LIPIcs.DISC.2023.7}, annote = {Keywords: distributed graph algorithms, locally checkable labelings, node-averaged complexity, trees} }

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**Published in:** LIPIcs, Volume 246, 36th International Symposium on Distributed Computing (DISC 2022)

We give practical, efficient algorithms that automatically determine the asymptotic distributed round complexity of a given locally checkable graph problem in the [Θ(log n), Θ(n)] region, in two settings. We present one algorithm for unrooted regular trees and another algorithm for rooted regular trees. The algorithms take the description of a locally checkable labeling problem as input, and the running time is polynomial in the size of the problem description. The algorithms decide if the problem is solvable in O(log n) rounds. If not, it is known that the complexity has to be Θ(n^{1/k}) for some k = 1, 2, ..., and in this case the algorithms also output the right value of the exponent k.
In rooted trees in the O(log n) case we can then further determine the exact complexity class by using algorithms from prior work; for unrooted trees the more fine-grained classification in the O(log n) region remains an open question.

Alkida Balliu, Sebastian Brandt, Yi-Jun Chang, Dennis Olivetti, Jan Studený, and Jukka Suomela. Efficient Classification of Locally Checkable Problems in Regular Trees. In 36th International Symposium on Distributed Computing (DISC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 246, pp. 8:1-8:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{balliu_et_al:LIPIcs.DISC.2022.8, author = {Balliu, Alkida and Brandt, Sebastian and Chang, Yi-Jun and Olivetti, Dennis and Studen\'{y}, Jan and Suomela, Jukka}, title = {{Efficient Classification of Locally Checkable Problems in Regular Trees}}, booktitle = {36th International Symposium on Distributed Computing (DISC 2022)}, pages = {8:1--8:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-255-6}, ISSN = {1868-8969}, year = {2022}, volume = {246}, editor = {Scheideler, Christian}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2022.8}, URN = {urn:nbn:de:0030-drops-171993}, doi = {10.4230/LIPIcs.DISC.2022.8}, annote = {Keywords: locally checkable labeling, locality, distributed computational complexity} }

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**Published in:** LIPIcs, Volume 246, 36th International Symposium on Distributed Computing (DISC 2022)

Locally Checkable Labeling (LCL) problems are graph problems in which a solution is correct if it satisfies some given constraints in the local neighborhood of each node. Example problems in this class include maximal matching, maximal independent set, and coloring problems. A successful line of research has been studying the complexities of LCL problems on paths/cycles, trees, and general graphs, providing many interesting results for the LOCAL model of distributed computing. In this work, we initiate the study of LCL problems in the low-space Massively Parallel Computation (MPC) model. In particular, on forests, we provide a method that, given the complexity of an LCL problem in the LOCAL model, automatically provides an exponentially faster algorithm for the low-space MPC setting that uses optimal global memory, that is, truly linear.
While restricting to forests may seem to weaken the result, we emphasize that all known (conditional) lower bounds for the MPC setting are obtained by lifting lower bounds obtained in the distributed setting in tree-like networks (either forests or high girth graphs), and hence the problems that we study are challenging already on forests. Moreover, the most important technical feature of our algorithms is that they use optimal global memory, that is, memory linear in the number of edges of the graph. In contrast, most of the state-of-the-art algorithms use more than linear global memory. Further, they typically start with a dense graph, sparsify it, and then solve the problem on the residual graph, exploiting the relative increase in global memory. On forests, this is not possible, because the given graph is already as sparse as it can be, and using optimal memory requires new solutions.

Alkida Balliu, Sebastian Brandt, Manuela Fischer, Rustam Latypov, Yannic Maus, Dennis Olivetti, and Jara Uitto. Exponential Speedup over Locality in MPC with Optimal Memory. In 36th International Symposium on Distributed Computing (DISC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 246, pp. 9:1-9:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{balliu_et_al:LIPIcs.DISC.2022.9, author = {Balliu, Alkida and Brandt, Sebastian and Fischer, Manuela and Latypov, Rustam and Maus, Yannic and Olivetti, Dennis and Uitto, Jara}, title = {{Exponential Speedup over Locality in MPC with Optimal Memory}}, booktitle = {36th International Symposium on Distributed Computing (DISC 2022)}, pages = {9:1--9:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-255-6}, ISSN = {1868-8969}, year = {2022}, volume = {246}, editor = {Scheideler, Christian}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2022.9}, URN = {urn:nbn:de:0030-drops-172003}, doi = {10.4230/LIPIcs.DISC.2022.9}, annote = {Keywords: Distributed computing, Locally checkable labeling problems, Trees, Massively Parallel Computation, Sublinear memory} }

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**Published in:** LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

We study connections between three different fields: distributed local algorithms, finitary factors of iid processes, and descriptive combinatorics. We focus on two central questions: Can we apply techniques from one of the areas to obtain results in another? Can we show that complexity classes coming from different areas contain precisely the same problems? We give an affirmative answer to both questions in the context of local problems on regular trees:
1) We extend the Borel determinacy technique of Marks [Marks - J. Am. Math. Soc. 2016] coming from descriptive combinatorics and adapt it to the area of distributed computing, thereby obtaining a more generally applicable lower bound technique in descriptive combinatorics and an entirely new lower bound technique for distributed algorithms. Using our new technique, we prove deterministic distributed Ω(log n)-round lower bounds for problems from a natural class of homomorphism problems. Interestingly, these lower bounds seem beyond the current reach of the powerful round elimination technique [Brandt - PODC 2019] responsible for all substantial locality lower bounds of the last years. Our key technical ingredient is a novel ID graph technique that we expect to be of independent interest; in fact, it has already played an important role in a new lower bound for the Lovász local lemma in the Local Computation Algorithms model from sequential computing [Brandt, Grunau, Rozhoň - PODC 2021].
2) We prove that a local problem admits a Baire measurable coloring if and only if it admits a local algorithm with local complexity O(log n), extending the classification of Baire measurable colorings of Bernshteyn [Bernshteyn - personal communication]. A key ingredient of the proof is a new and simple characterization of local problems that can be solved in O(log n) rounds. We complement this result by showing separations between complexity classes from distributed computing, finitary factors, and descriptive combinatorics. Most notably, the class of problems that allow a distributed algorithm with sublogarithmic randomized local complexity is incomparable with the class of problems with a Borel solution. We hope that our treatment will help to view all three perspectives as part of a common theory of locality, in which we follow the insightful paper of [Bernshteyn - arXiv 2004.04905].

Sebastian Brandt, Yi-Jun Chang, Jan Grebík, Christoph Grunau, Václav Rozhoň, and Zoltán Vidnyánszky. Local Problems on Trees from the Perspectives of Distributed Algorithms, Finitary Factors, and Descriptive Combinatorics. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 29:1-29:26, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{brandt_et_al:LIPIcs.ITCS.2022.29, author = {Brandt, Sebastian and Chang, Yi-Jun and Greb{\'\i}k, Jan and Grunau, Christoph and Rozho\v{n}, V\'{a}clav and Vidny\'{a}nszky, Zolt\'{a}n}, title = {{Local Problems on Trees from the Perspectives of Distributed Algorithms, Finitary Factors, and Descriptive Combinatorics}}, booktitle = {13th Innovations in Theoretical Computer Science Conference (ITCS 2022)}, pages = {29:1--29:26}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-217-4}, ISSN = {1868-8969}, year = {2022}, volume = {215}, editor = {Braverman, Mark}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.29}, URN = {urn:nbn:de:0030-drops-156259}, doi = {10.4230/LIPIcs.ITCS.2022.29}, annote = {Keywords: Distributed Algorithms, Descriptive Combinatorics} }

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Brief Announcement

**Published in:** LIPIcs, Volume 209, 35th International Symposium on Distributed Computing (DISC 2021)

We establish scalable Massively Parallel Computation (MPC) algorithms for a family of fundamental graph problems on trees. We give a general method that, for a wide range of LCL problems, turns their message passing counterparts into exponentially faster algorithms in the sublinear MPC model. In particular, we show that any LCL on trees that has a deterministic complexity of O(n) in the LOCAL model can be sped up to O(log n) (high-complexity regime) in the sublinear MPC model and similarly n^{o(1)} to O(log log n) (intermediate-complexity regime). We emphasize, that we work on bounded degree trees and all of our algorithms work in the sublinear MPC model, where local memory is O(n^δ) for δ < 1 and global memory is O(m).
For the high-complexity regime, one key ingredient is a novel pointer-chain technique and analysis that allows us to solve any solvable LCL on trees with a sublinear MPC algorithm with complexity O(log n). For the intermediate-complexity regime, we adapt the approach by Chang and Pettie [FOCS'17], who gave a canonical algorithm for solving LCL problems on trees in the LOCAL model. For the special case of 3-coloring trees, which is a natural LCL problem, we provide a conditional Ω(log log n) lower bound, implying that solving LCL problems on trees with deterministic LOCAL complexity n^{o(1)} requires Θ(log log n) deterministic time in the sublinear MPC model when using a natural family of component-stable algorithms.

Sebastian Brandt, Rustam Latypov, and Jara Uitto. Brief Announcement: Memory Efficient Massively Parallel Algorithms for LCL Problems on Trees. In 35th International Symposium on Distributed Computing (DISC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 209, pp. 50:1-50:4, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{brandt_et_al:LIPIcs.DISC.2021.50, author = {Brandt, Sebastian and Latypov, Rustam and Uitto, Jara}, title = {{Brief Announcement: Memory Efficient Massively Parallel Algorithms for LCL Problems on Trees}}, booktitle = {35th International Symposium on Distributed Computing (DISC 2021)}, pages = {50:1--50:4}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-210-5}, ISSN = {1868-8969}, year = {2021}, volume = {209}, editor = {Gilbert, Seth}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2021.50}, URN = {urn:nbn:de:0030-drops-148521}, doi = {10.4230/LIPIcs.DISC.2021.50}, annote = {Keywords: Distributed computing, Locally checkable labeling problems, Trees, Massively Parallel Computation, Sublinear memory, 3-coloring} }

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**Published in:** LIPIcs, Volume 179, 34th International Symposium on Distributed Computing (DISC 2020)

We study the problem of exploring an oriented grid with autonomous agents governed by finite automata. In the case of a 2-dimensional grid, the question how many agents are required to explore the grid, or equivalently, find a hidden treasure in the grid, is fully understood in both the synchronous and the semi-synchronous setting. For higher dimensions, Dobrev, Narayanan, Opatrny, and Pankratov [ICALP'19] showed very recently that, surprisingly, a (small) constant number of agents suffices to find the treasure, independent of the number of dimensions, thereby disproving a conjecture by Cohen, Emek, Louidor, and Uitto [SODA'17]. Dobrev et al. left as an open question whether their bounds on the number of agents can be improved. We answer this question in the affirmative for deterministic finite automata: we show that 3 synchronous and 4 semi-synchronous agents suffice to explore an n-dimensional grid for any constant n. The bounds are optimal and notably, the matching lower bounds already hold in the 2-dimensional case.
Our techniques can also be used to make progress on other open questions asked by Dobrev et al.: we prove that 4 synchronous and 5 semi-synchronous agents suffice for polynomial-time exploration, and we show that, under a natural assumption, 3 synchronous and 4 semi-synchronous agents suffice to explore unoriented grids of arbitrary dimension (which, again, is tight).

Sebastian Brandt, Julian Portmann, and Jara Uitto. Tight Bounds for Deterministic High-Dimensional Grid Exploration. In 34th International Symposium on Distributed Computing (DISC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 179, pp. 13:1-13:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{brandt_et_al:LIPIcs.DISC.2020.13, author = {Brandt, Sebastian and Portmann, Julian and Uitto, Jara}, title = {{Tight Bounds for Deterministic High-Dimensional Grid Exploration}}, booktitle = {34th International Symposium on Distributed Computing (DISC 2020)}, pages = {13:1--13:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-168-9}, ISSN = {1868-8969}, year = {2020}, volume = {179}, editor = {Attiya, Hagit}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2020.13}, URN = {urn:nbn:de:0030-drops-130911}, doi = {10.4230/LIPIcs.DISC.2020.13}, annote = {Keywords: Mobile agents, finite automata, grid search} }

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**Published in:** LIPIcs, Volume 179, 34th International Symposium on Distributed Computing (DISC 2020)

We present a complete classification of the deterministic distributed time complexity for a family of graph problems: binary labeling problems in trees. These are locally checkable problems that can be encoded with an alphabet of size two in the edge labeling formalism. Examples of binary labeling problems include sinkless orientation, sinkless and sourceless orientation, 2-vertex coloring, perfect matching, and the task of coloring edges red and blue such that all nodes are incident to at least one red and at least one blue edge. More generally, we can encode e.g. any cardinality constraints on indegrees and outdegrees.
We study the deterministic time complexity of solving a given binary labeling problem in trees, in the usual LOCAL model of distributed computing. We show that the complexity of any such problem is in one of the following classes: O(1), Θ(log n), Θ(n), or unsolvable. In particular, a problem that can be represented in the binary labeling formalism cannot have time complexity Θ(log^* n), and hence we know that e.g. any encoding of maximal matchings has to use at least three labels (which is tight).
Furthermore, given the description of any binary labeling problem, we can easily determine in which of the four classes it is and what is an asymptotically optimal algorithm for solving it. Hence the distributed time complexity of binary labeling problems is decidable, not only in principle, but also in practice: there is a simple and efficient algorithm that takes the description of a binary labeling problem and outputs its distributed time complexity.

Alkida Balliu, Sebastian Brandt, Yuval Efron, Juho Hirvonen, Yannic Maus, Dennis Olivetti, and Jukka Suomela. Classification of Distributed Binary Labeling Problems. In 34th International Symposium on Distributed Computing (DISC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 179, pp. 17:1-17:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{balliu_et_al:LIPIcs.DISC.2020.17, author = {Balliu, Alkida and Brandt, Sebastian and Efron, Yuval and Hirvonen, Juho and Maus, Yannic and Olivetti, Dennis and Suomela, Jukka}, title = {{Classification of Distributed Binary Labeling Problems}}, booktitle = {34th International Symposium on Distributed Computing (DISC 2020)}, pages = {17:1--17:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-168-9}, ISSN = {1868-8969}, year = {2020}, volume = {179}, editor = {Attiya, Hagit}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2020.17}, URN = {urn:nbn:de:0030-drops-130957}, doi = {10.4230/LIPIcs.DISC.2020.17}, annote = {Keywords: LOCAL model, graph problems, locally checkable labeling problems, distributed computational complexity} }

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Brief Announcement

**Published in:** LIPIcs, Volume 179, 34th International Symposium on Distributed Computing (DISC 2020)

We introduce a new graph problem, the token dropping game, and we show how to solve it efficiently in a distributed setting. We use the token dropping game as a tool to design an efficient distributed algorithm for the stable orientation problem, which is a special case of the more general locally optimal semi-matching problem. The prior work by Czygrinow et al. (DISC 2012) finds a locally optimal semi-matching in O(Δ⁵) rounds in graphs of maximum degree Δ, which directly implies an algorithm with the same runtime for stable orientations. We improve the runtime to O(Δ⁴) for stable orientations and prove a lower bound of Ω(Δ) rounds.

Sebastian Brandt, Barbara Keller, Joel Rybicki, Jukka Suomela, and Jara Uitto. Brief Announcement: Efficient Load-Balancing Through Distributed Token Dropping. In 34th International Symposium on Distributed Computing (DISC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 179, pp. 40:1-40:3, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{brandt_et_al:LIPIcs.DISC.2020.40, author = {Brandt, Sebastian and Keller, Barbara and Rybicki, Joel and Suomela, Jukka and Uitto, Jara}, title = {{Brief Announcement: Efficient Load-Balancing Through Distributed Token Dropping}}, booktitle = {34th International Symposium on Distributed Computing (DISC 2020)}, pages = {40:1--40:3}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-168-9}, ISSN = {1868-8969}, year = {2020}, volume = {179}, editor = {Attiya, Hagit}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2020.40}, URN = {urn:nbn:de:0030-drops-131182}, doi = {10.4230/LIPIcs.DISC.2020.40}, annote = {Keywords: distributed algorithms, graph problems, semi-matching} }

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**Published in:** LIPIcs, Volume 147, 26th International Symposium on Temporal Representation and Reasoning (TIME 2019)

Motivated by two industrial use cases that involve detecting events of interest in (asynchronous) time series from sensors in manufacturing rigs and gas turbines, we design an expressive rule language DslD equipped with interval aggregate functions (such as weighted average over a time interval), Allen’s interval relations and various metric constructs. We demonstrate how to model events in the uses cases in terms of DslD programs. We show that answering DslD queries in our use cases can be reduced to evaluating SQL queries. Our experiments with the use cases, carried out on the Apache Spark system, show that such SQL queries scale well on large real-world datasets.

Sebastian Brandt, Diego Calvanese, Elem Güzel Kalaycı, Roman Kontchakov, Benjamin Mörzinger, Vladislav Ryzhikov, Guohui Xiao, and Michael Zakharyaschev. Two-Dimensional Rule Language for Querying Sensor Log Data: A Framework and Use Cases. In 26th International Symposium on Temporal Representation and Reasoning (TIME 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 147, pp. 7:1-7:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{brandt_et_al:LIPIcs.TIME.2019.7, author = {Brandt, Sebastian and Calvanese, Diego and Kalayc{\i}, Elem G\"{u}zel and Kontchakov, Roman and M\"{o}rzinger, Benjamin and Ryzhikov, Vladislav and Xiao, Guohui and Zakharyaschev, Michael}, title = {{Two-Dimensional Rule Language for Querying Sensor Log Data: A Framework and Use Cases}}, booktitle = {26th International Symposium on Temporal Representation and Reasoning (TIME 2019)}, pages = {7:1--7:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-127-6}, ISSN = {1868-8969}, year = {2019}, volume = {147}, editor = {Gamper, Johann and Pinchinat, Sophie and Sciavicco, Guido}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TIME.2019.7}, URN = {urn:nbn:de:0030-drops-113658}, doi = {10.4230/LIPIcs.TIME.2019.7}, annote = {Keywords: Ontology-based data access, temporal logic, sensor log data} }

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**Published in:** LIPIcs, Volume 121, 32nd International Symposium on Distributed Computing (DISC 2018)

The landscape of the distributed time complexity is nowadays well-understood for subpolynomial complexities. When we look at deterministic algorithms in the LOCAL model and locally checkable problems (LCLs) in bounded-degree graphs, the following picture emerges:
- There are lots of problems with time complexities Theta(log^* n) or Theta(log n).
- It is not possible to have a problem with complexity between omega(log^* n) and o(log n).
- In general graphs, we can construct LCL problems with infinitely many complexities between omega(log n) and n^{o(1)}.
- In trees, problems with such complexities do not exist.
However, the high end of the complexity spectrum was left open by prior work. In general graphs there are problems with complexities of the form Theta(n^alpha) for any rational 0 < alpha <=1/2, while for trees only complexities of the form Theta(n^{1/k}) are known. No LCL problem with complexity between omega(sqrt{n}) and o(n) is known, and neither are there results that would show that such problems do not exist. We show that:
- In general graphs, we can construct LCL problems with infinitely many complexities between omega(sqrt{n}) and o(n).
- In trees, problems with such complexities do not exist.
Put otherwise, we show that any LCL with a complexity o(n) can be solved in time O(sqrt{n}) in trees, while the same is not true in general graphs.

Alkida Balliu, Sebastian Brandt, Dennis Olivetti, and Jukka Suomela. Almost Global Problems in the LOCAL Model. In 32nd International Symposium on Distributed Computing (DISC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 121, pp. 9:1-9:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{balliu_et_al:LIPIcs.DISC.2018.9, author = {Balliu, Alkida and Brandt, Sebastian and Olivetti, Dennis and Suomela, Jukka}, title = {{Almost Global Problems in the LOCAL Model}}, booktitle = {32nd International Symposium on Distributed Computing (DISC 2018)}, pages = {9:1--9:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-092-7}, ISSN = {1868-8969}, year = {2018}, volume = {121}, editor = {Schmid, Ulrich and Widder, Josef}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2018.9}, URN = {urn:nbn:de:0030-drops-97982}, doi = {10.4230/LIPIcs.DISC.2018.9}, annote = {Keywords: Distributed complexity theory, locally checkable labellings, LOCAL model} }

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**Published in:** LIPIcs, Volume 121, 32nd International Symposium on Distributed Computing (DISC 2018)

Recently, there has been a growing interest in grid exploration by agents with limited capabilities. We show that the grid cannot be explored by three semi-synchronous finite automata, answering an open question by Emek et al. [TCS'15] in the negative.
In the setting we consider, time is divided into discrete steps, where in each step, an adversarially selected subset of the agents executes one look-compute-move cycle. The agents operate according to a shared finite automaton, where every agent is allowed to have a distinct initial state. The only means of communication is to sense the states of the agents sharing the same grid cell. The agents are equipped with a global compass and whenever an agent moves, the destination cell of the movement is chosen by the agent's automaton from the set of neighboring grid cells. In contrast to the four agent protocol by Emek et al., we show that three agents do not suffice for grid exploration.

Sebastian Brandt, Jara Uitto, and Roger Wattenhofer. A Tight Lower Bound for Semi-Synchronous Collaborative Grid Exploration. In 32nd International Symposium on Distributed Computing (DISC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 121, pp. 13:1-13:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{brandt_et_al:LIPIcs.DISC.2018.13, author = {Brandt, Sebastian and Uitto, Jara and Wattenhofer, Roger}, title = {{A Tight Lower Bound for Semi-Synchronous Collaborative Grid Exploration}}, booktitle = {32nd International Symposium on Distributed Computing (DISC 2018)}, pages = {13:1--13:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-092-7}, ISSN = {1868-8969}, year = {2018}, volume = {121}, editor = {Schmid, Ulrich and Widder, Josef}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2018.13}, URN = {urn:nbn:de:0030-drops-98029}, doi = {10.4230/LIPIcs.DISC.2018.13}, annote = {Keywords: Finite automata, Graph exploration, Mobile robots} }

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**Published in:** LIPIcs, Volume 112, 26th Annual European Symposium on Algorithms (ESA 2018)

Cops and Robbers is a classic pursuit-evasion game played between a group of g cops and one robber on an undirected N-vertex graph G. We prove that the complexity of deciding the winner in the game under optimal play requires Omega (N^{g-o(1)}) time on instances with O(N log^2 N) edges, conditioned on the Strong Exponential Time Hypothesis. Moreover, the problem of calculating the minimum number of cops needed to win the game is 2^{Omega (sqrt{N})}, conditioned on the weaker Exponential Time Hypothesis. Our conditional lower bound comes very close to a conditional upper bound: if Meyniel's conjecture holds then the cop number can be decided in 2^{O(sqrt{N}log N)} time.
In recent years, the Strong Exponential Time Hypothesis has been used to obtain many lower bounds on classic combinatorial problems, such as graph diameter, LCS, EDIT-DISTANCE, and REGEXP matching. To our knowledge, these are the first conditional (S)ETH-hard lower bounds on a strategic game.

Sebastian Brandt, Seth Pettie, and Jara Uitto. Fine-grained Lower Bounds on Cops and Robbers. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 9:1-9:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{brandt_et_al:LIPIcs.ESA.2018.9, author = {Brandt, Sebastian and Pettie, Seth and Uitto, Jara}, title = {{Fine-grained Lower Bounds on Cops and Robbers}}, booktitle = {26th Annual European Symposium on Algorithms (ESA 2018)}, pages = {9:1--9:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-081-1}, ISSN = {1868-8969}, year = {2018}, volume = {112}, editor = {Azar, Yossi and Bast, Hannah and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2018.9}, URN = {urn:nbn:de:0030-drops-94725}, doi = {10.4230/LIPIcs.ESA.2018.9}, annote = {Keywords: Cops and Robbers} }

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**Published in:** LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)

For the game of Cops and Robbers, it is known that in 1-cop-win graphs, the cop can capture the robber in O(n) time, and that there exist graphs in which this capture time is tight. When k >= 2, a simple counting argument shows that in k-cop-win graphs, the capture time is at most O(n^{k + 1}), however, no non-trivial lower bounds were previously known; indeed, in their 2011 book, Bonato and Nowakowski ask whether this upper bound can be improved. In this paper, the question of Bonato and Nowakowski is answered on the negative, proving that the O(n^{k + 1}) bound is asymptotically tight for any constant k >= 2. This yields a surprising gap in the capture time complexities between the 1-cop and the 2-cop cases.

Sebastian Brandt, Yuval Emek, Jara Uitto, and Roger Wattenhofer. A Tight Lower Bound for the Capture Time of the Cops and Robbers Game. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 82:1-82:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{brandt_et_al:LIPIcs.ICALP.2017.82, author = {Brandt, Sebastian and Emek, Yuval and Uitto, Jara and Wattenhofer, Roger}, title = {{A Tight Lower Bound for the Capture Time of the Cops and Robbers Game}}, booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)}, pages = {82:1--82:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-041-5}, ISSN = {1868-8969}, year = {2017}, volume = {80}, editor = {Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.82}, URN = {urn:nbn:de:0030-drops-74134}, doi = {10.4230/LIPIcs.ICALP.2017.82}, annote = {Keywords: cops and robbers, capture time, lower bound} }

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