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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-dev.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 246, 36th International Symposium on Distributed Computing (DISC 2022)

A long line of research about connectivity in the Massively Parallel Computation model has culminated in the seminal works of Andoni et al. [FOCS'18] and Behnezhad et al. [FOCS'19]. They provide a randomized algorithm for low-space MPC with conjectured to be optimal round complexity O(log D + log log_{m/n} n) and O(m) space, for graphs on n vertices with m edges and diameter D. Surprisingly, a recent result of Coy and Czumaj [STOC'22] shows how to achieve the same deterministically. Unfortunately, however, their algorithm suffers from large local computation time.
We present a deterministic connectivity algorithm that matches all the parameters of the randomized algorithm and, in addition, significantly reduces the local computation time to nearly linear.
Our derandomization method is based on reducing the amount of randomness needed to allow for a simpler efficient search. While similar randomness reduction approaches have been used before, our result is not only strikingly simpler, but it is the first to have efficient local computation. This is why we believe it to serve as a starting point for the systematic development of computation-efficient derandomization approaches in low-memory MPC.

Manuela Fischer, Jeff Giliberti, and Christoph Grunau. Improved Deterministic Connectivity in Massively Parallel Computation. In 36th International Symposium on Distributed Computing (DISC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 246, pp. 22:1-22:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{fischer_et_al:LIPIcs.DISC.2022.22, author = {Fischer, Manuela and Giliberti, Jeff and Grunau, Christoph}, title = {{Improved Deterministic Connectivity in Massively Parallel Computation}}, booktitle = {36th International Symposium on Distributed Computing (DISC 2022)}, pages = {22:1--22:17}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2022.22}, URN = {urn:nbn:de:0030-drops-172138}, doi = {10.4230/LIPIcs.DISC.2022.22}, annote = {Keywords: Massively Parallel Computation, MPC, MapReduce, Deterministic Algorithms, Connectivity, Hitting Set, Maximum Matching, Derandomization} }

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

Sampling constitutes an important tool in a variety of areas: from machine learning and combinatorial optimization to computational physics and biology. A central class of sampling algorithms is the Markov Chain Monte Carlo method, based on the construction of a Markov chain with the desired sampling distribution as its stationary distribution. Many of the traditional Markov chains, such as the Glauber dynamics, do not scale well with increasing dimension. To address this shortcoming, we propose a simple local update rule based on the Glauber dynamics that leads to efficient parallel and distributed algorithms for sampling from Gibbs distributions.
Concretely, we present a Markov chain that mixes in O(log n) rounds when Dobrushin's condition for the Gibbs distribution is satisfied. This improves over the LubyGlauber algorithm by Feng, Sun, and Yin [PODC'17], which needs O(Delta log n) rounds, and their LocalMetropolis algorithm, which converges in O(log n) rounds but requires a considerably stronger mixing condition. Here, n denotes the number of nodes in the graphical model inducing the Gibbs distribution, and Delta its maximum degree. In particular, our method can sample a uniform proper coloring with alpha Delta colors in O(log n) rounds for any alpha >2, which almost matches the threshold of the sequential Glauber dynamics and improves on the alpha>2 + sqrt{2} threshold of Feng et al.

Manuela Fischer and Mohsen Ghaffari. A Simple Parallel and Distributed Sampling Technique: Local Glauber Dynamics. In 32nd International Symposium on Distributed Computing (DISC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 121, pp. 26:1-26:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{fischer_et_al:LIPIcs.DISC.2018.26, author = {Fischer, Manuela and Ghaffari, Mohsen}, title = {{A Simple Parallel and Distributed Sampling Technique: Local Glauber Dynamics}}, booktitle = {32nd International Symposium on Distributed Computing (DISC 2018)}, pages = {26:1--26:11}, 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.2018.26}, URN = {urn:nbn:de:0030-drops-98154}, doi = {10.4230/LIPIcs.DISC.2018.26}, annote = {Keywords: Distributed Graph Algorithms, Parallel Algorithms, Local Algorithms, Locality, Sampling, Glauber Dynamics, Coloring} }

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**Published in:** LIPIcs, Volume 91, 31st International Symposium on Distributed Computing (DISC 2017)

We present improved deterministic distributed algorithms for a number of well-studied matching problems, which are simpler, faster, more accurate, and/or more general than their known counterparts. The common denominator of these results is a deterministic distributed rounding method for certain linear programs, which is the first such rounding method, to our knowledge.
A sampling of our end results is as follows.
- An O(log^2 Delta log n)-round deterministic distributed algorithm for computing a maximal matching, in n-node graphs with maximum degree Delta. This is the first improvement in about 20 years over the celebrated O(log^4 n)-round algorithm of Hanckowiak, Karonski, and Panconesi [SODA'98, PODC'99].
- A deterministic distributed algorithm for computing a (2+epsilon)-approximation of maximum matching in O(log^2 Delta log(1/epsilon) + log^* n) rounds. This is exponentially faster than the classic O(Delta + log^* n)-round 2-approximation of Panconesi and Rizzi [DIST'01]. With some modifications, the algorithm can also find an epsilon-maximal matching which leaves only an epsilon-fraction of the edges on unmatched nodes.
- An O(log^2 Delta log(1/epsilon) + log^* n)-round deterministic distributed algorithm for computing a (2+epsilon)-approximation of a maximum weighted matching, and also for the more general problem of maximum weighted b-matching. These improve over the O(log^4 n log_(1+epsilon) W)-round (6+epsilon)-approximation algorithm of Panconesi and Sozio [DIST'10], where W denotes the maximum normalized weight.
- A deterministic local computation algorithm for a (2+epsilon)-approximation of maximum matching with 2^O(log^2 Delta) log^* n queries. This improves almost exponentially over the previous deterministic constant approximations which have query-complexity of 2^Omega(Delta log Delta) log^* n.

Manuela Fischer. Improved Deterministic Distributed Matching via Rounding. In 31st International Symposium on Distributed Computing (DISC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 91, pp. 17:1-17:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{fischer:LIPIcs.DISC.2017.17, author = {Fischer, Manuela}, title = {{Improved Deterministic Distributed Matching via Rounding}}, booktitle = {31st International Symposium on Distributed Computing (DISC 2017)}, pages = {17:1--17:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-053-8}, ISSN = {1868-8969}, year = {2017}, volume = {91}, editor = {Richa, Andr\'{e}a}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2017.17}, URN = {urn:nbn:de:0030-drops-80027}, doi = {10.4230/LIPIcs.DISC.2017.17}, annote = {Keywords: distributed graph algorithms, deterministic distributed algorithms, rounding linear programs, maximal matching, maximum matching approximation} }

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**Published in:** LIPIcs, Volume 91, 31st International Symposium on Distributed Computing (DISC 2017)

Locally Checkable Labeling (LCL) problems include essentially all the classic problems of LOCAL distributed algorithms. In a recent enlightening revelation, Chang and Pettie [FOCS'17] showed that any LCL (on bounded degree graphs) that has an o(log n)-round randomized algorithm can be solved in T_(LLL)(n) rounds, which is the randomized complexity of solving (a relaxed variant of) the Lovasz Local Lemma (LLL) on bounded degree n-node graphs. Currently, the best known upper bound on T_(LLL)(n) is O(log n), by Chung, Pettie, and Su [PODC'14], while the best known lower bound is Omega(log log n), by Brandt et al. [STOC'16]. Chang and Pettie conjectured that there should be an O(log log n)-round algorithm (on bounded degree graphs).
Making the first step of progress towards this conjecture, and providing a significant improvement on the algorithm of Chung et al. [PODC'14], we prove that T_(LLL)(n)= 2^O(sqrt(log log n)). Thus, any o(log n)-round randomized distributed algorithm for any LCL problem on bounded degree graphs can be automatically sped up to run in 2^O(sqrt(log log n)) rounds.
Using this improvement and a number of other ideas, we also improve the complexity of a number of graph coloring problems (in arbitrary degree graphs) from the O(log n)-round results of Chung, Pettie and Su [PODC'14] to 2^O(sqrt(log log n)). These problems include defective coloring, frugal coloring, and list vertex-coloring.

Manuela Fischer and Mohsen Ghaffari. Sublogarithmic Distributed Algorithms for Lovász Local Lemma, and the Complexity Hierarchy. In 31st International Symposium on Distributed Computing (DISC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 91, pp. 18:1-18:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{fischer_et_al:LIPIcs.DISC.2017.18, author = {Fischer, Manuela and Ghaffari, Mohsen}, title = {{Sublogarithmic Distributed Algorithms for Lov\'{a}sz Local Lemma, and the Complexity Hierarchy}}, booktitle = {31st International Symposium on Distributed Computing (DISC 2017)}, pages = {18:1--18:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-053-8}, ISSN = {1868-8969}, year = {2017}, volume = {91}, editor = {Richa, Andr\'{e}a}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2017.18}, URN = {urn:nbn:de:0030-drops-79732}, doi = {10.4230/LIPIcs.DISC.2017.18}, annote = {Keywords: Distributed Graph Algorithms, the Lov'\{a\}sz Local Lemma (LLL), Locally Checkable Labeling problems (LCL), Defective Coloring, Frugal Coloring, List Ve} }

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