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**Published in:** LIPIcs, Volume 283, 34th International Symposium on Algorithms and Computation (ISAAC 2023)

We study distribution testing in the standard access model and the conditional access model when the memory available to the testing algorithm is bounded. In both scenarios, we consider the samples appear in an online fashion. The goal is to test the properties of distribution using an optimal number of samples subject to a memory constraint on how many samples can be stored at a given time. First, we provide a trade-off between the sample complexity and the space complexity for testing identity when the samples are drawn according to the conditional access oracle. We then show that we can learn a succinct representation of a monotone distribution efficiently with a memory constraint on the number of samples that are stored that is almost optimal. We also show that the algorithm for monotone distributions can be extended to a larger class of decomposable distributions.

Sampriti Roy and Yadu Vasudev. Testing Properties of Distributions in the Streaming Model. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 56:1-56:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{roy_et_al:LIPIcs.ISAAC.2023.56, author = {Roy, Sampriti and Vasudev, Yadu}, title = {{Testing Properties of Distributions in the Streaming Model}}, booktitle = {34th International Symposium on Algorithms and Computation (ISAAC 2023)}, pages = {56:1--56:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-289-1}, ISSN = {1868-8969}, year = {2023}, volume = {283}, editor = {Iwata, Satoru and Kakimura, Naonori}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2023.56}, URN = {urn:nbn:de:0030-drops-193588}, doi = {10.4230/LIPIcs.ISAAC.2023.56}, annote = {Keywords: Property testing, distribution testing, streaming} }

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

We initiate the study of distributed graph algorithms under the presence of Byzantine nodes. We consider the fundamental problem of testing the connectivity of a graph in the congested clique model in a Byzantine setting. We are given a n-vertex (arbitrary) graph G embedded in a n-node congested clique where an arbitrary subset of B nodes of the clique of size up to (1/3-ε)n (for any arbitrary small constant ε > 0) can be Byzantine. We consider the full information model where Byzantine nodes can behave arbitrarily, collude with each other, and have unlimited computational power and full knowledge of the states and actions of the honest nodes, including random choices made up to the current round.
Our main result is an efficient randomized distributed algorithm that is able to correctly distinguish between two contrasting cases: (1) the graph G⧵ B (i.e., the graph induced by the removal of the vertices assigned to the Byzantine nodes in the clique) is connected or (2) the graph G is far from connected, i.e., it has at least 2|B|+1 connected components. Our algorithm runs in O(polylog n) rounds in the congested clique model and guarantees that all honest nodes will decide on the correct case with high probability. Since Byzantine nodes can lie about the vertices assigned to them, we show that this is essentially the best possible that can be done by any algorithm. Our result can be viewed also in the spirit of property testing, where our algorithm is able to distinguish between two contrasting cases while giving no guarantees if the graph falls in the grey area (i.e., neither of the cases occur).
Our work is a step towards robust and secure distributed graph computation that can output meaningful results even in the presence of a large number of faulty or malicious nodes.

John Augustine, Anisur Rahaman Molla, Gopal Pandurangan, and Yadu Vasudev. Byzantine Connectivity Testing in the Congested Clique. In 36th International Symposium on Distributed Computing (DISC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 246, pp. 7:1-7:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{augustine_et_al:LIPIcs.DISC.2022.7, author = {Augustine, John and Molla, Anisur Rahaman and Pandurangan, Gopal and Vasudev, Yadu}, title = {{Byzantine Connectivity Testing in the Congested Clique}}, booktitle = {36th International Symposium on Distributed Computing (DISC 2022)}, pages = {7:1--7:21}, 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.2022.7}, URN = {urn:nbn:de:0030-drops-171987}, doi = {10.4230/LIPIcs.DISC.2022.7}, annote = {Keywords: Byzantine protocols, distributed graph algorithms, congested clique, graph connectivity, fault-tolerant computation, randomized algorithms} }

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**Published in:** LIPIcs, Volume 117, 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)

We study property testing in the distributed model and extend its setting from testing with one-sided error to testing with two-sided error. In particular, we develop a two-sided error property tester for general graphs with round complexity O(log(n) / (epsilon Phi^2)) in the CONGEST model, which accepts graphs with conductance Phi and rejects graphs that are epsilon-far from having conductance at least Phi^2 / 1000 with constant probability. Our main insight is that one can start poly(n) random walks from a few random vertices without violating the congestion and unite the results to obtain a consistent answer from all vertices. For connected graphs, this is even possible when the number of vertices is unknown. We also obtain a matching Omega(log n) lower bound for the LOCAL and CONGEST models by an indistinguishability argument. Although the power of vertex labels that arises from two-sided error might seem to be much stronger than in the sequential query model, we can show that this is not the case.

Hendrik Fichtenberger and Yadu Vasudev. A Two-Sided Error Distributed Property Tester For Conductance. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 19:1-19:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{fichtenberger_et_al:LIPIcs.MFCS.2018.19, author = {Fichtenberger, Hendrik and Vasudev, Yadu}, title = {{A Two-Sided Error Distributed Property Tester For Conductance}}, booktitle = {43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)}, pages = {19:1--19:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-086-6}, ISSN = {1868-8969}, year = {2018}, volume = {117}, editor = {Potapov, Igor and Spirakis, Paul and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2018.19}, URN = {urn:nbn:de:0030-drops-96019}, doi = {10.4230/LIPIcs.MFCS.2018.19}, annote = {Keywords: property testing, distributed algorithms, conductance} }

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**Published in:** LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

We consider one-sided error property testing of F-minor freeness in bounded-degree graphs for any finite family of graphs F that contains a minor of K_{2,k}, the k-circus graph, or the (k x 2)-grid for any k in N. This includes, for instance, testing whether a graph is outerplanar or a cactus graph. The query complexity of our algorithm in terms of the number of vertices in the graph, n, is O~(n^{2/3} / epsilon^5). Czumaj et al. (2014) showed that cycle-freeness and C_k-minor freeness can be tested with query complexity O~(sqrt{n}) by using random walks, and that testing H-minor freeness for any H that contains a cycles requires Omega(sqrt{n}) queries. In contrast to these results, we analyze the structure of the graph and show that either we can find a subgraph of sublinear size that includes the forbidden minor H, or we can find a pair of disjoint subsets of vertices whose edge-cut is large, which induces an H-minor.

Hendrik Fichtenberger, Reut Levi, Yadu Vasudev, and Maximilian Wötzel. A Sublinear Tester for Outerplanarity (and Other Forbidden Minors) With One-Sided Error. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 52:1-52:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{fichtenberger_et_al:LIPIcs.ICALP.2018.52, author = {Fichtenberger, Hendrik and Levi, Reut and Vasudev, Yadu and W\"{o}tzel, Maximilian}, title = {{A Sublinear Tester for Outerplanarity (and Other Forbidden Minors) With One-Sided Error}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {52:1--52:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.52}, URN = {urn:nbn:de:0030-drops-90563}, doi = {10.4230/LIPIcs.ICALP.2018.52}, annote = {Keywords: graph property testing, minor-free graphs} }

Document

**Published in:** LIPIcs, Volume 66, 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)

Distribution testing deals with what information can be deduced about an unknown distribution over {1,...,n}, where the algorithm is only allowed to obtain a relatively small number of independent samples from the distribution. In the extended conditional sampling model, the algorithm is also allowed to obtain samples from the restriction of the original distribution on subsets of {1,...,n}.
In 2015, Canonne, Diakonikolas, Gouleakis and Rubinfeld unified several previous results, and showed that for any property of distributions satisfying a "decomposability" criterion, there exists an algorithm (in the basic model) that can distinguish with high probability distributions satisfying the property from distributions that are far from it in variation distance.
We present here a more efficient yet simpler algorithm for the basic model, as well as very efficient algorithms for the conditional model, which until now was not investigated under the umbrella of decomposable properties. Additionally, we provide an algorithm for the conditional model that handles a much larger class of properties.
Our core mechanism is a way of efficiently producing an interval-partition of {1,...,n} that satisfies a "fine-grain" quality. We show that with such a partition at hand we can directly move forward with testing individual intervals, instead of first searching for the "correct" partition of {1,...,n}.

Eldar Fischer, Oded Lachish, and Yadu Vasudev. Improving and Extending the Testing of Distributions for Shape-Restricted Properties. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 31:1-31:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{fischer_et_al:LIPIcs.STACS.2017.31, author = {Fischer, Eldar and Lachish, Oded and Vasudev, Yadu}, title = {{Improving and Extending the Testing of Distributions for Shape-Restricted Properties}}, booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)}, pages = {31:1--31:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-028-6}, ISSN = {1868-8969}, year = {2017}, volume = {66}, editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.31}, URN = {urn:nbn:de:0030-drops-70024}, doi = {10.4230/LIPIcs.STACS.2017.31}, annote = {Keywords: conditional sampling, distribution testing, property testing, statistics} }

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