Document

**Published in:** LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)

The online manipulation-resilient testing model, proposed by Kalemaj, Raskhodnikova and Varma (ITCS 2022 and Theory of Computing 2023), studies property testing in situations where access to the input degrades continuously and adversarially. Specifically, after each query made by the tester is answered, the adversary can intervene and either erase or corrupt t data points. In this work, we investigate a more nuanced version of the online model in order to overcome old and new impossibility results for the original model. We start by presenting an optimal tester for linearity and a lower bound for low-degree testing of Boolean functions in the original model. We overcome the lower bound by allowing batch queries, where the tester gets a group of queries answered between manipulations of the data. Our batch size is small enough so that function values for a single batch on their own give no information about whether the function is of low degree. Finally, to overcome the impossibility results of Kalemaj et al. for sortedness and the Lipschitz property of sequences, we extend the model to include t < 1, i.e., adversaries that make less than one erasure per query. For sortedness, we characterize the rate of erasures for which online testing can be performed, exhibiting a sharp transition from optimal query complexity to impossibility of testability (with any number of queries). Our online tester works for a general class of local properties of sequences. One feature of our results is that we get new (and in some cases, simpler) optimal algorithms for several properties in the standard property testing model.

Omri Ben-Eliezer, Esty Kelman, Uri Meir, and Sofya Raskhodnikova. Property Testing with Online Adversaries. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 11:1-11:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{beneliezer_et_al:LIPIcs.ITCS.2024.11, author = {Ben-Eliezer, Omri and Kelman, Esty and Meir, Uri and Raskhodnikova, Sofya}, title = {{Property Testing with Online Adversaries}}, booktitle = {15th Innovations in Theoretical Computer Science Conference (ITCS 2024)}, pages = {11:1--11:25}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-309-6}, ISSN = {1868-8969}, year = {2024}, volume = {287}, editor = {Guruswami, Venkatesan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.11}, URN = {urn:nbn:de:0030-drops-195395}, doi = {10.4230/LIPIcs.ITCS.2024.11}, annote = {Keywords: Linearity testing, low-degree testing, Reed-Muller codes, testing properties of sequences, erasure-resilience, corruption-resilience} }

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RANDOM

**Published in:** LIPIcs, Volume 275, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)

We investigate algorithms for testing whether an image is connected. Given a proximity parameter ε ∈ (0,1) and query access to a black-and-white image represented by an n×n matrix of Boolean pixel values, a (1-sided error) connectedness tester accepts if the image is connected and rejects with probability at least 2/3 if the image is ε-far from connected. We show that connectedness can be tested nonadaptively with O(1/ε²) queries and adaptively with O(1/ε^{3/2} √{log1/ε}) queries. The best connectedness tester to date, by Berman, Raskhodnikova, and Yaroslavtsev (STOC 2014) had query complexity O(1/ε² log 1/ε) and was adaptive. We also prove that every nonadaptive, 1-sided error tester for connectedness must make Ω(1/ε log 1/ε) queries.

Piotr Berman, Meiram Murzabulatov, Sofya Raskhodnikova, and Dragos Ristache. Testing Connectedness of Images. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 66:1-66:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{berman_et_al:LIPIcs.APPROX/RANDOM.2023.66, author = {Berman, Piotr and Murzabulatov, Meiram and Raskhodnikova, Sofya and Ristache, Dragos}, title = {{Testing Connectedness of Images}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)}, pages = {66:1--66:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-296-9}, ISSN = {1868-8969}, year = {2023}, volume = {275}, editor = {Megow, Nicole and Smith, Adam}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2023.66}, URN = {urn:nbn:de:0030-drops-188918}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2023.66}, annote = {Keywords: Property testing, sublinear-algorithms, lower bounds, connectivity, graphs} }

Document

Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

We generalize the celebrated isoperimetric inequality of Khot, Minzer, and Safra (SICOMP 2018) for Boolean functions to the case of real-valued functions f:{0,1}^d → ℝ. Our main tool in the proof of the generalized inequality is a new Boolean decomposition that represents every real-valued function f over an arbitrary partially ordered domain as a collection of Boolean functions over the same domain, roughly capturing the distance of f to monotonicity and the structure of violations of f to monotonicity.
We apply our generalized isoperimetric inequality to improve algorithms for testing monotonicity and approximating the distance to monotonicity for real-valued functions. Our tester for monotonicity has query complexity Õ(min(r √d,d)), where r is the size of the image of the input function. (The best previously known tester makes O(d) queries, as shown by Chakrabarty and Seshadhri (STOC 2013).) Our tester is nonadaptive and has 1-sided error. We prove a matching lower bound for nonadaptive, 1-sided error testers for monotonicity. We also show that the distance to monotonicity of real-valued functions that are α-far from monotone can be approximated nonadaptively within a factor of O(√{d log d}) with query complexity polynomial in 1/α and the dimension d. This query complexity is known to be nearly optimal for nonadaptive algorithms even for the special case of Boolean functions. (The best previously known distance approximation algorithm for real-valued functions, by Fattal and Ron (TALG 2010) achieves O(d log r)-approximation.)

Hadley Black, Iden Kalemaj, and Sofya Raskhodnikova. Isoperimetric Inequalities for Real-Valued Functions with Applications to Monotonicity Testing. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 25:1-25:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{black_et_al:LIPIcs.ICALP.2023.25, author = {Black, Hadley and Kalemaj, Iden and Raskhodnikova, Sofya}, title = {{Isoperimetric Inequalities for Real-Valued Functions with Applications to Monotonicity Testing}}, booktitle = {50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)}, pages = {25:1--25:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-278-5}, ISSN = {1868-8969}, year = {2023}, volume = {261}, editor = {Etessami, Kousha and Feige, Uriel and Puppis, Gabriele}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.25}, URN = {urn:nbn:de:0030-drops-180774}, doi = {10.4230/LIPIcs.ICALP.2023.25}, annote = {Keywords: Isoperimetric inequalities, property testing, monotonicity testing} }

Document

Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

Many deployments of differential privacy in industry are in the local model, where each party releases its private information via a differentially private randomizer. We study triangle counting in the noninteractive and interactive local model with edge differential privacy (that, intuitively, requires that the outputs of the algorithm on graphs that differ in one edge be indistinguishable). In this model, each party’s local view consists of the adjacency list of one vertex.
In the noninteractive model, we prove that additive Ω(n²) error is necessary, where n is the number of nodes. This lower bound is our main technical contribution. It uses a reconstruction attack with a new class of linear queries and a novel mix-and-match strategy of running the local randomizers with different completions of their adjacency lists. It matches the additive error of the algorithm based on Randomized Response, proposed by Imola, Murakami and Chaudhuri (USENIX2021) and analyzed by Imola, Murakami and Chaudhuri (CCS2022) for constant ε. We use a different postprocessing of Randomized Response and provide tight bounds on the variance of the resulting algorithm.
In the interactive setting, we prove a lower bound of Ω(n^{3/2}) on the additive error. Previously, no hardness results were known for interactive, edge-private algorithms in the local model, except for those that follow trivially from the results for the central model. Our work significantly improves on the state of the art in differentially private graph analysis in the local model.

Talya Eden, Quanquan C. Liu, Sofya Raskhodnikova, and Adam Smith. Triangle Counting with Local Edge Differential Privacy. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 52:1-52:21, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{eden_et_al:LIPIcs.ICALP.2023.52, author = {Eden, Talya and Liu, Quanquan C. and Raskhodnikova, Sofya and Smith, Adam}, title = {{Triangle Counting with Local Edge Differential Privacy}}, booktitle = {50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)}, pages = {52:1--52:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-278-5}, ISSN = {1868-8969}, year = {2023}, volume = {261}, editor = {Etessami, Kousha and Feige, Uriel and Puppis, Gabriele}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.52}, URN = {urn:nbn:de:0030-drops-181048}, doi = {10.4230/LIPIcs.ICALP.2023.52}, annote = {Keywords: local differential privacy, reconstruction attacks, lower bounds, triangle counting} }

Document

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

We initiate the study of sublinear-time algorithms that access their input via an online adversarial erasure oracle. After answering each query to the input object, such an oracle can erase t input values. Our goal is to understand the complexity of basic computational tasks in extremely adversarial situations, where the algorithm’s access to data is blocked during the execution of the algorithm in response to its actions. Specifically, we focus on property testing in the model with online erasures. We show that two fundamental properties of functions, linearity and quadraticity, can be tested for constant t with asymptotically the same complexity as in the standard property testing model. For linearity testing, we prove tight bounds in terms of t, showing that the query complexity is Θ(log t). In contrast to linearity and quadraticity, some other properties, including sortedness and the Lipschitz property of sequences, cannot be tested at all, even for t = 1. Our investigation leads to a deeper understanding of the structure of violations of linearity and other widely studied properties.

Iden Kalemaj, Sofya Raskhodnikova, and Nithin Varma. Sublinear-Time Computation in the Presence of Online Erasures. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 90:1-90:25, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{kalemaj_et_al:LIPIcs.ITCS.2022.90, author = {Kalemaj, Iden and Raskhodnikova, Sofya and Varma, Nithin}, title = {{Sublinear-Time Computation in the Presence of Online Erasures}}, booktitle = {13th Innovations in Theoretical Computer Science Conference (ITCS 2022)}, pages = {90:1--90:25}, 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.2022.90}, URN = {urn:nbn:de:0030-drops-156867}, doi = {10.4230/LIPIcs.ITCS.2022.90}, annote = {Keywords: Randomized algorithms, property testing, Fourier analysis, linear functions, quadratic functions, Lipschitz and monotone functions, sorted sequences} }

Document

**Published in:** LIPIcs, Volume 185, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)

We investigate sublinear-time algorithms that take partially erased graphs represented by adjacency lists as input. Our algorithms make degree and neighbor queries to the input graph and work with a specified fraction of adversarial erasures in adjacency entries. We focus on two computational tasks: testing if a graph is connected or ε-far from connected and estimating the average degree. For testing connectedness, we discover a threshold phenomenon: when the fraction of erasures is less than ε, this property can be tested efficiently (in time independent of the size of the graph); when the fraction of erasures is at least ε, then a number of queries linear in the size of the graph representation is required. Our erasure-resilient algorithm (for the special case with no erasures) is an improvement over the previously known algorithm for connectedness in the standard property testing model and has optimal dependence on the proximity parameter ε. For estimating the average degree, our results provide an "interpolation" between the query complexity for this computational task in the model with no erasures in two different settings: with only degree queries, investigated by Feige (SIAM J. Comput. `06), and with degree queries and neighbor queries, investigated by Goldreich and Ron (Random Struct. Algorithms `08) and Eden et al. (ICALP `17). We conclude with a discussion of our model and open questions raised by our work.

Amit Levi, Ramesh Krishnan S. Pallavoor, Sofya Raskhodnikova, and Nithin Varma. Erasure-Resilient Sublinear-Time Graph Algorithms. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 80:1-80:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{levi_et_al:LIPIcs.ITCS.2021.80, author = {Levi, Amit and Pallavoor, Ramesh Krishnan S. and Raskhodnikova, Sofya and Varma, Nithin}, title = {{Erasure-Resilient Sublinear-Time Graph Algorithms}}, booktitle = {12th Innovations in Theoretical Computer Science Conference (ITCS 2021)}, pages = {80:1--80:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-177-1}, ISSN = {1868-8969}, year = {2021}, volume = {185}, editor = {Lee, James R.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.80}, URN = {urn:nbn:de:0030-drops-136192}, doi = {10.4230/LIPIcs.ITCS.2021.80}, annote = {Keywords: Graph property testing, Computing with incomplete information, Approximating graph parameters} }

Document

**Published in:** LIPIcs, Volume 124, 10th Innovations in Theoretical Computer Science Conference (ITCS 2019)

We initiate the study of the role of erasures in local decoding and use our understanding to prove a separation between erasure-resilient and tolerant property testing. Local decoding in the presence of errors has been extensively studied, but has not been considered explicitly in the presence of erasures.
Motivated by applications in property testing, we begin our investigation with local list decoding in the presence of erasures. We prove an analog of a famous result of Goldreich and Levin on local list decodability of the Hadamard code. Specifically, we show that the Hadamard code is locally list decodable in the presence of a constant fraction of erasures, arbitrary close to 1, with list sizes and query complexity better than in the Goldreich-Levin theorem. We use this result to exhibit a property which is testable with a number of queries independent of the length of the input in the presence of erasures, but requires a number of queries that depends on the input length, n, for tolerant testing. We further study approximate locally list decodable codes that work against erasures and use them to strengthen our separation by constructing a property which is testable with a constant number of queries in the presence of erasures, but requires n^{Omega(1)} queries for tolerant testing.
Next, we study the general relationship between local decoding in the presence of errors and in the presence of erasures. We observe that every locally (uniquely or list) decodable code that works in the presence of errors also works in the presence of twice as many erasures (with the same parameters up to constant factors). We show that there is also an implication in the other direction for locally decodable codes (with unique decoding): specifically, that the existence of a locally decodable code that works in the presence of erasures implies the existence of a locally decodable code that works in the presence of errors and has related parameters. However, it remains open whether there is an implication in the other direction for locally list decodable codes. We relate this question to other open questions in local decoding.

Sofya Raskhodnikova, Noga Ron-Zewi, and Nithin Varma. Erasures vs. Errors in Local Decoding and Property Testing. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 63:1-63:21, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{raskhodnikova_et_al:LIPIcs.ITCS.2019.63, author = {Raskhodnikova, Sofya and Ron-Zewi, Noga and Varma, Nithin}, title = {{Erasures vs. Errors in Local Decoding and Property Testing}}, booktitle = {10th Innovations in Theoretical Computer Science Conference (ITCS 2019)}, pages = {63:1--63:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-095-8}, ISSN = {1868-8969}, year = {2019}, volume = {124}, editor = {Blum, Avrim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2019.63}, URN = {urn:nbn:de:0030-drops-101568}, doi = {10.4230/LIPIcs.ITCS.2019.63}, annote = {Keywords: Error-correcting codes, probabilistically checkable proofs (PCPs) of proximity, Hadamard code, local list decoding, tolerant testing} }

Document

Brief Announcement

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

We describe work in progress on providing a separation between erasure-resilient and tolerant property testing. Specifically, we are able to exhibit a property which is testable (with the number of queries independent of the length of the input) in the presence of erasures, but is not testable tolerantly.

Sofya Raskhodnikova and Nithin Varma. Brief Announcement: Erasure-Resilience Versus Tolerance to Errors. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 111:1-111:3, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{raskhodnikova_et_al:LIPIcs.ICALP.2018.111, author = {Raskhodnikova, Sofya and Varma, Nithin}, title = {{Brief Announcement: Erasure-Resilience Versus Tolerance to Errors}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {111:1--111:3}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.111}, URN = {urn:nbn:de:0030-drops-91154}, doi = {10.4230/LIPIcs.ICALP.2018.111}, annote = {Keywords: Property testing, erasures, tolerance to errors, model separation} }

Document

**Published in:** LIPIcs, Volume 67, 8th Innovations in Theoretical Computer Science Conference (ITCS 2017)

We investigate the parameters in terms of which the complexity of sublinear-time algorithms should be expressed. Our goal is to find input parameters that are tailored to the combinatorics of the specific problem being studied and design algorithms that run faster when these parameters are small. This direction enables us to surpass the (worst-case) lower bounds, expressed in terms of the input size, for several problems. Our aim is to develop a similar level of understanding of the complexity of sublinear-time algorithms to the one that was enabled by research in parameterized complexity for classical algorithms.
Specifically, we focus on testing properties of functions. By parameterizing the query complexity in terms of the size r of the image of the input function, we obtain testers for monotonicity and convexity of functions of the form f:[n]\to \mathbb{R} with query complexity O(\log r), with no dependence on n. The result for monotonicity circumvents the \Omega(\log n) lower bound by Fischer (Inf. Comput., 2004) for this problem. We present several other parameterized testers, providing compelling evidence that expressing the query complexity of property testers in terms of the input size is not always the best choice.

Ramesh Krishnan S. Pallavoor, Sofya Raskhodnikova, and Nithin Varma. Parameterized Property Testing of Functions. In 8th Innovations in Theoretical Computer Science Conference (ITCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 67, pp. 12:1-12:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{pallavoor_et_al:LIPIcs.ITCS.2017.12, author = {Pallavoor, Ramesh Krishnan S. and Raskhodnikova, Sofya and Varma, Nithin}, title = {{Parameterized Property Testing of Functions}}, booktitle = {8th Innovations in Theoretical Computer Science Conference (ITCS 2017)}, pages = {12:1--12:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-029-3}, ISSN = {1868-8969}, year = {2017}, volume = {67}, editor = {Papadimitriou, Christos H.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2017.12}, URN = {urn:nbn:de:0030-drops-81479}, doi = {10.4230/LIPIcs.ITCS.2017.12}, annote = {Keywords: Sublinear algorithms, property testing, parameterization, monotonicity, convexity} }

Document

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

We study the problem of testing unateness of functions f:{0,1}^d -> R. We give an O(d/\epsilon . log(d/\epsilon))-query nonadaptive tester and an O(d/\epsilon)-query adaptive tester and show that both testers are optimal for a fixed distance parameter \epsilon. Previously known unateness testers worked only for Boolean functions, and their query complexity had worse dependence on the dimension both for the adaptive and the nonadaptive case. Moreover, no lower bounds for testing unateness were known. We generalize our results to obtain optimal unateness testers for functions f:[n]^d -> R.
Our results establish that adaptivity helps with testing unateness of real-valued functions on domains of the form {0,1}^d and, more generally, [n]^d. This stands in contrast to the situation for monotonicity testing where there is no adaptivity gap for functions f:[n]^d -> R.

Roksana Baleshzar, Deeparnab Chakrabarty, Ramesh Krishnan S. Pallavoor, Sofya Raskhodnikova, and C. Seshadhri. Optimal Unateness Testers for Real-Valued Functions: Adaptivity Helps. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 5:1-5:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{baleshzar_et_al:LIPIcs.ICALP.2017.5, author = {Baleshzar, Roksana and Chakrabarty, Deeparnab and Pallavoor, Ramesh Krishnan S. and Raskhodnikova, Sofya and Seshadhri, C.}, title = {{Optimal Unateness Testers for Real-Valued Functions: Adaptivity Helps}}, booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)}, pages = {5:1--5:14}, 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.2017.5}, URN = {urn:nbn:de:0030-drops-74844}, doi = {10.4230/LIPIcs.ICALP.2017.5}, annote = {Keywords: Property testing, unate and monotone functions} }

Document

**Published in:** LIPIcs, Volume 65, 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016)

We investigate testing of properties of 2-dimensional figures that consist of a black object on a white background. Given a parameter epsilon in (0,1/2), a tester for a specified property has to accept with probability at least 2/3 if the input figure satisfies the property and reject with probability at least 2/3 if it does not. In general, property testers can query the color of any point in the input figure.
We study the power of testers that get access only to uniform samples from the input figure. We show that for the property of being a half-plane, the uniform testers are as powerful as general testers: they require only O(1/epsilon) samples. In contrast, we prove that convexity can be tested with O(1/epsilon) queries by testers that can make queries of their choice while uniform testers for this property require Omega(1/epsilon^{5/4}) samples. Previously, the fastest known tester for convexity needed Theta(1/epsilon^{4/3}) queries.

Piotr Berman, Meiram Murzabulatov, and Sofya Raskhodnikova. The Power and Limitations of Uniform Samples in Testing Properties of Figures. In 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 65, pp. 45:1-45:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{berman_et_al:LIPIcs.FSTTCS.2016.45, author = {Berman, Piotr and Murzabulatov, Meiram and Raskhodnikova, Sofya}, title = {{The Power and Limitations of Uniform Samples in Testing Properties of Figures}}, booktitle = {36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016)}, pages = {45:1--45:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-027-9}, ISSN = {1868-8969}, year = {2016}, volume = {65}, editor = {Lal, Akash and Akshay, S. and Saurabh, Saket and Sen, Sandeep}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2016.45}, URN = {urn:nbn:de:0030-drops-68803}, doi = {10.4230/LIPIcs.FSTTCS.2016.45}, annote = {Keywords: Property testing, randomized algorithms, being a half-plane, convexity} }

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**Published in:** LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

We initiate a systematic study of tolerant testers of image properties or, equivalently, algorithms that approximate the distance from a given image to the desired property (that is, the smallest fraction of pixels that need to change in the image to ensure that the image satisfies the desired property). Image processing is a particularly compelling area of applications for sublinear-time algorithms and, specifically, property testing. However, for testing algorithms to reach their full potential in image processing, they have to be tolerant, which allows them to be resilient to noise. Prior to this work, only one tolerant testing algorithm for an image property (image partitioning) has been published.
We design efficient approximation algorithms for the following fundamental questions: What fraction of pixels have to be changed in an image so that it becomes a half-plane? a representation of a convex object? a representation of a connected object? More precisely, our algorithms approximate the distance to three basic properties (being a half-plane, convexity, and connectedness) within a small additive error epsilon, after reading a number of pixels polynomial in 1/epsilon and independent of the size of the image. The running time of the testers for half-plane and convexity is also polynomial in 1/epsilon. Tolerant testers for these three properties were not investigated previously. For convexity and connectedness, even the existence of distance approximation algorithms with query complexity independent of the input size is not implied by previous work. (It does not follow from the VC-dimension bounds, since VC dimension of convexity and connectedness, even in two dimensions, depends on the input size. It also does not follow from the existence of non-tolerant testers.)
Our algorithms require very simple access to the input: uniform random samples for the half-plane property and convexity, and samples from uniformly random blocks for connectedness. However, the analysis of the algorithms, especially for convexity, requires many geometric and combinatorial insights. For example, in the analysis of the algorithm for convexity, we define a set of reference polygons P_{epsilon} such that (1) every convex image has a nearby polygon in P_{epsilon} and (2) one can use dynamic programming to quickly compute the smallest empirical distance to a polygon in P_{epsilon}. This construction might be of independent interest.

Piotr Berman, Meiram Murzabulatov, and Sofya Raskhodnikova. Tolerant Testers of Image Properties. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 90:1-90:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{berman_et_al:LIPIcs.ICALP.2016.90, author = {Berman, Piotr and Murzabulatov, Meiram and Raskhodnikova, Sofya}, title = {{Tolerant Testers of Image Properties}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {90:1--90:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-013-2}, ISSN = {1868-8969}, year = {2016}, volume = {55}, editor = {Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.90}, URN = {urn:nbn:de:0030-drops-61959}, doi = {10.4230/LIPIcs.ICALP.2016.90}, annote = {Keywords: Computational geometry, convexity, half-plane, connectedness, propertytesting, tolerant property testing} }

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**Published in:** LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

Property testers form an important class of sublinear algorithms. In the standard property testing model, an algorithm accesses the input function f:D -> R via an oracle. With very few exceptions, all property testers studied in this model rely on the oracle to provide function values at all queried domain points. However, in many realistic situations, the oracle may be unable to reveal the function values at some domain points due to privacy concerns, or when some of the values get erased by mistake or by an adversary. The testers do not learn anything useful about the property by querying those erased points. Moreover, the knowledge of a tester may enable an adversary to erase some of the values so as to increase the query complexity of the tester arbitrarily or, in some cases, make the tester entirely useless.
In this work, we initiate a study of property testers that are resilient to the presence of adversarially erased function values. An alpha-erasure-resilient epsilon-tester is given parameters alpha, epsilon in (0,1), along with oracle access to a function f such that at most an alpha fraction of function values have been erased. The tester does not know whether a value is erased until it queries the corresponding domain point. The tester has to accept with high probability if there is a way to assign values to the erased points such that the resulting function satisfies the desired property P. It has to reject with high probability if, for every assignment of values to the erased points, the resulting function has to be changed in at least an epsilon-fraction of the non-erased domain points to satisfy P.
We design erasure-resilient property testers for a large class of properties. For some properties, it is possible to obtain erasure-resilient testers by simply using standard testers as a black box. However, there are more challenging properties for which all known testers rely on querying a specific point. If this point is erased, all these testers break. We give efficient erasure-resilient testers for several important classes of such properties of functions including monotonicity, the Lipschitz property, and convexity. Finally, we show a separation between the standard testing and erasure-resilient testing. Specifically, we describe a property that can be epsilon-tested with O(1/epsilon) queries in the standard model, whereas testing it in the erasure-resilient model requires number of queries polynomial in the input size.

Kashyap Dixit, Sofya Raskhodnikova, Abhradeep Thakurta, and Nithin Varma. Erasure-Resilient Property Testing. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 91:1-91:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{dixit_et_al:LIPIcs.ICALP.2016.91, author = {Dixit, Kashyap and Raskhodnikova, Sofya and Thakurta, Abhradeep and Varma, Nithin}, title = {{Erasure-Resilient Property Testing}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {91:1--91:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-013-2}, ISSN = {1868-8969}, year = {2016}, volume = {55}, editor = {Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.91}, URN = {urn:nbn:de:0030-drops-61947}, doi = {10.4230/LIPIcs.ICALP.2016.91}, annote = {Keywords: Randomized algorithms, property testing, error correction, monotoneand Lipschitz functions} }

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**Published in:** LIPIcs, Volume 51, 32nd International Symposium on Computational Geometry (SoCG 2016)

We consider the following basic geometric problem: Given epsilon in (0,1/2), a 2-dimensional figure that consists of a black object and a white background is epsilon-far from convex if it differs in at least an epsilon fraction of the area from every figure where the black object is convex. How many uniform and independent samples from a figure that is epsilon-far from convex are needed to detect a violation of convexity with probability at least 2/3? This question arises in the context of designing property testers for convexity. Specifically, a (1-sided error) tester for convexity gets samples from the figure, labeled by their color; it always accepts if the black object is convex; it rejects with probability at least 2/3 if the figure is epsilon-far from convex.
We show that Theta(epsilon^{-4/3}) uniform samples are necessary and sufficient for detecting a violation of convexity in an epsilon-far figure and, equivalently, for testing convexity of figures with 1-sided error. Our testing algorithm runs in time O(epsilon^{-4/3}) and thus beats the Omega(epsilon^{-3/2}) sample lower bound for learning convex figures under the uniform distribution from the work of Schmeltz (Data Structures and Efficient Algorithms,1992). It shows that, with uniform samples, we can check if a set is approximately convex much faster than we can find an approximate representation of a convex set.

Piotr Berman, Meiram Murzabulatov, and Sofya Raskhodnikova. Testing Convexity of Figures Under the Uniform Distribution. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 17:1-17:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{berman_et_al:LIPIcs.SoCG.2016.17, author = {Berman, Piotr and Murzabulatov, Meiram and Raskhodnikova, Sofya}, title = {{Testing Convexity of Figures Under the Uniform Distribution}}, booktitle = {32nd International Symposium on Computational Geometry (SoCG 2016)}, pages = {17:1--17:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-009-5}, ISSN = {1868-8969}, year = {2016}, volume = {51}, editor = {Fekete, S\'{a}ndor and Lubiw, Anna}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2016.17}, URN = {urn:nbn:de:0030-drops-59094}, doi = {10.4230/LIPIcs.SoCG.2016.17}, annote = {Keywords: Convex sets, 2D geometry, randomized algorithms, property testing} }

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**Published in:** LIPIcs, Volume 8, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010)

A spanner of a graph is a sparse subgraph that approximately preserves distances in the original graph. More precisely, a subgraph $H = (V,E_H)$ is a $k$-spanner of a graph $G=(V,E)$ if for every pair of vertices $u,v \in V$, the shortest path distance $dist_H(u,v)$ from $u$ to $v$ in $H$ is at most $k.dist_G(u,v)$. We focus on spanners of directed graphs and a related notion of transitive-closure spanners. The latter captures the idea that a spanner should have a small diameter but preserve the connectivity of the original graph. We study the computational problem of finding the sparsest $k$-spanner (resp., $k$-TC-spanner) of a given directed graph, which we refer to as DIRECTED $k$-SPANNER (resp., $k$-TC-SPANNER). We improve all known approximation algorithms for these problems for $k\geq 3$. (For $k=2$, the current ratios are tight, assuming P$\neq$NP.) Along the way, we prove several structural results about the size of the sparsest spanners of directed graphs.

Piotr Berman, Sofya Raskhodnikova, and Ge Ruan. Finding Sparser Directed Spanners. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010). Leibniz International Proceedings in Informatics (LIPIcs), Volume 8, pp. 424-435, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2010)

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@InProceedings{berman_et_al:LIPIcs.FSTTCS.2010.424, author = {Berman, Piotr and Raskhodnikova, Sofya and Ruan, Ge}, title = {{Finding Sparser Directed Spanners}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010)}, pages = {424--435}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-23-1}, ISSN = {1868-8969}, year = {2010}, volume = {8}, editor = {Lodaya, Kamal and Mahajan, Meena}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2010.424}, URN = {urn:nbn:de:0030-drops-28830}, doi = {10.4230/LIPIcs.FSTTCS.2010.424}, annote = {Keywords: Approximation algorithms, directed graphs, spanners} }

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