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

We study the problems of testing and learning high-dimensional discrete convex sets. The simplest high-dimensional discrete domain where convexity is a non-trivial property is the ternary hypercube, {-1,0,1}ⁿ. The goal of this work is to understand structural combinatorial properties of convex sets in this domain and to determine the complexity of the testing and learning problems. We obtain the following results.
Structural: We prove nearly tight bounds on the edge boundary of convex sets in {0,±1}ⁿ, showing that the maximum edge boundary of a convex set is Õ(n^{3/4})⋅3ⁿ, or equivalently that every convex set has influence Õ(n^{3/4}) and a convex set exists with influence Ω(n^{3/4}).
Learning and sample-based testing: We prove upper and lower bounds of 3^{Õ(n^{3/4})} and 3^{Ω(√n)} for the task of learning convex sets under the uniform distribution from random examples. The analysis of the learning algorithm relies on our upper bound on the influence. Both the upper and lower bound also hold for the problem of sample-based testing with two-sided error. For sample-based testing with one-sided error we show that the sample-complexity is 3^{Θ(n)}.
Testing with queries: We prove nearly matching upper and lower bounds of 3^{Θ̃(√n)} for one-sided error testing of convex sets with non-adaptive queries.

Hadley Black, Eric Blais, and Nathaniel Harms. Testing and Learning Convex Sets in the Ternary Hypercube. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 15:1-15:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{black_et_al:LIPIcs.ITCS.2024.15, author = {Black, Hadley and Blais, Eric and Harms, Nathaniel}, title = {{Testing and Learning Convex Sets in the Ternary Hypercube}}, booktitle = {15th Innovations in Theoretical Computer Science Conference (ITCS 2024)}, pages = {15:1--15:21}, 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.2024.15}, URN = {urn:nbn:de:0030-drops-195435}, doi = {10.4230/LIPIcs.ITCS.2024.15}, annote = {Keywords: Property testing, learning theory, convex sets, testing convexity, fluctuation} }

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**Published in:** LIPIcs, Volume 186, 24th International Conference on Database Theory (ICDT 2021)

Recent beyond worst-case optimal join algorithms Minesweeper and its generalization Tetris have brought the theory of indexing and join processing together by developing a geometric framework for joins. These algorithms take as input an index ℬ, referred to as a box cover, that stores output gaps that can be inferred from traditional indexes, such as B+ trees or tries, on the input relations. The performances of these algorithms highly depend on the certificate of ℬ, which is the smallest subset of gaps in ℬ whose union covers all of the gaps in the output space of a query Q. Different box covers can have different size certificates and the sizes of both the box covers and certificates highly depend on the ordering of the domain values of the attributes in Q. We study how to generate box covers that contain small size certificates to guarantee efficient runtimes for these algorithms. First, given a query Q over a set of relations of size N and a fixed set of domain orderings for the attributes, we give a Õ(N)-time algorithm called GAMB which generates a box cover for Q that is guaranteed to contain the smallest size certificate across any box cover for Q. Second, we show that finding a domain ordering to minimize the box cover size and certificate is NP-hard through a reduction from the 2 consecutive block minimization problem on boolean matrices. Our third contribution is a Õ(N)-time approximation algorithm called ADORA to compute domain orderings, under which one can compute a box cover of size Õ(K^r), where K is the minimum box cover for Q under any domain ordering and r is the maximum arity of any relation. This guarantees certificates of size Õ(K^r). We combine ADORA and GAMB with Tetris to form a new algorithm we call TetrisReordered, which provides several new beyond worst-case bounds. On infinite families of queries, TetrisReordered’s runtimes are unboundedly better than the bounds stated in prior work.

Kaleb Alway, Eric Blais, and Semih Salihoglu. Box Covers and Domain Orderings for Beyond Worst-Case Join Processing. In 24th International Conference on Database Theory (ICDT 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 186, pp. 3:1-3:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{alway_et_al:LIPIcs.ICDT.2021.3, author = {Alway, Kaleb and Blais, Eric and Salihoglu, Semih}, title = {{Box Covers and Domain Orderings for Beyond Worst-Case Join Processing}}, booktitle = {24th International Conference on Database Theory (ICDT 2021)}, pages = {3:1--3:23}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-179-5}, ISSN = {1868-8969}, year = {2021}, volume = {186}, editor = {Yi, Ke and Wei, Zhewei}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICDT.2021.3}, URN = {urn:nbn:de:0030-drops-137114}, doi = {10.4230/LIPIcs.ICDT.2021.3}, annote = {Keywords: Beyond worst-case join algorithms, Tetris, Box covers, Domain orderings} }

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RANDOM

**Published in:** LIPIcs, Volume 176, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)

A body K ⊂ ℝⁿ is convex if and only if the line segment between any two points in K is completely contained within K or, equivalently, if and only if the convex hull of a set of points in K is contained within K. We show that neither of those characterizations of convexity are robust: there are bodies in ℝⁿ that are far from convex - in the sense that the volume of the symmetric difference between the set K and any convex set C is a constant fraction of the volume of K - for which a line segment between two randomly chosen points x,y ∈ K or the convex hull of a random set X of points in K is completely contained within K except with exponentially small probability. These results show that any algorithms for testing convexity based on the natural line segment and convex hull tests have exponential query complexity.

Eric Blais and Abhinav Bommireddi. On Testing and Robust Characterizations of Convexity. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 18:1-18:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{blais_et_al:LIPIcs.APPROX/RANDOM.2020.18, author = {Blais, Eric and Bommireddi, Abhinav}, title = {{On Testing and Robust Characterizations of Convexity}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)}, pages = {18:1--18:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-164-1}, ISSN = {1868-8969}, year = {2020}, volume = {176}, editor = {Byrka, Jaros{\l}aw and Meka, Raghu}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2020.18}, URN = {urn:nbn:de:0030-drops-126214}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2020.18}, annote = {Keywords: Convexity, Line segment test, Convex hull test, Intersecting cones} }

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**Published in:** LIPIcs, Volume 137, 34th Computational Complexity Conference (CCC 2019)

We establish two results regarding the query complexity of bounded-error randomized algorithms.
Bounded-error separation theorem. There exists a total function f : {0,1}^n -> {0,1} whose epsilon-error randomized query complexity satisfies overline{R}_epsilon(f) = Omega(R(f) * log 1/epsilon).
Strong direct sum theorem. For every function f and every k >= 2, the randomized query complexity of computing k instances of f simultaneously satisfies overline{R}_epsilon(f^k) = Theta(k * overline{R}_{epsilon/k}(f)).
As a consequence of our two main results, we obtain an optimal superlinear direct-sum-type theorem for randomized query complexity: there exists a function f for which R(f^k) = Theta(k log k * R(f)). This answers an open question of Drucker (2012). Combining this result with the query-to-communication complexity lifting theorem of Göös, Pitassi, and Watson (2017), this also shows that there is a total function whose public-coin randomized communication complexity satisfies R^{cc}(f^k) = Theta(k log k * R^{cc}(f)), answering a question of Feder, Kushilevitz, Naor, and Nisan (1995).

Eric Blais and Joshua Brody. Optimal Separation and Strong Direct Sum for Randomized Query Complexity. In 34th Computational Complexity Conference (CCC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 137, pp. 29:1-29:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{blais_et_al:LIPIcs.CCC.2019.29, author = {Blais, Eric and Brody, Joshua}, title = {{Optimal Separation and Strong Direct Sum for Randomized Query Complexity}}, booktitle = {34th Computational Complexity Conference (CCC 2019)}, pages = {29:1--29:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-116-0}, ISSN = {1868-8969}, year = {2019}, volume = {137}, editor = {Shpilka, Amir}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2019.29}, URN = {urn:nbn:de:0030-drops-108511}, doi = {10.4230/LIPIcs.CCC.2019.29}, annote = {Keywords: Decision trees, query complexity, communication complexity} }

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Complete Volume

**Published in:** LIPIcs, Volume 116, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)

LIPIcs, Volume 116, APPROX/RANDOM'18, Complete Volume

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@Proceedings{blais_et_al:LIPIcs.APPROX-RANDOM.2018, title = {{LIPIcs, Volume 116, APPROX/RANDOM'18, Complete Volume}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-085-9}, ISSN = {1868-8969}, year = {2018}, volume = {116}, editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018}, URN = {urn:nbn:de:0030-drops-97254}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2018}, annote = {Keywords: Mathematics of computing, Theory of computation} }

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Front Matter

**Published in:** LIPIcs, Volume 116, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)

Front Matter, Table of Contents, Preface, Conference Organization

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 0:i-0:xvi, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{blais_et_al:LIPIcs.APPROX-RANDOM.2018.0, author = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David}, title = {{Front Matter, Table of Contents, Preface, Conference Organization}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)}, pages = {0:i--0:xvi}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-085-9}, ISSN = {1868-8969}, year = {2018}, volume = {116}, editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.0}, URN = {urn:nbn:de:0030-drops-94043}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.0}, annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization} }

Document

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

We show that for any constant \epsilon > 0 and p \ge 1, it is possible to distinguish functions f : \{0,1\}^n \to [0,1] that are submodular from those that are \epsilon-far from every submodular function in \ell_p distance with a constant number of queries.
More generally, we extend the testing-by-implicit-learning framework of Diakonikolas et al.(2007) to show that every property of real-valued functions that is well-approximated in \ell_2 distance by a class of k-juntas for some k = O(1) can be tested in the \ell_p-testing model with a constant number of queries. This result, combined with a recent junta theorem of Feldman and \Vondrak (2016), yields the constant-query testability of submodularity. It also yields constant-query testing algorithms for a variety of other natural properties of valuation functions, including fractionally additive (XOS) functions, OXS functions, unit demand functions, coverage functions, and self-bounding functions.

Eric Blais and Abhinav Bommireddi. Testing Submodularity and Other Properties of Valuation Functions. In 8th Innovations in Theoretical Computer Science Conference (ITCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 67, pp. 33:1-33:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{blais_et_al:LIPIcs.ITCS.2017.33, author = {Blais, Eric and Bommireddi, Abhinav}, title = {{Testing Submodularity and Other Properties of Valuation Functions}}, booktitle = {8th Innovations in Theoretical Computer Science Conference (ITCS 2017)}, pages = {33:1--33:17}, 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.2017.33}, URN = {urn:nbn:de:0030-drops-81619}, doi = {10.4230/LIPIcs.ITCS.2017.33}, annote = {Keywords: Property testing, Testing by implicit learning, Self-bounding functions} }

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**Published in:** LIPIcs, Volume 79, 32nd Computational Complexity Conference (CCC 2017)

We present a new methodology for proving distribution testing lower bounds, establishing a connection between distribution testing and the simultaneous message passing (SMP) communication model. Extending the framework of Blais, Brody, and Matulef (Computational Complexity, 2012), we show a simple way to reduce (private-coin) SMP problems to distribution testing problems. This method allows us to prove new distribution testing lower bounds, as well as to provide simple proofs of known lower bounds.
Our main result is concerned with testing identity to a specific distribution p, given as a parameter. In a recent and influential work, Valiant and Valiant (FOCS, 2014) showed that the sample complexity of the aforementioned problem is closely related to the 2/3-quasinorm of p. We obtain alternative bounds on the complexity of this problem in terms of an arguably more intuitive measure and using simpler proofs. More specifically, we prove that the sample complexity is essentially determined by a fundamental operator in the theory of interpolation of Banach spaces, known as Peetre's K-functional. We show that this quantity is closely related to the size of the effective support of p (loosely speaking, the number of supported elements that constitute the vast majority of the mass of p). This result, in turn, stems from an unexpected connection to functional analysis and refined concentration of measure inequalities, which arise naturally in our reduction.

Eric Blais, Clément L. Canonne, and Tom Gur. Distribution Testing Lower Bounds via Reductions from Communication Complexity. In 32nd Computational Complexity Conference (CCC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 79, pp. 28:1-28:40, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{blais_et_al:LIPIcs.CCC.2017.28, author = {Blais, Eric and Canonne, Cl\'{e}ment L. and Gur, Tom}, title = {{Distribution Testing Lower Bounds via Reductions from Communication Complexity}}, booktitle = {32nd Computational Complexity Conference (CCC 2017)}, pages = {28:1--28:40}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-040-8}, ISSN = {1868-8969}, year = {2017}, volume = {79}, editor = {O'Donnell, Ryan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2017.28}, URN = {urn:nbn:de:0030-drops-75366}, doi = {10.4230/LIPIcs.CCC.2017.28}, annote = {Keywords: Distribution testing, communication complexity, lower bounds, simultaneous message passing, functional analysis} }

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**Published in:** LIPIcs, Volume 40, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)

Monotone Boolean functions, and the monotone Boolean circuits that compute them, have been intensively studied in complexity theory. In this paper we study the structure of Boolean functions in terms of the minimum number of negations in any circuit computing them, a complexity measure that interpolates between monotone functions and the class of all functions. We study this generalization of monotonicity from the vantage point of learning theory, establishing nearly matching upper and lower bounds on the uniform-distribution learnability of circuits in terms of the number of negations they contain. Our upper bounds are based on a new structural characterization of negation-limited circuits that extends a classical result of A.A. Markov. Our lower bounds, which employ Fourier-analytic tools from hardness amplification, give new results even for circuits with no negations (i.e. monotone functions).

Eric Blais, Clément L. Canonne, Igor C. Oliveira, Rocco A. Servedio, and Li-Yang Tan. Learning Circuits with few Negations. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 512-527, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{blais_et_al:LIPIcs.APPROX-RANDOM.2015.512, author = {Blais, Eric and Canonne, Cl\'{e}ment L. and Oliveira, Igor C. and Servedio, Rocco A. and Tan, Li-Yang}, title = {{Learning Circuits with few Negations}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)}, pages = {512--527}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-89-7}, ISSN = {1868-8969}, year = {2015}, volume = {40}, editor = {Garg, Naveen and Jansen, Klaus and Rao, Anup and Rolim, Jos\'{e} D. P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2015.512}, URN = {urn:nbn:de:0030-drops-53214}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2015.512}, annote = {Keywords: Boolean functions, monotonicity, negations, PAC learning} }

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**Published in:** LIPIcs, Volume 28, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)

The Hamming distance function Ham_{n,d} returns 1 on all pairs of inputs x and y that differ in at most d coordinates and returns 0 otherwise. We initiate the study of the information complexity of the Hamming distance function.
We give a new optimal lower bound for the information complexity of the Ham_{n,d} function in the small-error regime where the protocol is required to err with probability at most epsilon < d/n. We also give a new conditional lower bound for the information complexity of Ham_{n,d} that is optimal in all regimes. These results imply the first new lower bounds on the communication complexity of the Hamming distance function for the shared randomness two-way communication model since Pang and El-Gamal (1986). These results also imply new lower bounds in the areas of property testing and parity decision tree complexity.

Eric Blais, Joshua Brody, and Badih Ghazi. The Information Complexity of Hamming Distance. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 28, pp. 465-489, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{blais_et_al:LIPIcs.APPROX-RANDOM.2014.465, author = {Blais, Eric and Brody, Joshua and Ghazi, Badih}, title = {{The Information Complexity of Hamming Distance}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)}, pages = {465--489}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-74-3}, ISSN = {1868-8969}, year = {2014}, volume = {28}, editor = {Jansen, Klaus and Rolim, Jos\'{e} and Devanur, Nikhil R. and Moore, Cristopher}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2014.465}, URN = {urn:nbn:de:0030-drops-47174}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2014.465}, annote = {Keywords: Hamming distance, communication complexity, information complexity} }