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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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**Published in:** LIPIcs, Volume 93, 37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2017)

In this work, we continue the examination of the role non-adaptivity plays in maintaining dynamic data structures, initiated by Brody and Larsen. We consider non-adaptive data structures for predecessor search in the w-bit cell probe model. In this problem, the goal is to dynamically maintain a subset T of up to n elements from
{1, ..., m}, while supporting insertions, deletions, and a predecessor query Pred(x), which returns the largest element in T that is less than or equal to x. Predecessor search is one of the most well-studied data structure problems. For this problem, using non-adaptivity comes at a steep price. We provide exponential cell probe complexity separations between (i) adaptive and non-adaptive data structures and (ii) non-adaptive and memoryless data structures for predecessor search.
A classic data structure of van Emde Boas solves dynamic predecessor search in log(log(m)) probes; this data structure is adaptive. For dynamic data structures which make non-adaptive updates, we show the cell probe complexity is O(log(m)/log(w/log(m))). We also give a nearly-matching Omega(log(m)/log(w)) lower bound. We also give an m/w lower bound for memoryless data structures.
Our lower bound technique is tailored to non-adaptive (as opposed to memoryless) updates and might be of independent interest.

Joseph Boninger, Joshua Brody, and Owen Kephart. Non-Adaptive Data Structure Bounds for Dynamic Predecessor. In 37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 93, pp. 20:1-20:12, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{boninger_et_al:LIPIcs.FSTTCS.2017.20, author = {Boninger, Joseph and Brody, Joshua and Kephart, Owen}, title = {{Non-Adaptive Data Structure Bounds for Dynamic Predecessor}}, booktitle = {37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2017)}, pages = {20:1--20:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-055-2}, ISSN = {1868-8969}, year = {2018}, volume = {93}, editor = {Lokam, Satya and Ramanujam, R.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2017.20}, URN = {urn:nbn:de:0030-drops-83892}, doi = {10.4230/LIPIcs.FSTTCS.2017.20}, annote = {Keywords: dynamic data structures, lower bounds, predecessor search, non-adaptivity} }

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

We initiate a study of a relaxed version of the standard Erdos-Renyi random graph model, where each edge may depend on a few other edges. We call such graphs "dependent random graphs". Our main result in this direction is a thorough understanding of the clique number of dependent random graphs. We also obtain bounds for the chromatic number. Surprisingly, many of the standard properties of random graphs also hold in this relaxed setting. We show that with high probability, a dependent random graph will contain a clique of size ((1-o(1))log(n))/log(1/p), and the chromatic number will be at most (nlog(1/(1-p)))/log(n). We expect these results to be of independent interest. As an application and second main result, we give a new communication protocol for the k-player Multi-Party Pointer Jumping problem (MPJk) in the number-on-the-forehead (NOF) model. Multi-Party Pointer Jumping is one of the canonical NOF communication problems, yet even for three players, its communication complexity is not well understood. Our protocol for MPJ3 costs O((n * log(log(n)))/log(n)) communication, improving on a bound from [BrodyChakrabarti08]. We extend our protocol to the non-Boolean pointer jumping problem, achieving an upper bound which is o(n) for any k >= 4 players. This is the first o(n) protocol and improves on a bound of Damm, Jukna, and Sgall, which has stood for almost twenty years.

Joshua Brody and Mario Sanchez. Dependent Random Graphs and Multi-Party Pointer Jumping. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 606-624, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{brody_et_al:LIPIcs.APPROX-RANDOM.2015.606, author = {Brody, Joshua and Sanchez, Mario}, title = {{Dependent Random Graphs and Multi-Party Pointer Jumping}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)}, pages = {606--624}, 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.2015.606}, URN = {urn:nbn:de:0030-drops-53266}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2015.606}, annote = {Keywords: random graphs, communication complexity, number-on-the-forehead model, pointer jumping} }

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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} }

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

The EQUALITY problem is usually one’s first encounter with communication complexity and is one of the most fundamental problems in the field. Although its deterministic and randomized communication complexity were settled decades ago, we find several new things to say about the problem by focusing on three subtle aspects. The first is to consider the expected communication cost (at a worst-case input) for a protocol that uses limited interaction—i.e., a bounded number of rounds of communication—and whose error probability is zero or close to it. The second is to treat the false negative error rate separately from the false positive error rate. The third is to consider the information cost of such protocols. We obtain asymptotically optimal rounds-versus-cost tradeoffs for EQUALITY: both expected communication cost and information cost scale as Theta(log log ... log n), with r-1 logs, where r is the number of rounds. These bounds hold even when the false negative rate approaches 1. For the case of zero-error communication cost, we obtain essentially matching bounds, up to a tiny additive constant. We also provide some applications.

Joshua Brody, Amit Chakrabarti, Ranganath Kondapally, David P. Woodruff, and Grigory Yaroslavtsev. Certifying Equality With Limited Interaction. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 28, pp. 545-581, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{brody_et_al:LIPIcs.APPROX-RANDOM.2014.545, author = {Brody, Joshua and Chakrabarti, Amit and Kondapally, Ranganath and Woodruff, David P. and Yaroslavtsev, Grigory}, title = {{Certifying Equality With Limited Interaction}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)}, pages = {545--581}, 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.2014.545}, URN = {urn:nbn:de:0030-drops-47229}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2014.545}, annote = {Keywords: equality, communication complexity, information complexity} }

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**Published in:** LIPIcs, Volume 1, 25th International Symposium on Theoretical Aspects of Computer Science (2008)

We study the one-way number-on-the-forehead (NOF) communication
complexity of the $k$-layer pointer jumping problem with $n$
vertices per layer. This classic problem, which has connections to
many aspects of complexity theory, has seen a recent burst of
research activity, seemingly preparing the ground for an
$Omega(n)$ lower bound, for constant $k$. Our first result is a
surprising sublinear --- i.e., $o(n)$ --- upper bound for the
problem that holds for $k ge 3$, dashing hopes for such a lower
bound.
A closer look at the protocol achieving the upper bound shows that
all but one of the players involved are collapsing, i.e., their
messages depend only on the composition of the layers ahead of
them. We consider protocols for the pointer jumping problem where
all players are collapsing. Our second result shows that a strong
$n - O(log n)$ lower bound does hold in this case. Our third
result is another upper bound showing that nontrivial protocols for
(a non-Boolean version of) pointer jumping are possible even when
all players are collapsing.
Our lower bound result uses a novel proof technique, different from
those of earlier lower bounds that had an information-theoretic
flavor. We hope this is useful in further study of the problem.

Joshua Brody and Amit Chakrabarti. Sublinear Communication Protocols for Multi-Party Pointer Jumping and a Related Lower Bound. In 25th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 1, pp. 145-156, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2008)

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@InProceedings{brody_et_al:LIPIcs.STACS.2008.1341, author = {Brody, Joshua and Chakrabarti, Amit}, title = {{Sublinear Communication Protocols for Multi-Party Pointer Jumping and a Related Lower Bound}}, booktitle = {25th International Symposium on Theoretical Aspects of Computer Science}, pages = {145--156}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-06-4}, ISSN = {1868-8969}, year = {2008}, volume = {1}, editor = {Albers, Susanne and Weil, Pascal}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2008.1341}, URN = {urn:nbn:de:0030-drops-13415}, doi = {10.4230/LIPIcs.STACS.2008.1341}, annote = {Keywords: Communication complexity, pointer jumping, number on the forehead} }

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