10 Search Results for "Brody, Joshua"


Document
A Technique for Hardness Amplification Against AC⁰

Authors: William M. Hoza

Published in: LIPIcs, Volume 300, 39th Computational Complexity Conference (CCC 2024)


Abstract
We study hardness amplification in the context of two well-known "moderate" average-case hardness results for AC⁰ circuits. First, we investigate the extent to which AC⁰ circuits of depth d can approximate AC⁰ circuits of some larger depth d + k. The case k = 1 is resolved by Håstad, Rossman, Servedio, and Tan’s celebrated average-case depth hierarchy theorem (JACM 2017). Our contribution is a significantly stronger correlation bound when k ≥ 3. Specifically, we show that there exists a linear-size AC⁰_{d + k} circuit h : {0, 1}ⁿ → {0, 1} such that for every AC⁰_d circuit g, either g has size exp(n^{Ω(1/d)}), or else g agrees with h on at most a (1/2 + ε)-fraction of inputs where ε = exp(-(1/d) ⋅ Ω(log n)^{k-1}). For comparison, Håstad, Rossman, Servedio, and Tan’s result has ε = n^{-Θ(1/d)}. Second, we consider the majority function. It is well known that the majority function is moderately hard for AC⁰ circuits (and stronger classes). Our contribution is a stronger correlation bound for the XOR of t copies of the n-bit majority function, denoted MAJ_n^{⊕ t}. We show that if g is an AC⁰_d circuit of size S, then g agrees with MAJ_n^{⊕ t} on at most a (1/2 + ε)-fraction of inputs, where ε = (O(log S)^{d - 1} / √n)^t. To prove these results, we develop a hardness amplification technique that is tailored to a specific type of circuit lower bound proof. In particular, one way to show that a function h is moderately hard for AC⁰ circuits is to (a) design some distribution over random restrictions or random projections, (b) show that AC⁰ circuits simplify to shallow decision trees under these restrictions/projections, and finally (c) show that after applying the restriction/projection, h is moderately hard for shallow decision trees with respect to an appropriate distribution. We show that (roughly speaking) if h can be proven to be moderately hard by a proof with that structure, then XORing multiple copies of h amplifies its hardness. Our analysis involves a new kind of XOR lemma for decision trees, which might be of independent interest.

Cite as

William M. Hoza. A Technique for Hardness Amplification Against AC⁰. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 1:1-1:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{hoza:LIPIcs.CCC.2024.1,
  author =	{Hoza, William M.},
  title =	{{A Technique for Hardness Amplification Against AC⁰}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{1:1--1:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-331-7},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{300},
  editor =	{Santhanam, Rahul},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.1},
  URN =		{urn:nbn:de:0030-drops-203977},
  doi =		{10.4230/LIPIcs.CCC.2024.1},
  annote =	{Keywords: Bounded-depth circuits, average-case lower bounds, hardness amplification, XOR lemmas}
}
Document
A Strong Direct Sum Theorem for Distributional Query Complexity

Authors: Guy Blanc, Caleb Koch, Carmen Strassle, and Li-Yang Tan

Published in: LIPIcs, Volume 300, 39th Computational Complexity Conference (CCC 2024)


Abstract
Consider the expected query complexity of computing the k-fold direct product f^{⊗ k} of a function f to error ε with respect to a distribution μ^k. One strategy is to sequentially compute each of the k copies to error ε/k with respect to μ and apply the union bound. We prove a strong direct sum theorem showing that this naive strategy is essentially optimal. In particular, computing a direct product necessitates a blowup in both query complexity and error. Strong direct sum theorems contrast with results that only show a blowup in query complexity or error but not both. There has been a long line of such results for distributional query complexity, dating back to (Impagliazzo, Raz, Wigderson 1994) and (Nisan, Rudich, Saks 1994), but a strong direct sum theorem that holds for all functions in the standard query model had been elusive. A key idea in our work is the first use of the Hardcore Theorem (Impagliazzo 1995) in the context of query complexity. We prove a new resilience lemma that accompanies it, showing that the hardcore of f^{⊗k} is likely to remain dense under arbitrary partitions of the input space.

Cite as

Guy Blanc, Caleb Koch, Carmen Strassle, and Li-Yang Tan. A Strong Direct Sum Theorem for Distributional Query Complexity. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 16:1-16:30, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{blanc_et_al:LIPIcs.CCC.2024.16,
  author =	{Blanc, Guy and Koch, Caleb and Strassle, Carmen and Tan, Li-Yang},
  title =	{{A Strong Direct Sum Theorem for Distributional Query Complexity}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{16:1--16:30},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-331-7},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{300},
  editor =	{Santhanam, Rahul},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.16},
  URN =		{urn:nbn:de:0030-drops-204123},
  doi =		{10.4230/LIPIcs.CCC.2024.16},
  annote =	{Keywords: Query complexity, direct product theorem, hardcore theorem}
}
Document
Track A: Algorithms, Complexity and Games
Optimal Non-Adaptive Cell Probe Dictionaries and Hashing

Authors: Kasper Green Larsen, Rasmus Pagh, Giuseppe Persiano, Toniann Pitassi, Kevin Yeo, and Or Zamir

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)


Abstract
We present a simple and provably optimal non-adaptive cell probe data structure for the static dictionary problem. Our data structure supports storing a set of n key-value pairs from [u]× [u] using s words of space and answering key lookup queries in t = O(lg(u/n)/lg(s/n)) non-adaptive probes. This generalizes a solution to the membership problem (i.e., where no values are associated with keys) due to Buhrman et al. We also present matching lower bounds for the non-adaptive static membership problem in the deterministic setting. Our lower bound implies that both our dictionary algorithm and the preceding membership algorithm are optimal, and in particular that there is an inherent complexity gap in these problems between no adaptivity and one round of adaptivity (with which hashing-based algorithms solve these problems in constant time). Using the ideas underlying our data structure, we also obtain the first implementation of a n-wise independent family of hash functions with optimal evaluation time in the cell probe model.

Cite as

Kasper Green Larsen, Rasmus Pagh, Giuseppe Persiano, Toniann Pitassi, Kevin Yeo, and Or Zamir. Optimal Non-Adaptive Cell Probe Dictionaries and Hashing. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 104:1-104:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{larsen_et_al:LIPIcs.ICALP.2024.104,
  author =	{Larsen, Kasper Green and Pagh, Rasmus and Persiano, Giuseppe and Pitassi, Toniann and Yeo, Kevin and Zamir, Or},
  title =	{{Optimal Non-Adaptive Cell Probe Dictionaries and Hashing}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{104:1--104:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.104},
  URN =		{urn:nbn:de:0030-drops-202471},
  doi =		{10.4230/LIPIcs.ICALP.2024.104},
  annote =	{Keywords: non-adaptive, cell probe, dictionary, hashing}
}
Document
Searching for Cryptogenography Upper Bounds via Sum of Square Programming

Authors: Dominik Scheder, Shuyang Tang, and Jiaheng Zhang

Published in: LIPIcs, Volume 149, 30th International Symposium on Algorithms and Computation (ISAAC 2019)


Abstract
Cryptogenography is a secret-leaking game in which one of n players is holding a secret to be leaked. The n players engage in communication as to (1) reveal the secret while (2) keeping the identity of the secret holder as obscure as possible. All communication is public, and no computational hardness assumptions are made, i.e., the setting is purely information theoretic. Brody, Jakobsen, Scheder, and Winkler [Joshua Brody et al., 2014] formally defined this problem, showed that it has an equivalent geometric characterization, and gave upper and lower bounds for the case in which the n players want to leak a single bit. Surprisingly, even the easiest case, where two players want to leak a secret consisting of a single bit, is not completely understood. Doerr and Künnemann [Benjamin Doerr and Marvin Künnemann, 2016] showed how to automatically search for good protocols using a computer, thus finding an improved protocol for the 1-bit two-player case. In this work, we show how the search for upper bounds (impossibility results) can be formulated as a Sum of Squares program. We implement this idea for the 1-bit two-player case and significantly improve the previous upper bound from 47/128 = 0.3671875 to 0.35183.

Cite as

Dominik Scheder, Shuyang Tang, and Jiaheng Zhang. Searching for Cryptogenography Upper Bounds via Sum of Square Programming. In 30th International Symposium on Algorithms and Computation (ISAAC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 149, pp. 31:1-31:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{scheder_et_al:LIPIcs.ISAAC.2019.31,
  author =	{Scheder, Dominik and Tang, Shuyang and Zhang, Jiaheng},
  title =	{{Searching for Cryptogenography Upper Bounds via Sum of Square Programming}},
  booktitle =	{30th International Symposium on Algorithms and Computation (ISAAC 2019)},
  pages =	{31:1--31:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-130-6},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{149},
  editor =	{Lu, Pinyan and Zhang, Guochuan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2019.31},
  URN =		{urn:nbn:de:0030-drops-115276},
  doi =		{10.4230/LIPIcs.ISAAC.2019.31},
  annote =	{Keywords: Communication Complexity, Secret Leaking, Sum of Squares Programming}
}
Document
Optimal Separation and Strong Direct Sum for Randomized Query Complexity

Authors: Eric Blais and Joshua Brody

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


Abstract
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).

Cite as

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}
}
Document
Non-Adaptive Data Structure Bounds for Dynamic Predecessor

Authors: Joseph Boninger, Joshua Brody, and Owen Kephart

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


Abstract
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.

Cite as

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}
}
Document
Dependent Random Graphs and Multi-Party Pointer Jumping

Authors: Joshua Brody and Mario Sanchez

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


Abstract
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.

Cite as

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}
}
Document
The Information Complexity of Hamming Distance

Authors: Eric Blais, Joshua Brody, and Badih Ghazi

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


Abstract
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.

Cite as

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}
}
Document
Certifying Equality With Limited Interaction

Authors: Joshua Brody, Amit Chakrabarti, Ranganath Kondapally, David P. Woodruff, and Grigory Yaroslavtsev

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


Abstract
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.

Cite as

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}
}
Document
Sublinear Communication Protocols for Multi-Party Pointer Jumping and a Related Lower Bound

Authors: Joshua Brody and Amit Chakrabarti

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


Abstract
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.

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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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