8 Search Results for "Sherif, Suhail"

Document
DOI: 10.4230/LIPIcs.ITCS.2024.58
An Improved Protocol for ExactlyN with More Than 3 Players

Authors: Lianna Hambardzumyan, Toniann Pitassi, Suhail Sherif, Morgan Shirley, and Adi Shraibman

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

Abstract
The ExactlyN problem in the number-on-forehead (NOF) communication setting asks k players, each of whom can see every input but their own, if the k input numbers add up to N. Introduced by Chandra, Furst and Lipton in 1983, ExactlyN is important for its role in understanding the strength of randomness in communication complexity with many players. It is also tightly connected to the field of combinatorics: its k-party NOF communication complexity is related to the size of the largest corner-free subset in [N]^{k-1}. In 2021, Linial and Shraibman gave more efficient protocols for ExactlyN for 3 players. As an immediate consequence, this also gave a new construction of larger corner-free subsets in [N]². Later that year Green gave a further refinement to their argument. These results represent the first improvements to the highest-order term for k = 3 since the famous work of Behrend in 1946. In this paper we give a corresponding improvement to the highest-order term for k > 3, the first since Rankin in 1961. That is, we give a more efficient protocol for ExactlyN as well as larger corner-free sets in higher dimensions. Nearly all previous results in this line of research approached the problem from the combinatorics perspective, implicitly resulting in non-constructive protocols for ExactlyN. Approaching the problem from the communication complexity point of view and constructing explicit protocols for ExactlyN was key to the improvements in the k = 3 setting. As a further contribution we provide explicit protocols for ExactlyN for any number of players which serves as a base for our improvement.
Cite as

Lianna Hambardzumyan, Toniann Pitassi, Suhail Sherif, Morgan Shirley, and Adi Shraibman. An Improved Protocol for ExactlyN with More Than 3 Players. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 58:1-58:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{hambardzumyan_et_al:LIPIcs.ITCS.2024.58,
  author =	{Hambardzumyan, Lianna and Pitassi, Toniann and Sherif, Suhail and Shirley, Morgan and Shraibman, Adi},
  title =	{{An Improved Protocol for ExactlyN with More Than 3 Players}},
  booktitle =	{15th Innovations in Theoretical Computer Science Conference (ITCS 2024)},
  pages =	{58:1--58:23},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.58},
  URN =		{urn:nbn:de:0030-drops-195868},
  doi =		{10.4230/LIPIcs.ITCS.2024.58},
  annote =	{Keywords: Corner-free sets, number-on-forehead communication}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2023.33
Lifting to Parity Decision Trees via Stifling

Authors: Arkadev Chattopadhyay, Nikhil S. Mande, Swagato Sanyal, and Suhail Sherif

Published in: LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)

Abstract
We show that the deterministic decision tree complexity of a (partial) function or relation f lifts to the deterministic parity decision tree (PDT) size complexity of the composed function/relation f∘g as long as the gadget g satisfies a property that we call stifling. We observe that several simple gadgets of constant size, like Indexing on 3 input bits, Inner Product on 4 input bits, Majority on 3 input bits and random functions, satisfy this property. It can be shown that existing randomized communication lifting theorems ([Göös, Pitassi, Watson. SICOMP'20], [Chattopadhyay et al. SICOMP'21]) imply PDT-size lifting. However there are two shortcomings of this approach: first they lift randomized decision tree complexity of f, which could be exponentially smaller than its deterministic counterpart when either f is a partial function or even a total search problem. Second, the size of the gadgets in such lifting theorems are as large as logarithmic in the size of the input to f. Reducing the gadget size to a constant is an important open problem at the frontier of current research. Our result shows that even a random constant-size gadget does enable lifting to PDT size. Further, it also yields the first systematic way of turning lower bounds on the width of tree-like resolution proofs of the unsatisfiability of constant-width CNF formulas to lower bounds on the size of tree-like proofs in the resolution with parity system, i.e., Res(⊕), of the unsatisfiability of closely related constant-width CNF formulas.
Cite as

Arkadev Chattopadhyay, Nikhil S. Mande, Swagato Sanyal, and Suhail Sherif. Lifting to Parity Decision Trees via Stifling. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 33:1-33:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{chattopadhyay_et_al:LIPIcs.ITCS.2023.33,
  author =	{Chattopadhyay, Arkadev and Mande, Nikhil S. and Sanyal, Swagato and Sherif, Suhail},
  title =	{{Lifting to Parity Decision Trees via Stifling}},
  booktitle =	{14th Innovations in Theoretical Computer Science Conference (ITCS 2023)},
  pages =	{33:1--33:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-263-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{251},
  editor =	{Tauman Kalai, Yael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.33},
  URN =		{urn:nbn:de:0030-drops-175362},
  doi =		{10.4230/LIPIcs.ITCS.2023.33},
  annote =	{Keywords: Decision trees, parity decision trees, lifting theorems}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2023.89
The Strength of Equality Oracles in Communication

Authors: Toniann Pitassi, Morgan Shirley, and Adi Shraibman

Published in: LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)

Abstract
It is well-known that randomized communication protocols are more powerful than deterministic protocols. In particular the Equality function requires Ω(n) deterministic communication complexity but has efficient randomized protocols. Previous work of Chattopadhyay, Lovett and Vinyals shows that randomized communication is strictly stronger than what can be solved by deterministic protocols equipped with an Equality oracle. Despite this separation, we are far from understanding the exact strength of Equality oracles in the context of communication complexity. In this work we focus on nondeterminisic communication equipped with an Equality oracle, which is a subclass of Merlin-Arthur communication. We show that this inclusion is strict by proving that the previously-studied Integer Inner Product function, which can be efficiently computed even with bounded-error randomness, cannot be computed using sublinear communication in the nondeterministic Equality model. To prove this we give a new matrix-theoretic characterization of the nondeterministic Equality model: specifically, there is a tight connection between this model and a covering number based on the blocky matrices of Hambardzumyan, Hatami, and Hatami, as well as a natural variant of the Gamma-2 factorization norm. Similar equivalences are shown for the unambiguous nondeterministic model with Equality oracles. A bonus result arises from these proofs: for the studied communication models, a single Equality oracle call suffices without loss of generality. Our results allow us to prove a separation between deterministic and unambiguous nondeterminism in the presence of Equality oracles. This stands in contrast to the result of Yannakakis which shows that these models are polynomially-related without oracles. We suggest a number of intriguing open questions along this direction of inquiry, as well as others that arise from our work.
Cite as

Toniann Pitassi, Morgan Shirley, and Adi Shraibman. The Strength of Equality Oracles in Communication. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 89:1-89:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{pitassi_et_al:LIPIcs.ITCS.2023.89,
  author =	{Pitassi, Toniann and Shirley, Morgan and Shraibman, Adi},
  title =	{{The Strength of Equality Oracles in Communication}},
  booktitle =	{14th Innovations in Theoretical Computer Science Conference (ITCS 2023)},
  pages =	{89:1--89:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-263-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{251},
  editor =	{Tauman Kalai, Yael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.89},
  URN =		{urn:nbn:de:0030-drops-175927},
  doi =		{10.4230/LIPIcs.ITCS.2023.89},
  annote =	{Keywords: Factorization norm, blocky rank, Merlin-Arthur}
}
Document
DOI: 10.4230/LIPIcs.STACS.2022.49
One-Way Communication Complexity and Non-Adaptive Decision Trees

Authors: Nikhil S. Mande, Swagato Sanyal, and Suhail Sherif

Published in: LIPIcs, Volume 219, 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)

Abstract
We study the relationship between various one-way communication complexity measures of a composed function with the analogous decision tree complexity of the outer function. We consider two gadgets: the AND function on 2 inputs, and the Inner Product on a constant number of inputs. More generally, we show the following when the gadget is Inner Product on 2b input bits for all b ≥ 2, denoted IP. - If f is a total Boolean function that depends on all of its n input bits, then the bounded-error one-way quantum communication complexity of f∘IP equals Ω(n(b-1)). - If f is a partial Boolean function, then the deterministic one-way communication complexity of f∘IP is at least Ω(b ⋅ 𝖣_{dt}^ → (f)), where 𝖣_{dt}^ → (f) denotes non-adaptive decision tree complexity of f. To prove our quantum lower bound, we first show a lower bound on the VC-dimension of f∘IP. We then appeal to a result of Klauck [STOC'00], which immediately yields our quantum lower bound. Our deterministic lower bound relies on a combinatorial result independently proven by Ahlswede and Khachatrian [Adv. Appl. Math.'98], and Frankl and Tokushige [Comb.'99]. It is known due to a result of Montanaro and Osborne [arXiv'09] that the deterministic one-way communication complexity of f∘XOR equals the non-adaptive parity decision tree complexity of f. In contrast, we show the following when the inner gadget is the AND function on 2 input bits. - There exists a function for which even the quantum non-adaptive AND decision tree complexity of f is exponentially large in the deterministic one-way communication complexity of f∘AND. - However, for symmetric functions f, the non-adaptive AND decision tree complexity of f is at most quadratic in the (even two-way) communication complexity of f∘AND. In view of the first bullet, a lower bound on non-adaptive AND decision tree complexity of f does not lift to a lower bound on one-way communication complexity of f∘AND. The proof of the first bullet above uses the well-studied Odd-Max-Bit function. For the second bullet, we first observe a connection between the one-way communication complexity of f and the Möbius sparsity of f, and then give a lower bound on the Möbius sparsity of symmetric functions. An upper bound on the non-adaptive AND decision tree complexity of symmetric functions follows implicitly from prior work on combinatorial group testing; for the sake of completeness, we include a proof of this result. It is well known that the rank of the communication matrix of a function F is an upper bound on its deterministic one-way communication complexity. This bound is known to be tight for some F. However, in our final result we show that this is not the case when F = f∘AND. More precisely we show that for all f, the deterministic one-way communication complexity of F = f∘AND is at most (rank(M_{F}))(1 - Ω(1)), where M_{F} denotes the communication matrix of F.
Cite as

Nikhil S. Mande, Swagato Sanyal, and Suhail Sherif. One-Way Communication Complexity and Non-Adaptive Decision Trees. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 49:1-49:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{mande_et_al:LIPIcs.STACS.2022.49,
  author =	{Mande, Nikhil S. and Sanyal, Swagato and Sherif, Suhail},
  title =	{{One-Way Communication Complexity and Non-Adaptive Decision Trees}},
  booktitle =	{39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
  pages =	{49:1--49:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-222-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{219},
  editor =	{Berenbrink, Petra and Monmege, Benjamin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.49},
  URN =		{urn:nbn:de:0030-drops-158598},
  doi =		{10.4230/LIPIcs.STACS.2022.49},
  annote =	{Keywords: Decision trees, communication complexity, composed Boolean functions}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2021.13
Towards Stronger Counterexamples to the Log-Approximate-Rank Conjecture

Authors: Arkadev Chattopadhyay, Ankit Garg, and Suhail Sherif

Published in: LIPIcs, Volume 213, 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)

Abstract
We give improved separations for the query complexity analogue of the log-approximate-rank conjecture i.e. we show that there are a plethora of total Boolean functions on n input bits, each of which has approximate Fourier sparsity at most O(n³) and randomized parity decision tree complexity Θ(n). This improves upon the recent work of Chattopadhyay, Mande and Sherif [Chattopadhyay et al., 2020] both qualitatively (in terms of designing a large number of examples) and quantitatively (shrinking the gap from quartic to cubic). We leave open the problem of proving a randomized communication complexity lower bound for XOR compositions of our examples. A linear lower bound would lead to new and improved refutations of the log-approximate-rank conjecture. Moreover, if any of these compositions had even a sub-linear cost randomized communication protocol, it would demonstrate that randomized parity decision tree complexity does not lift to randomized communication complexity in general (with the XOR gadget).
Cite as

Arkadev Chattopadhyay, Ankit Garg, and Suhail Sherif. Towards Stronger Counterexamples to the Log-Approximate-Rank Conjecture. In 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 213, pp. 13:1-13:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{chattopadhyay_et_al:LIPIcs.FSTTCS.2021.13,
  author =	{Chattopadhyay, Arkadev and Garg, Ankit and Sherif, Suhail},
  title =	{{Towards Stronger Counterexamples to the Log-Approximate-Rank Conjecture}},
  booktitle =	{41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)},
  pages =	{13:1--13:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-215-0},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{213},
  editor =	{Boja\'{n}czyk, Miko{\l}aj and Chekuri, Chandra},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2021.13},
  URN =		{urn:nbn:de:0030-drops-155245},
  doi =		{10.4230/LIPIcs.FSTTCS.2021.13},
  annote =	{Keywords: Approximate Rank, Randomized Parity Decision Trees, Randomized Communication Complexity, XOR functions, Subspace Designs}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2021.53
No Quantum Speedup over Gradient Descent for Non-Smooth Convex Optimization

Authors: Ankit Garg, Robin Kothari, Praneeth Netrapalli, and Suhail Sherif

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

Abstract
We study the first-order convex optimization problem, where we have black-box access to a (not necessarily smooth) function f:ℝⁿ → ℝ and its (sub)gradient. Our goal is to find an ε-approximate minimum of f starting from a point that is distance at most R from the true minimum. If f is G-Lipschitz, then the classic gradient descent algorithm solves this problem with O((GR/ε)²) queries. Importantly, the number of queries is independent of the dimension n and gradient descent is optimal in this regard: No deterministic or randomized algorithm can achieve better complexity that is still independent of the dimension n. In this paper we reprove the randomized lower bound of Ω((GR/ε)²) using a simpler argument than previous lower bounds. We then show that although the function family used in the lower bound is hard for randomized algorithms, it can be solved using O(GR/ε) quantum queries. We then show an improved lower bound against quantum algorithms using a different set of instances and establish our main result that in general even quantum algorithms need Ω((GR/ε)²) queries to solve the problem. Hence there is no quantum speedup over gradient descent for black-box first-order convex optimization without further assumptions on the function family.
Cite as

Ankit Garg, Robin Kothari, Praneeth Netrapalli, and Suhail Sherif. No Quantum Speedup over Gradient Descent for Non-Smooth Convex Optimization. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 53:1-53:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{garg_et_al:LIPIcs.ITCS.2021.53,
  author =	{Garg, Ankit and Kothari, Robin and Netrapalli, Praneeth and Sherif, Suhail},
  title =	{{No Quantum Speedup over Gradient Descent for Non-Smooth Convex Optimization}},
  booktitle =	{12th Innovations in Theoretical Computer Science Conference (ITCS 2021)},
  pages =	{53:1--53:20},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.53},
  URN =		{urn:nbn:de:0030-drops-135921},
  doi =		{10.4230/LIPIcs.ITCS.2021.53},
  annote =	{Keywords: Quantum algorithms, Gradient descent, Convex optimization}
}
Document
DOI: 10.4230/LIPIcs.CCC.2019.14
Equality Alone Does not Simulate Randomness

Authors: Arkadev Chattopadhyay, Shachar Lovett, and Marc Vinyals

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

Abstract
The canonical problem that gives an exponential separation between deterministic and randomized communication complexity in the classical two-party communication model is "Equality". In this work we show that even allowing access to an "Equality" oracle, deterministic protocols remain exponentially weaker than randomized ones. More precisely, we exhibit a total function on n bits with randomized one-sided communication complexity O(log n), but such that every deterministic protocol with access to "Equality" oracle needs Omega(n) cost to compute it. Additionally we exhibit a natural and strict infinite hierarchy within BPP, starting with the class P^{EQ} at its bottom.
Cite as

Arkadev Chattopadhyay, Shachar Lovett, and Marc Vinyals. Equality Alone Does not Simulate Randomness. In 34th Computational Complexity Conference (CCC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 137, pp. 14:1-14:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{chattopadhyay_et_al:LIPIcs.CCC.2019.14,
  author =	{Chattopadhyay, Arkadev and Lovett, Shachar and Vinyals, Marc},
  title =	{{Equality Alone Does not Simulate Randomness}},
  booktitle =	{34th Computational Complexity Conference (CCC 2019)},
  pages =	{14:1--14:11},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2019.14},
  URN =		{urn:nbn:de:0030-drops-108368},
  doi =		{10.4230/LIPIcs.CCC.2019.14},
  annote =	{Keywords: Communication lower bound, derandomization}
}
Document
DOI: 10.4230/LIPIcs.STACS.2019.50
Lifting Theorems for Equality

Authors: Bruno Loff and Sagnik Mukhopadhyay

Published in: LIPIcs, Volume 126, 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)

Abstract
We show a deterministic simulation (or lifting) theorem for composed problems f o Eq_n where the inner function (the gadget) is Equality on n bits. When f is a total function on p bits, it is easy to show via a rank argument that the communication complexity of f o Eq_n is Omega(deg(f) * n). However, there is a surprising counter-example of a partial function f on p bits, such that any completion f' of f has deg(f') = Omega(p), and yet f o Eq_n has communication complexity O(n). Nonetheless, we are able to show that the communication complexity of f o Eq_n is at least D(f) * n for a complexity measure D(f) which is closely related to the AND-query complexity of f and is lower-bounded by the logarithm of the leaf complexity of f. As a corollary, we also obtain lifting theorems for the set-disjointness gadget, and a lifting theorem in the context of parity decision-trees, for the NOR gadget. As an application, we prove a tight lower-bound for the deterministic communication complexity of the communication problem, where Alice and Bob are each given p-many n-bit strings, with the promise that either all of the strings are distinct, or all-but-one of the strings are distinct, and they wish to know which is the case. We show that the complexity of this problem is Theta(p * n).
Cite as

Bruno Loff and Sagnik Mukhopadhyay. Lifting Theorems for Equality. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 50:1-50:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{loff_et_al:LIPIcs.STACS.2019.50,
  author =	{Loff, Bruno and Mukhopadhyay, Sagnik},
  title =	{{Lifting Theorems for Equality}},
  booktitle =	{36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)},
  pages =	{50:1--50:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-100-9},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{126},
  editor =	{Niedermeier, Rolf and Paul, Christophe},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2019.50},
  URN =		{urn:nbn:de:0030-drops-102892},
  doi =		{10.4230/LIPIcs.STACS.2019.50},
  annote =	{Keywords: Communication complexity, Query complexity, Simulation theorem, Equality function}
}
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