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

We introduce semi-direct sum theorem as a framework for proving asymmetric communication lower bounds for the functions of the form V_{i=1}^n f(x,y_i). Utilizing tools developed in proving direct sum theorem for information complexity, we show that if the function is of the form V_{i=1}^n f(x,y_i) where Alice is given x and Bob is given y_i's, it suffices to prove a lower bound for a single f(x,y_i). This opens a new avenue of attack other than the conventional combinatorial technique (i.e. "richness lemma" from [Miltersen et al., 1995]) for proving randomized lower bounds for asymmetric communication for functions of such form.
As the main technical result and an application of semi-direct sum framework, we prove an information lower bound on c-approximate Nearest Neighbor (ANN) under l_infty which implies that the algorithm of [Indyk, 2001] for c-approximate Nearest Neighbor under l_infty is optimal even under randomization for both decision tree and cell probe data structure model (under certain parameter assumption for the latter). In particular, this shows that randomization cannot improve [Indyk, 2001] under decision tree model. Previously only a deterministic lower bound was known by [Andoni et al., 2008] and randomized lower bound for cell probe model by [Kapralov and Panigrahy, 2012]. We suspect further applications of our framework in exhibiting randomized asymmetric communication lower bounds for big data applications.

Mark Braverman and Young Kun Ko. Semi-Direct Sum Theorem and Nearest Neighbor under l_infty. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 6:1-6:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{braverman_et_al:LIPIcs.APPROX-RANDOM.2018.6, author = {Braverman, Mark and Ko, Young Kun}, title = {{Semi-Direct Sum Theorem and Nearest Neighbor under l\underlineinfty}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)}, pages = {6:1--6:17}, 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.2018.6}, URN = {urn:nbn:de:0030-drops-94101}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.6}, annote = {Keywords: Asymmetric Communication Lower Bound, Data Structure Lower Bound, Nearest Neighbor Search} }

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**Published in:** LIPIcs, Volume 94, 9th Innovations in Theoretical Computer Science Conference (ITCS 2018)

We introduce a generalization of the standard framework for studying the difficulty of two-prover games. Specifically, we study the model where Alice and Bob are allowed to communicate (with information constraints) - in contrast to the usual two-prover game where they are not allowed to communicate after receiving their respective input. We study the trade-off between the information cost of the protocol and the achieved value of the game after the protocol.
In particular, we show the connection of this trade-off and the amortized behavior of the game (i.e. repeated value of the game).
We show that if one can win the game with at least (1 - \epsilon)-probability by communicating at most \epsilon bits of information,
then one can win n copies with probability at least 2^{-O(\epsilon n)}. This gives an intuitive explanation why Raz's counter-example to strong parallel repetition [Raz2008] (the odd cycle game) is a counter-example to strong parallel repetition - one can win the odd-cycle game on a cycle of length $m$ by communicating O(m^{-2})-bits where m is the number of vertices.
Conversely, for projection games, we show that if one can win n copies with probability larger than (1-\epsilon)^n,
then one can win one copy with at least (1 - O(\epsilon))-probability by communicating O(\epsilon) bits of information.
By showing the equivalence between information value and amortized value, we give an alternative direction for further works in studying amortized behavior of the two-prover games.
The main technical tool is the "Chi-Squared Lemma" which bounds the information cost of the protocol in terms of Chi-Squared distance,
instead of usual divergence. This avoids the square loss from using Pinsker's Inequality.

Mark Braverman and Young Kun Ko. Information Value of Two-Prover Games. In 9th Innovations in Theoretical Computer Science Conference (ITCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 94, pp. 12:1-12:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{braverman_et_al:LIPIcs.ITCS.2018.12, author = {Braverman, Mark and Ko, Young Kun}, title = {{Information Value of Two-Prover Games}}, booktitle = {9th Innovations in Theoretical Computer Science Conference (ITCS 2018)}, pages = {12:1--12:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-060-6}, ISSN = {1868-8969}, year = {2018}, volume = {94}, editor = {Karlin, Anna R.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2018.12}, URN = {urn:nbn:de:0030-drops-83466}, doi = {10.4230/LIPIcs.ITCS.2018.12}, annote = {Keywords: Two Prover Game, Parallel Repetition, Odd-Cycle Game, Amortized Value of the Game} }

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