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**Published in:** LIPIcs, Volume 153, 23rd International Conference on Principles of Distributed Systems (OPODIS 2019)

In this work, we propose FLECKS, an algorithm which implements atomic memory objects in a multi-writer multi-reader (MWMR) setting in asynchronous networks and server failures. FLECKS substantially reduces storage and communication costs over its replication-based counterparts by employing erasure-codes. FLECKS outperforms the previously proposed algorithms in terms of the metrics that to deliver good performance such as storage cost per object, communication cost a high fault-tolerance of clients and servers, guaranteed liveness of operation, and a given number of communication rounds per operation, etc. We provide proofs for liveness and atomicity properties of FLECKS and derive worst-case latency bounds for the operations. We implemented and deployed FLECKS in cloud-based clusters and demonstrate that FLECKS has substantially lower storage and bandwidth costs, and significantly lower latency of operations than the replication-based mechanisms.

Kishori M. Konwar, N. Prakash, Muriel Médard, and Nancy Lynch. Fast Lean Erasure-Coded Atomic Memory Object. In 23rd International Conference on Principles of Distributed Systems (OPODIS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 153, pp. 12:1-12:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{konwar_et_al:LIPIcs.OPODIS.2019.12, author = {Konwar, Kishori M. and Prakash, N. and M\'{e}dard, Muriel and Lynch, Nancy}, title = {{Fast Lean Erasure-Coded Atomic Memory Object}}, booktitle = {23rd International Conference on Principles of Distributed Systems (OPODIS 2019)}, pages = {12:1--12:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-133-7}, ISSN = {1868-8969}, year = {2020}, volume = {153}, editor = {Felber, Pascal and Friedman, Roy and Gilbert, Seth and Miller, Avery}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.OPODIS.2019.12}, URN = {urn:nbn:de:0030-drops-117988}, doi = {10.4230/LIPIcs.OPODIS.2019.12}, annote = {Keywords: Atomicity, Distributed Storage System, Erasure-codes} }

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**Published in:** LIPIcs, Volume 70, 20th International Conference on Principles of Distributed Systems (OPODIS 2016)

Erasure codes offer an efficient way to decrease storage and communication costs while implementing atomic memory service in asynchronous distributed storage systems. In this paper, we provide erasure-code-based algorithms having the additional ability to perform background repair of crashed nodes. A repair operation of a node in the crashed state is triggered externally, and is carried out by the concerned node via message exchanges with other active nodes in the system. Upon completion of repair, the node re-enters active state, and resumes participation in ongoing and future read, write, and repair operations. To guarantee liveness and atomicity simultaneously, existing works assume either the presence of nodes with stable storage, or presence of nodes that never crash during the execution. We demand neither of these; instead we consider a natural, yet practical network stability condition N1 that only restricts the number of nodes in the crashed/repair state during broadcast of any message.
We present an erasure-code based algorithm RADON_{C} that is always live, and guarantees atomicity as long as condition N1 holds. In situations when the number of concurrent writes is limited, RADON_{C} has significantly improved storage and communication cost over a replication-based algorithm RADON_{R}, which also works under N1. We further show how a slightly stronger network stability condition N2 can be used to construct algorithms that never violate atomicity. The guarantee of atomicity comes at the expense of having an additional phase during the read and write operations.

Kishori M. Konwar, N. Prakash, Nancy A. Lynch, and Muriel Médard. RADON: Repairable Atomic Data Object in Networks. In 20th International Conference on Principles of Distributed Systems (OPODIS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 70, pp. 28:1-28:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{konwar_et_al:LIPIcs.OPODIS.2016.28, author = {Konwar, Kishori M. and Prakash, N. and Lynch, Nancy A. and M\'{e}dard, Muriel}, title = {{RADON: Repairable Atomic Data Object in Networks}}, booktitle = {20th International Conference on Principles of Distributed Systems (OPODIS 2016)}, pages = {28:1--28:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-031-6}, ISSN = {1868-8969}, year = {2017}, volume = {70}, editor = {Fatourou, Panagiota and Jim\'{e}nez, Ernesto and Pedone, Fernando}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.OPODIS.2016.28}, URN = {urn:nbn:de:0030-drops-70970}, doi = {10.4230/LIPIcs.OPODIS.2016.28}, annote = {Keywords: Atomicity, repair, fault-tolerance, storage cost, erasure codes} }

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**Published in:** LIPIcs, Volume 112, 26th Annual European Symposium on Algorithms (ESA 2018)

Let F_{q} be the finite field of size q and let l: F_{q}^{n} -> F_{q} be a linear function. We introduce the Learning From Subset problem LFS(q,n,d) of learning l, given samples u in F_{q}^{n} from a special distribution depending on l: the probability of sampling u is a function of l(u) and is non zero for at most d values of l(u). We provide a randomized algorithm for LFS(q,n,d) with sample complexity (n+d)^{O(d)} and running time polynomial in log q and (n+d)^{O(d)}. Our algorithm generalizes and improves upon previous results [Friedl et al., 2014; Gábor Ivanyos, 2008] that had provided algorithms for LFS(q,n,q-1) with running time (n+q)^{O(q)}. We further present applications of our result to the Hidden Multiple Shift problem HMS(q,n,r) in quantum computation where the goal is to determine the hidden shift s given oracle access to r shifted copies of an injective function f: Z_{q}^{n} -> {0, 1}^{l}, that is we can make queries of the form f_{s}(x,h) = f(x-hs) where h can assume r possible values. We reduce HMS(q,n,r) to LFS(q,n, q-r+1) to obtain a polynomial time algorithm for HMS(q,n,r) when q=n^{O(1)} is prime and q-r=O(1). The best known algorithms [Andrew M. Childs and Wim van Dam, 2007; Friedl et al., 2014] for HMS(q,n,r) with these parameters require exponential time.

Gábor Ivanyos, Anupam Prakash, and Miklos Santha. On Learning Linear Functions from Subset and Its Applications in Quantum Computing. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 66:1-66:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{ivanyos_et_al:LIPIcs.ESA.2018.66, author = {Ivanyos, G\'{a}bor and Prakash, Anupam and Santha, Miklos}, title = {{On Learning Linear Functions from Subset and Its Applications in Quantum Computing}}, booktitle = {26th Annual European Symposium on Algorithms (ESA 2018)}, pages = {66:1--66:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-081-1}, ISSN = {1868-8969}, year = {2018}, volume = {112}, editor = {Azar, Yossi and Bast, Hannah and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2018.66}, URN = {urn:nbn:de:0030-drops-95299}, doi = {10.4230/LIPIcs.ESA.2018.66}, annote = {Keywords: Learning from subset, hidden shift problem, quantum algorithms, linearization} }

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

A recommendation system uses the past purchases or ratings of n products by a group of m users, in order to provide personalized recommendations to individual users. The information is modeled as an m \times n preference matrix which is assumed to have a good rank-k approximation, for a small constant k.
In this work, we present a quantum algorithm for recommendation systems that has running time O(\text{poly}(k)\text{polylog}(mn)). All known classical algorithms for recommendation systems that work through reconstructing an approximation of the preference matrix run in time polynomial in the matrix dimension. Our algorithm provides good recommendations by sampling efficiently from an approximation of the preference matrix, without reconstructing the entire matrix. For this, we design an efficient quantum procedure to project a given vector onto the row space of a given matrix. This is the first algorithm for recommendation systems that runs in time polylogarithmic in the dimensions of the matrix and provides an example of a quantum machine learning algorithm for a real world application.

Iordanis Kerenidis and Anupam Prakash. Quantum Recommendation Systems. In 8th Innovations in Theoretical Computer Science Conference (ITCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 67, pp. 49:1-49:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{kerenidis_et_al:LIPIcs.ITCS.2017.49, author = {Kerenidis, Iordanis and Prakash, Anupam}, title = {{Quantum Recommendation Systems}}, booktitle = {8th Innovations in Theoretical Computer Science Conference (ITCS 2017)}, pages = {49:1--49:21}, 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.2017.49}, URN = {urn:nbn:de:0030-drops-81541}, doi = {10.4230/LIPIcs.ITCS.2017.49}, annote = {Keywords: Recommendation systems, quantum machine learning, singular value estimation, matrix sampling, quantum algorithms.} }

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**Published in:** LIPIcs, Volume 50, 31st Conference on Computational Complexity (CCC 2016)

We study how well functions over the boolean hypercube of the form f_k(x)=(|x|-k)(|x|-k-1) can be approximated by sums of squares of low-degree polynomials, obtaining good bounds for the case of approximation in l_{infinity}-norm as well as in l_1-norm. We describe three complexity-theoretic applications: (1) a proof that the recent breakthrough lower bound of Lee, Raghavendra, and Steurer [Lee/Raghavendra/Steurer, STOC 2015] on the positive semidefinite extension complexity of the correlation and TSP polytopes cannot be improved further by showing better sum-of-squares degree lower bounds on l_1-approximation of f_k; (2) a proof that Grigoriev's lower bound on the degree of Positivstellensatz refutations for the knapsack problem is optimal, answering an open question from [Grigoriev, Comp. Compl. 2001]; (3) bounds on the query complexity of quantum algorithms whose expected output approximates such functions.

Troy Lee, Anupam Prakash, Ronald de Wolf, and Henry Yuen. On the Sum-of-Squares Degree of Symmetric Quadratic Functions. In 31st Conference on Computational Complexity (CCC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 50, pp. 17:1-17:31, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{lee_et_al:LIPIcs.CCC.2016.17, author = {Lee, Troy and Prakash, Anupam and de Wolf, Ronald and Yuen, Henry}, title = {{On the Sum-of-Squares Degree of Symmetric Quadratic Functions}}, booktitle = {31st Conference on Computational Complexity (CCC 2016)}, pages = {17:1--17:31}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-008-8}, ISSN = {1868-8969}, year = {2016}, volume = {50}, editor = {Raz, Ran}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2016.17}, URN = {urn:nbn:de:0030-drops-58383}, doi = {10.4230/LIPIcs.CCC.2016.17}, annote = {Keywords: Sum-of-squares degree, approximation theory, Positivstellensatz refutations of knapsack, quantum query complexity in expectation, extension complexity} }