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Complete Volume

**Published in:** LIPIcs, Volume 267, 4th Conference on Information-Theoretic Cryptography (ITC 2023)

LIPIcs, Volume 267, ITC 2023, Complete Volume

4th Conference on Information-Theoretic Cryptography (ITC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 267, pp. 1-358, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@Proceedings{chung:LIPIcs.ITC.2023, title = {{LIPIcs, Volume 267, ITC 2023, Complete Volume}}, booktitle = {4th Conference on Information-Theoretic Cryptography (ITC 2023)}, pages = {1--358}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-271-6}, ISSN = {1868-8969}, year = {2023}, volume = {267}, editor = {Chung, Kai-Min}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITC.2023}, URN = {urn:nbn:de:0030-drops-183272}, doi = {10.4230/LIPIcs.ITC.2023}, annote = {Keywords: LIPIcs, Volume 267, ITC 2023, Complete Volume} }

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Front Matter

**Published in:** LIPIcs, Volume 267, 4th Conference on Information-Theoretic Cryptography (ITC 2023)

Front Matter, Table of Contents, Preface, Conference Organization

4th Conference on Information-Theoretic Cryptography (ITC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 267, pp. 0:i-0:xii, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{chung:LIPIcs.ITC.2023.0, author = {Chung, Kai-Min}, title = {{Front Matter, Table of Contents, Preface, Conference Organization}}, booktitle = {4th Conference on Information-Theoretic Cryptography (ITC 2023)}, pages = {0:i--0:xii}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-271-6}, ISSN = {1868-8969}, year = {2023}, volume = {267}, editor = {Chung, Kai-Min}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITC.2023.0}, URN = {urn:nbn:de:0030-drops-183280}, doi = {10.4230/LIPIcs.ITC.2023.0}, annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization} }

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**Published in:** LIPIcs, Volume 264, 38th Computational Complexity Conference (CCC 2023)

Hamiltonian simulation is one of the most important problems in the field of quantum computing. There have been extended efforts on designing algorithms for faster simulation, and the evolution time T for the simulation greatly affect algorithm runtime as expected. While there are some specific types of Hamiltonians that can be fast-forwarded, i.e., simulated within time o(T), for some large classes of Hamiltonians (e.g., all local/sparse Hamiltonians), existing simulation algorithms require running time at least linear in the evolution time T. On the other hand, while there exist lower bounds of Ω(T) circuit size for some large classes of Hamiltonian, these lower bounds do not rule out the possibilities of Hamiltonian simulation with large but "low-depth" circuits by running things in parallel. As a result, physical systems with system size scaling with T can potentially do a fast-forwarding simulation. Therefore, it is intriguing whether we can achieve fast Hamiltonian simulation with the power of parallelism.
In this work, we give a negative result for the above open problem in various settings. In the oracle model, we prove that there are time-independent sparse Hamiltonians that cannot be simulated via an oracle circuit of depth o(T). In the plain model, relying on the random oracle heuristic, we show that there exist time-independent local Hamiltonians and time-dependent geometrically local Hamiltonians on n qubits that cannot be simulated via an oracle circuit of depth o(T/n^c), where the Hamiltonians act on n qubits, and c is a constant. Lastly, we generalize the above results and show that any simulators that are geometrically local Hamiltonians cannot do the simulation much faster than parallel quantum algorithms.

Nai-Hui Chia, Kai-Min Chung, Yao-Ching Hsieh, Han-Hsuan Lin, Yao-Ting Lin, and Yu-Ching Shen. On the Impossibility of General Parallel Fast-Forwarding of Hamiltonian Simulation. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 33:1-33:45, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{chia_et_al:LIPIcs.CCC.2023.33, author = {Chia, Nai-Hui and Chung, Kai-Min and Hsieh, Yao-Ching and Lin, Han-Hsuan and Lin, Yao-Ting and Shen, Yu-Ching}, title = {{On the Impossibility of General Parallel Fast-Forwarding of Hamiltonian Simulation}}, booktitle = {38th Computational Complexity Conference (CCC 2023)}, pages = {33:1--33:45}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-282-2}, ISSN = {1868-8969}, year = {2023}, volume = {264}, editor = {Ta-Shma, Amnon}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.33}, URN = {urn:nbn:de:0030-drops-183038}, doi = {10.4230/LIPIcs.CCC.2023.33}, annote = {Keywords: Hamiltonian simulation, Depth lower bound, Parallel query lower bound} }

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**Published in:** LIPIcs, Volume 197, 16th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2021)

The probably approximately correct (PAC) model [Leslie G. Valiant, 1984] is a well studied model in classical learning theory. Here, we generalize the PAC model from concepts of Boolean functions to quantum channels, introducing PAC model for learning quantum channels, and give two sample efficient algorithms that are analogous to the classical "Occam’s razor" result [Blumer et al., 1987]. The classical Occam’s razor algorithm is done trivially by excluding any concepts not compatible with the input-output pairs one gets, but such an approach is not immediately possible with a concept class of quantum channels, because the outputs are unknown quantum states from the quantum channel.
To study the quantum state learning problem associated with PAC learning quantum channels, we focus on the special case where the channels all have constant output. In this special case, learning the channels reduce to a problem of learning quantum states that is similar to the well known quantum state discrimination problem [Joonwoo Bae and Leong-Chuan Kwek, 2017], but with the extra twist that we allow ε-trace-distance-error in the output. We call this problem Approximate State Discrimination, which we believe is a natural problem that is of independent interest.
We give two algorithms for learning quantum channels in PAC model. The first algorithm has sample complexity
O((log|C| + log(1/ δ))/(ε²)),
but only works when the outputs are pure states, where C is the concept class, ε is the error of the output, and δ is the probability of failure of the algorithm. The second algorithm has sample complexity
O((log³|C|(log|C|+log(1/ δ)))/(ε²)),
and work for mixed state outputs. Some implications of our results are that we can PAC-learn a polynomial sized quantum circuit in polynomial samples, and approximate state discrimination can be solved in polynomial samples even when the size of the input set is exponential in the number of qubits, exponentially better than a naive state tomography.

Kai-Min Chung and Han-Hsuan Lin. Sample Efficient Algorithms for Learning Quantum Channels in PAC Model and the Approximate State Discrimination Problem. In 16th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 197, pp. 3:1-3:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{chung_et_al:LIPIcs.TQC.2021.3, author = {Chung, Kai-Min and Lin, Han-Hsuan}, title = {{Sample Efficient Algorithms for Learning Quantum Channels in PAC Model and the Approximate State Discrimination Problem}}, booktitle = {16th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2021)}, pages = {3:1--3:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-198-6}, ISSN = {1868-8969}, year = {2021}, volume = {197}, editor = {Hsieh, Min-Hsiu}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2021.3}, URN = {urn:nbn:de:0030-drops-139984}, doi = {10.4230/LIPIcs.TQC.2021.3}, annote = {Keywords: PAC learning, Quantum PAC learning, Sample Complexity, Approximate State Discrimination, Quantum information} }

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**Published in:** LIPIcs, Volume 163, 1st Conference on Information-Theoretic Cryptography (ITC 2020)

Function inversion is the problem that given a random function f: [M] → [N], we want to find pre-image of any image f^{-1}(y) in time T. In this work, we revisit this problem under the preprocessing model where we can compute some auxiliary information or advice of size S that only depends on f but not on y. It is a well-studied problem in the classical settings, however, it is not clear how quantum algorithms can solve this task any better besides invoking Grover’s algorithm [Grover, 1996], which does not leverage the power of preprocessing.
Nayebi et al. [Nayebi et al., 2015] proved a lower bound ST² ≥ ̃Ω(N) for quantum algorithms inverting permutations, however, they only consider algorithms with classical advice. Hhan et al. [Minki Hhan et al., 2019] subsequently extended this lower bound to fully quantum algorithms for inverting permutations. In this work, we give the same asymptotic lower bound to fully quantum algorithms for inverting functions for fully quantum algorithms under the regime where M = O(N).
In order to prove these bounds, we generalize the notion of quantum random access code, originally introduced by Ambainis et al. [Ambainis et al., 1999], to the setting where we are given a list of (not necessarily independent) random variables, and we wish to compress them into a variable-length encoding such that we can retrieve a random element just using the encoding with high probability. As our main technical contribution, we give a nearly tight lower bound (for a wide parameter range) for this generalized notion of quantum random access codes, which may be of independent interest.

Kai-Min Chung, Tai-Ning Liao, and Luowen Qian. Lower Bounds for Function Inversion with Quantum Advice. In 1st Conference on Information-Theoretic Cryptography (ITC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 163, pp. 8:1-8:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{chung_et_al:LIPIcs.ITC.2020.8, author = {Chung, Kai-Min and Liao, Tai-Ning and Qian, Luowen}, title = {{Lower Bounds for Function Inversion with Quantum Advice}}, booktitle = {1st Conference on Information-Theoretic Cryptography (ITC 2020)}, pages = {8:1--8:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-151-1}, ISSN = {1868-8969}, year = {2020}, volume = {163}, editor = {Tauman Kalai, Yael and Smith, Adam D. and Wichs, Daniel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITC.2020.8}, URN = {urn:nbn:de:0030-drops-121134}, doi = {10.4230/LIPIcs.ITC.2020.8}, annote = {Keywords: Cryptanalysis, Data Structures, Quantum Query Complexity} }

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**Published in:** LIPIcs, Volume 151, 11th Innovations in Theoretical Computer Science Conference (ITCS 2020)

Massively Parallel Computation (MPC) is a model of computation widely believed to best capture realistic parallel computing architectures such as large-scale MapReduce and Hadoop clusters. Motivated by the fact that many data analytics tasks performed on these platforms involve sensitive user data, we initiate the theoretical exploration of how to leverage MPC architectures to enable efficient, privacy-preserving computation over massive data. Clearly if a computation task does not lend itself to an efficient implementation on MPC even without security, then we cannot hope to compute it efficiently on MPC with security. We show, on the other hand, that any task that can be efficiently computed on MPC can also be securely computed with comparable efficiency. Specifically, we show the following results:
- any MPC algorithm can be compiled to a communication-oblivious counterpart while asymptotically preserving its round and space complexity, where communication-obliviousness ensures that any network intermediary observing the communication patterns learn no information about the secret inputs;
- assuming the existence of Fully Homomorphic Encryption with a suitable notion of compactness and other standard cryptographic assumptions, any MPC algorithm can be compiled to a secure counterpart that defends against an adversary who controls not only intermediate network routers but additionally up to 1/3 - η fraction of machines (for an arbitrarily small constant η) - moreover, this compilation preserves the round complexity tightly, and preserves the space complexity upto a multiplicative security parameter related blowup.
As an initial exploration of this important direction, our work suggests new definitions and proposes novel protocols that blend algorithmic and cryptographic techniques.

T-H. Hubert Chan, Kai-Min Chung, Wei-Kai Lin, and Elaine Shi. MPC for MPC: Secure Computation on a Massively Parallel Computing Architecture. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 75:1-75:52, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{chan_et_al:LIPIcs.ITCS.2020.75, author = {Chan, T-H. Hubert and Chung, Kai-Min and Lin, Wei-Kai and Shi, Elaine}, title = {{MPC for MPC: Secure Computation on a Massively Parallel Computing Architecture}}, booktitle = {11th Innovations in Theoretical Computer Science Conference (ITCS 2020)}, pages = {75:1--75:52}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-134-4}, ISSN = {1868-8969}, year = {2020}, volume = {151}, editor = {Vidick, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2020.75}, URN = {urn:nbn:de:0030-drops-117600}, doi = {10.4230/LIPIcs.ITCS.2020.75}, annote = {Keywords: massively parallel computation, secure multi-party computation} }

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**Published in:** LIPIcs, Volume 124, 10th Innovations in Theoretical Computer Science Conference (ITCS 2019)

Spiking Neural Networks (SNN) are mathematical models in neuroscience to describe the dynamics among a set of neurons that interact with each other by firing instantaneous signals, a.k.a., spikes. Interestingly, a recent advance in neuroscience [Barrett-Denève-Machens, NIPS 2013] showed that the neurons' firing rate, i.e., the average number of spikes fired per unit of time, can be characterized by the optimal solution of a quadratic program defined by the parameters of the dynamics. This indicated that SNN potentially has the computational power to solve non-trivial quadratic programs. However, the results were justified empirically without rigorous analysis.
We put this into the context of natural algorithms and aim to investigate the algorithmic power of SNN. Especially, we emphasize on giving rigorous asymptotic analysis on the performance of SNN in solving optimization problems. To enforce a theoretical study, we first identify a simplified SNN model that is tractable for analysis. Next, we confirm the empirical observation in the work of Barrett et al. by giving an upper bound on the convergence rate of SNN in solving the quadratic program. Further, we observe that in the case where there are infinitely many optimal solutions, SNN tends to converge to the one with smaller l_1 norm. We give an affirmative answer to our finding by showing that SNN can solve the l_1 minimization problem under some regular conditions.
Our main technical insight is a dual view of the SNN dynamics, under which SNN can be viewed as a new natural primal-dual algorithm for the l_1 minimization problem. We believe that the dual view is of independent interest and may potentially find interesting interpretation in neuroscience.

Chi-Ning Chou, Kai-Min Chung, and Chi-Jen Lu. On the Algorithmic Power of Spiking Neural Networks. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 26:1-26:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{chou_et_al:LIPIcs.ITCS.2019.26, author = {Chou, Chi-Ning and Chung, Kai-Min and Lu, Chi-Jen}, title = {{On the Algorithmic Power of Spiking Neural Networks}}, booktitle = {10th Innovations in Theoretical Computer Science Conference (ITCS 2019)}, pages = {26:1--26:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-095-8}, ISSN = {1868-8969}, year = {2019}, volume = {124}, editor = {Blum, Avrim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2019.26}, URN = {urn:nbn:de:0030-drops-101191}, doi = {10.4230/LIPIcs.ITCS.2019.26}, annote = {Keywords: Spiking Neural Networks, Natural Algorithms, l\underline1 Minimization} }

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**Published in:** LIPIcs, Volume 33, 30th Conference on Computational Complexity (CCC 2015)

We present two parallel repetition theorems for the entangled value of multi-player, one-round free games (games where the inputs come from a product distribution). Our first theorem shows that for a k-player free game G with entangled value val^*(G) = 1 - epsilon, the n-fold repetition of G has entangled value val^*(G^(\otimes n)) at most (1 - epsilon^(3/2))^(Omega(n/sk^4)), where s is the answer length of any player. In contrast, the best known parallel repetition theorem for the classical value of two-player free games is val(G^(\otimes n)) <= (1 - epsilon^2)^(Omega(n/s)), due to Barak, et al. (RANDOM 2009). This suggests the possibility of a separation between the behavior of entangled and classical free games under parallel repetition.
Our second theorem handles the broader class of free games G where the players can output (possibly entangled) quantum states. For such games, the repeated entangled value is upper bounded by (1 - epsilon^2)^(Omega(n/sk^2)). We also show that the dependence of the exponent on k is necessary: we exhibit a k-player free game G and n >= 1 such that val^*(G^(\otimes n)) >= val^*(G)^(n/k).
Our analysis exploits the novel connection between communication protocols and quantum parallel repetition, first explored by Chailloux and Scarpa (ICALP 2014). We demonstrate that better communication protocols yield better parallel repetition theorems: in particular, our first theorem crucially uses a quantum search protocol by Aaronson and Ambainis, which gives a quadratic Grover speed-up for distributed search problems. Finally, our results apply to a broader class of games than were previously considered before; in particular, we obtain the first parallel repetition theorem for entangled games involving more than two players, and for games involving quantum outputs.

Kai-Min Chung, Xiaodi Wu, and Henry Yuen. Parallel Repetition for Entangled k-player Games via Fast Quantum Search. In 30th Conference on Computational Complexity (CCC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 33, pp. 512-536, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{chung_et_al:LIPIcs.CCC.2015.512, author = {Chung, Kai-Min and Wu, Xiaodi and Yuen, Henry}, title = {{Parallel Repetition for Entangled k-player Games via Fast Quantum Search}}, booktitle = {30th Conference on Computational Complexity (CCC 2015)}, pages = {512--536}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-81-1}, ISSN = {1868-8969}, year = {2015}, volume = {33}, editor = {Zuckerman, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2015.512}, URN = {urn:nbn:de:0030-drops-50727}, doi = {10.4230/LIPIcs.CCC.2015.512}, annote = {Keywords: Parallel repetition, quantum entanglement, communication complexity} }

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**Published in:** LIPIcs, Volume 14, 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)

We prove the first Chernoff-Hoeffding bounds for general nonreversible finite-state Markov chains based on the standard L_1 (variation distance) mixing-time of the chain. Specifically, consider an ergodic Markov chain M and a weight function f: [n] -> [0,1] on the state space [n] of M with mean mu = E_{v <- pi}[f(v)], where pi is the stationary distribution of M. A t-step random walk (v_1,...,v_t) on M starting from the stationary distribution pi has expected total weight E[X] = mu t, where X = sum_{i=1}^t f(v_i). Let T be the L_1 mixing-time of M. We show that the probability of X deviating from its mean by a multiplicative factor of delta, i.e., Pr [ |X - mu t| >= delta mu t ], is at most exp(-Omega( delta^2 mu t / T )) for 0 <= delta <= 1, and exp(-Omega( delta mu t / T )) for delta > 1. In fact, the bounds hold even if the weight functions f_i's for i in [t] are distinct, provided that all of them have the same mean mu.
We also obtain a simplified proof for the Chernoff-Hoeffding bounds based on the spectral expansion lambda of M, which is the square root of the second largest eigenvalue (in absolute value) of M tilde{M}, where tilde{M} is the time-reversal Markov chain of M. We show that the probability Pr [ |X - mu t| >= delta mu t ] is at most exp(-Omega( delta^2 (1-lambda) mu t )) for 0 <= delta <= 1, and exp(-Omega( delta (1-lambda) mu t )) for delta > 1.
Both of our results extend to continuous-time Markov chains, and to the case where the walk starts from an arbitrary distribution x, at a price of a multiplicative factor depending on the distribution x in the concentration bounds.

Kai-Min Chung, Henry Lam, Zhenming Liu, and Michael Mitzenmacher. Chernoff-Hoeffding Bounds for Markov Chains: Generalized and Simplified. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 124-135, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{chung_et_al:LIPIcs.STACS.2012.124, author = {Chung, Kai-Min and Lam, Henry and Liu, Zhenming and Mitzenmacher, Michael}, title = {{Chernoff-Hoeffding Bounds for Markov Chains: Generalized and Simplified}}, booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)}, pages = {124--135}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-35-4}, ISSN = {1868-8969}, year = {2012}, volume = {14}, editor = {D\"{u}rr, Christoph and Wilke, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.124}, URN = {urn:nbn:de:0030-drops-34374}, doi = {10.4230/LIPIcs.STACS.2012.124}, annote = {Keywords: probabilistic analysis, tail bounds, Markov chains} }

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**Published in:** LIPIcs, Volume 5, 27th International Symposium on Theoretical Aspects of Computer Science (2010)

In their seminal work, Alon, Matias, and Szegedy introduced several sketching techniques, including showing that $4$-wise independence is sufficient to obtain good approximations of the second frequency moment. In this work, we show that their sketching technique can be extended to product domains $[n]^k$ by using the product of $4$-wise independent functions on $[n]$.
Our work extends that of Indyk and McGregor, who showed the result for $k = 2$. Their primary motivation was the problem of identifying correlations in data streams. In their model, a stream of pairs $(i,j) \in [n]^2$ arrive, giving a joint distribution $(X,Y)$, and they find approximation algorithms for how close the joint distribution is to the product of the marginal distributions under various metrics, which naturally corresponds to how close $X$ and $Y$ are to being independent. By using our technique, we obtain a new result for the problem of approximating the $\ell_2$ distance between the joint distribution and the product of the marginal distributions for $k$-ary vectors, instead of just pairs, in a single pass. Our analysis gives a randomized algorithm that is a $(1\pm \epsilon)$ approximation (with probability $1-\delta$) that requires space logarithmic in $n$ and $m$ and proportional to $3^k$.

Vladimir Braverman, Kai-Min Chung, Zhenming Liu, Michael Mitzenmacher, and Rafail Ostrovsky. AMS Without 4-Wise Independence on Product Domains. In 27th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 5, pp. 119-130, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{braverman_et_al:LIPIcs.STACS.2010.2449, author = {Braverman, Vladimir and Chung, Kai-Min and Liu, Zhenming and Mitzenmacher, Michael and Ostrovsky, Rafail}, title = {{AMS Without 4-Wise Independence on Product Domains}}, booktitle = {27th International Symposium on Theoretical Aspects of Computer Science}, pages = {119--130}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-16-3}, ISSN = {1868-8969}, year = {2010}, volume = {5}, editor = {Marion, Jean-Yves and Schwentick, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2010.2449}, URN = {urn:nbn:de:0030-drops-24496}, doi = {10.4230/LIPIcs.STACS.2010.2449}, annote = {Keywords: Data Streams, Randomized Algorithms, Streaming Algorithms, Independence, Sketches} }