Document

**Published in:** LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

We consider the problem of scheduling to minimize mean response time in M/G/1 queues where only estimated job sizes (processing times) are known to the scheduler, where a job of true size s has estimated size in the interval [β s, α s] for some α ≥ β > 0. We evaluate each scheduling policy by its approximation ratio, which we define to be the ratio between its mean response time and that of Shortest Remaining Processing Time (SRPT), the optimal policy when true sizes are known. Our question: is there a scheduling policy that (a) has approximation ratio near 1 when α and β are near 1, (b) has approximation ratio bounded by some function of α and β even when they are far from 1, and (c) can be implemented without knowledge of α and β?
We first show that naively running SRPT using estimated sizes in place of true sizes is not such a policy: its approximation ratio can be arbitrarily large for any fixed β < 1. We then provide a simple variant of SRPT for estimated sizes that satisfies criteria (a), (b), and (c). In particular, we prove its approximation ratio approaches 1 uniformly as α and β approach 1. This is the first result showing this type of convergence for M/G/1 scheduling.
We also study the Preemptive Shortest Job First (PSJF) policy, a cousin of SRPT. We show that, unlike SRPT, naively running PSJF using estimated sizes in place of true sizes satisfies criteria (b) and (c), as well as a weaker version of (a).

Ziv Scully, Isaac Grosof, and Michael Mitzenmacher. Uniform Bounds for Scheduling with Job Size Estimates. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 114:1-114:30, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{scully_et_al:LIPIcs.ITCS.2022.114, author = {Scully, Ziv and Grosof, Isaac and Mitzenmacher, Michael}, title = {{Uniform Bounds for Scheduling with Job Size Estimates}}, booktitle = {13th Innovations in Theoretical Computer Science Conference (ITCS 2022)}, pages = {114:1--114:30}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-217-4}, ISSN = {1868-8969}, year = {2022}, volume = {215}, editor = {Braverman, Mark}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.114}, URN = {urn:nbn:de:0030-drops-157108}, doi = {10.4230/LIPIcs.ITCS.2022.114}, annote = {Keywords: Scheduling, queueing systems, algorithms with predictions, shortest remaining processing time (SRPT), preemptive shortest job first (PSJF), M/G/1 queue} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

We consider the fundamental problem of communicating an estimate of a real number x ∈ [0,1] using a single bit. A sender that knows x chooses a value X ∈ {0,1} to transmit. In turn, a receiver estimates x based on the value of X. The goal is to minimize the cost, defined as the worst-case (over the choice of x) expected squared error.
We first overview common biased and unbiased estimation approaches and prove their optimality when no shared randomness is allowed. We then show how a small amount of shared randomness, which can be as low as a single bit, reduces the cost in both cases. Specifically, we derive lower bounds on the cost attainable by any algorithm with unrestricted use of shared randomness and propose optimal and near-optimal solutions that use a small number of shared random bits. Finally, we discuss open problems and future directions.

Ran Ben Basat, Michael Mitzenmacher, and Shay Vargaftik. How to Send a Real Number Using a Single Bit (And Some Shared Randomness). In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 25:1-25:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{benbasat_et_al:LIPIcs.ICALP.2021.25, author = {Ben Basat, Ran and Mitzenmacher, Michael and Vargaftik, Shay}, title = {{How to Send a Real Number Using a Single Bit (And Some Shared Randomness)}}, booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)}, pages = {25:1--25:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-195-5}, ISSN = {1868-8969}, year = {2021}, volume = {198}, editor = {Bansal, Nikhil and Merelli, Emanuela and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.25}, URN = {urn:nbn:de:0030-drops-140942}, doi = {10.4230/LIPIcs.ICALP.2021.25}, annote = {Keywords: Randomized Algorithms, Approximation Algorithms, Shared Randomness, Distributed Protocols, Estimation, Subtractive Dithering} }

Document

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

In many traditional job scheduling settings, it is assumed that one knows the time it will take for a job to complete service. In such cases, strategies such as shortest job first can be used to improve performance in terms of measures such as the average time a job waits in the system. We consider the setting where the service time is not known, but is predicted by for example a machine learning algorithm. Our main result is the derivation, under natural assumptions, of formulae for the performance of several strategies for queueing systems that use predictions for service times in order to schedule jobs. As part of our analysis, we suggest the framework of the "price of misprediction," which offers a measure of the cost of using predicted information.

Michael Mitzenmacher. Scheduling with Predictions and the Price of Misprediction. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 14:1-14:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{mitzenmacher:LIPIcs.ITCS.2020.14, author = {Mitzenmacher, Michael}, title = {{Scheduling with Predictions and the Price of Misprediction}}, booktitle = {11th Innovations in Theoretical Computer Science Conference (ITCS 2020)}, pages = {14:1--14:18}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2020.14}, URN = {urn:nbn:de:0030-drops-116996}, doi = {10.4230/LIPIcs.ITCS.2020.14}, annote = {Keywords: Queues, shortest remaining processing time, algorithms with predictions, scheduling} }

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Complete Volume

**Published in:** OASIcs, Volume 69, 2nd Symposium on Simplicity in Algorithms (SOSA 2019)

OASIcs, Volume 69, SOSA'19, Complete Volume

2nd Symposium on Simplicity in Algorithms (SOSA 2019). Open Access Series in Informatics (OASIcs), Volume 69, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@Proceedings{fineman_et_al:OASIcs.SOSA.2019, title = {{OASIcs, Volume 69, SOSA'19, Complete Volume}}, booktitle = {2nd Symposium on Simplicity in Algorithms (SOSA 2019)}, series = {Open Access Series in Informatics (OASIcs)}, ISBN = {978-3-95977-099-6}, ISSN = {2190-6807}, year = {2019}, volume = {69}, editor = {Fineman, Jeremy T. and Mitzenmacher, Michael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/OASIcs.SOSA.2019}, URN = {urn:nbn:de:0030-drops-101683}, doi = {10.4230/OASIcs.SOSA.2019}, annote = {Keywords: Theory of computation, Design and analysis of algorithms} }

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Front Matter

**Published in:** OASIcs, Volume 69, 2nd Symposium on Simplicity in Algorithms (SOSA 2019)

Front Matter, Table of Contents, Preface, Conference Organization

2nd Symposium on Simplicity in Algorithms (SOSA 2019). Open Access Series in Informatics (OASIcs), Volume 69, pp. 0:i-0:x, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{fineman_et_al:OASIcs.SOSA.2019.0, author = {Fineman, Jeremy T. and Mitzenmacher, Michael}, title = {{Front Matter, Table of Contents, Preface, Conference Organization}}, booktitle = {2nd Symposium on Simplicity in Algorithms (SOSA 2019)}, pages = {0:i--0:x}, series = {Open Access Series in Informatics (OASIcs)}, ISBN = {978-3-95977-099-6}, ISSN = {2190-6807}, year = {2019}, volume = {69}, editor = {Fineman, Jeremy T. and Mitzenmacher, Michael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/OASIcs.SOSA.2019.0}, URN = {urn:nbn:de:0030-drops-100263}, doi = {10.4230/OASIcs.SOSA.2019.0}, annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization} }

Document

**Published in:** LIPIcs, Volume 101, 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018)

In k-ary cuckoo hashing, each of cn objects is associated with k random buckets in a hash table of size n. An l-orientation is an assignment of objects to associated buckets such that each bucket receives at most l objects. Several works have determined load thresholds c^* = c^*(k,l) for k-ary cuckoo hashing; that is, for c < c^* an l-orientation exists with high probability, and for c > c^* no l-orientation exists with high probability.
A natural variant of k-ary cuckoo hashing utilizes double hashing, where, when the buckets are numbered 0,1,...,n-1, the k choices of random buckets form an arithmetic progression modulo n. Double hashing simplifies implementation and requires less randomness, and it has been shown that double hashing has the same behavior as fully random hashing in several other data structures that similarly use multiple hashes for each object. Interestingly, previous work has come close to but has not fully shown that the load threshold for k-ary cuckoo hashing is the same when using double hashing as when using fully random hashing. Specifically, previous work has shown that the thresholds for both settings coincide, except that for double hashing it was possible that o(n) objects would have been left unplaced. Here we close this open question by showing the thresholds are indeed the same, by providing a combinatorial argument that reconciles this stubborn difference.

Michael Mitzenmacher, Konstantinos Panagiotou, and Stefan Walzer. Load Thresholds for Cuckoo Hashing with Double Hashing. In 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 101, pp. 29:1-29:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{mitzenmacher_et_al:LIPIcs.SWAT.2018.29, author = {Mitzenmacher, Michael and Panagiotou, Konstantinos and Walzer, Stefan}, title = {{Load Thresholds for Cuckoo Hashing with Double Hashing}}, booktitle = {16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018)}, pages = {29:1--29:9}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-068-2}, ISSN = {1868-8969}, year = {2018}, volume = {101}, editor = {Eppstein, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2018.29}, URN = {urn:nbn:de:0030-drops-88557}, doi = {10.4230/LIPIcs.SWAT.2018.29}, annote = {Keywords: Cuckoo Hashing, Double Hashing, Load Thresholds, Hypergraph Orientability} }

Document

**Published in:** Dagstuhl Reports, Volume 7, Issue 5 (2018)

This report documents the program and the topics discussed of the 4-day
Dagstuhl Seminar 17181 "Theory and Applications of Hashing",
which took place May 1-5, 2017. Four long and eighteen short talks
covered a wide and diverse range of topics within the theme of the workshop.
The program left sufficient space for informal discussions among the 40 participants.

Martin Dietzfelbinger, Michael Mitzenmacher, Rasmus Pagh, David P. Woodruff, and Martin Aumüller. Theory and Applications of Hashing (Dagstuhl Seminar 17181). In Dagstuhl Reports, Volume 7, Issue 5, pp. 1-21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@Article{dietzfelbinger_et_al:DagRep.7.5.1, author = {Dietzfelbinger, Martin and Mitzenmacher, Michael and Pagh, Rasmus and Woodruff, David P. and Aum\"{u}ller, Martin}, title = {{Theory and Applications of Hashing (Dagstuhl Seminar 17181)}}, pages = {1--21}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2017}, volume = {7}, number = {5}, editor = {Dietzfelbinger, Martin and Mitzenmacher, Michael and Pagh, Rasmus and Woodruff, David P. and Aum\"{u}ller, Martin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/DagRep.7.5.1}, URN = {urn:nbn:de:0030-drops-82788}, doi = {10.4230/DagRep.7.5.1}, annote = {Keywords: connections to complexity theory, data streaming applications, hash function construction and analysis, hashing primitives, information retrieval applications, locality-sensitive hashing, machine learning applications} }

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Complete Volume

**Published in:** LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

LIPIcs, Volume 55, ICALP'16, Complete Volume

43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@Proceedings{chatzigiannakis_et_al:LIPIcs.ICALP.2016, title = {{LIPIcs, Volume 55, ICALP'16, Complete Volume}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-013-2}, ISSN = {1868-8969}, year = {2016}, volume = {55}, editor = {Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016}, URN = {urn:nbn:de:0030-drops-65844}, doi = {10.4230/LIPIcs.ICALP.2016}, annote = {Keywords: Theory of Computation} }

Document

Front Matter

**Published in:** LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

Front Matter, Table of Contents, Preface, Organization, List of Authors

43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 0:i-0:xliv, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{chatzigiannakis_et_al:LIPIcs.ICALP.2016.0, author = {Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide}, title = {{Front Matter, Table of Contents, Preface, Organization, List of Authors}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {0:i--0:xliv}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-013-2}, ISSN = {1868-8969}, year = {2016}, volume = {55}, editor = {Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.0}, URN = {urn:nbn:de:0030-drops-61917}, doi = {10.4230/LIPIcs.ICALP.2016.0}, annote = {Keywords: Front Matter, Table of Contents, Preface, Organization, List of Authors} }

Document

**Published in:** LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

We study novel variations of Voronoi games and associated random processes that we call Voronoi choice games. These games provide a rich framework for studying questions regarding the power of small numbers of choices in multi-player, competitive scenarios, and they further lead to many interesting, non-trivial random processes that appear worthy of study.
As an example of the type of problem we study, suppose a group of n miners (or players) are staking land claims through the following process: each miner has m associated points independently and uniformly distributed on an underlying space (such as the unit circle, the unit square, or the unit torus), so the kth miner will have associated points p_{k1}, p_{k2}, ..., p_{km}. We generally here think of m as being a small constant, such as 2. Each miner chooses one of these points as the base point for their claim. Each miner obtains mining rights for the area of the square that is closest to their chosen base; that is, they obtain the Voronoi cell corresponding to their chosen point in the Voronoi diagram of the n chosen points. Each player's goal is simply to maximize the amount of land under their control. What can we say about the players’ strategy and the equilibria of such games?
In our main result, we derive bounds on the expected number of pure Nash equilibria for a variation of the 1-dimensional game on the circle where a player owns the arc starting from their point and moving clockwise to the next point. This result uses interesting properties of random arc lengths on circles, and demonstrates the challenges in analyzing these kinds of problems. We also provide several other related results. In particular, for the 1-dimensional game on the circle, we show that a pure Nash equilibrium always exists when each player owns the part of the circle nearest to their point, but it is NP-hard to determine whether a pure Nash equilibrium exists in the variant when each player owns the arc starting from their point clockwise to the next point. This last result, in part, motivates our examination of the random setting.

Meena Boppana, Rani Hod, Michael Mitzenmacher, and Tom Morgan. Voronoi Choice Games. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 23:1-23:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{boppana_et_al:LIPIcs.ICALP.2016.23, author = {Boppana, Meena and Hod, Rani and Mitzenmacher, Michael and Morgan, Tom}, title = {{Voronoi Choice Games}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {23:1--23:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-013-2}, ISSN = {1868-8969}, year = {2016}, volume = {55}, editor = {Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.23}, URN = {urn:nbn:de:0030-drops-63022}, doi = {10.4230/LIPIcs.ICALP.2016.23}, annote = {Keywords: Voronoi games, correlated equilibria, power of two choices, Hotelling model} }

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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-dev.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-dev.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} }

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