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**Published in:** Dagstuhl Reports, Volume 13, Issue 2 (2023)

This report documents the program and the outcomes of Dagstuhl Seminar 23071 "From Big Data Theory to Big Data Practice". Some recent advances in the theory of algorithms for big data - sublinear/local algorithms, streaming algorithms and external memory algorithms - have translated into impressive improvements in practice, whereas others have remained stubbornly resistant to useful implementations. This seminar aimed to glean lessons for those aspect of these algorithms that have led to practical implementation to see if the lessons learned can both improve the implementations of other theoretical ideas and to help guide the next generation of theoretical advances.

Martin Farach-Colton, Fabian Daniel Kuhn, Ronitt Rubinfeld, and Przemysław Uznański. From Big Data Theory to Big Data Practice (Dagstuhl Seminar 23071). In Dagstuhl Reports, Volume 13, Issue 2, pp. 33-46, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@Article{farachcolton_et_al:DagRep.13.2.33, author = {Farach-Colton, Martin and Kuhn, Fabian Daniel and Rubinfeld, Ronitt and Uzna\'{n}ski, Przemys{\l}aw}, title = {{From Big Data Theory to Big Data Practice (Dagstuhl Seminar 23071)}}, pages = {33--46}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2023}, volume = {13}, number = {2}, editor = {Farach-Colton, Martin and Kuhn, Fabian Daniel and Rubinfeld, Ronitt and Uzna\'{n}ski, Przemys{\l}aw}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagRep.13.2.33}, URN = {urn:nbn:de:0030-drops-191809}, doi = {10.4230/DagRep.13.2.33}, annote = {Keywords: external memory, local algorithms, sublinear algorithms} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

Interval scheduling is a basic problem in the theory of algorithms and a classical task in combinatorial optimization. We develop a set of techniques for partitioning and grouping jobs based on their starting and ending times, that enable us to view an instance of interval scheduling on many jobs as a union of multiple interval scheduling instances, each containing only a few jobs. Instantiating these techniques in dynamic and local settings of computation leads to several new results.
For (1+ε)-approximation of job scheduling of n jobs on a single machine, we develop a fully dynamic algorithm with O((log n)/ε) update and O(log n) query worst-case time. Further, we design a local computation algorithm that uses only O((log N)/ε) queries when all jobs are length at least 1 and have starting/ending times within [0,N]. Our techniques are also applicable in a setting where jobs have rewards/weights. For this case we design a fully dynamic deterministic algorithm whose worst-case update and query time are poly(log n,1/ε). Equivalently, this is the first algorithm that maintains a (1+ε)-approximation of the maximum independent set of a collection of weighted intervals in poly(log n,1/ε) time updates/queries. This is an exponential improvement in 1/ε over the running time of a randomized algorithm of Henzinger, Neumann, and Wiese [SoCG, 2020], while also removing all dependence on the values of the jobs' starting/ending times and rewards, as well as removing the need for any randomness.
We also extend our approaches for interval scheduling on a single machine to examine the setting with M machines.

Spencer Compton, Slobodan Mitrović, and Ronitt Rubinfeld. New Partitioning Techniques and Faster Algorithms for Approximate Interval Scheduling. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 45:1-45:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{compton_et_al:LIPIcs.ICALP.2023.45, author = {Compton, Spencer and Mitrovi\'{c}, Slobodan and Rubinfeld, Ronitt}, title = {{New Partitioning Techniques and Faster Algorithms for Approximate Interval Scheduling}}, booktitle = {50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)}, pages = {45:1--45:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-278-5}, ISSN = {1868-8969}, year = {2023}, volume = {261}, editor = {Etessami, Kousha and Feige, Uriel and Puppis, Gabriele}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.45}, URN = {urn:nbn:de:0030-drops-180978}, doi = {10.4230/LIPIcs.ICALP.2023.45}, annote = {Keywords: interval scheduling, dynamic algorithms, local computation algorithms} }

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APPROX

**Published in:** LIPIcs, Volume 245, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)

Over the last two decades, frameworks for distributed-memory parallel computation, such as MapReduce, Hadoop, Spark and Dryad, have gained significant popularity with the growing prevalence of large network datasets. The Massively Parallel Computation (MPC) model is the de-facto standard for studying graph algorithms in these frameworks theoretically. Subgraph counting is one such fundamental problem in analyzing massive graphs, with the main algorithmic challenges centering on designing methods which are both scalable and accurate.
Given a graph G = (V, E) with n vertices, m edges and T triangles, our first result is an algorithm that outputs a (1+ε)-approximation to T, with asymptotically optimal round and total space complexity provided any S ≥ max{(√ m, n²/m)} space per machine and assuming T = Ω(√{m/n}). Our result gives a quadratic improvement on the bound on T over previous works. We also provide a simple extension of our result to counting any subgraph of k size for constant k ≥ 1. Our second result is an O_δ(log log n)-round algorithm for exactly counting the number of triangles, whose total space usage is parametrized by the arboricity α of the input graph. We extend this result to exactly counting k-cliques for any constant k. Finally, we prove that a recent result of Bera, Pashanasangi and Seshadhri (ITCS 2020) for exactly counting all subgraphs of size at most 5 can be implemented in the MPC model in Õ_δ(√{log n}) rounds, O(n^δ) space per machine and O(mα³) total space.
In addition to our theoretical results, we simulate our triangle counting algorithms in real-world graphs obtained from the Stanford Network Analysis Project (SNAP) database. Our results show that both our approximate and exact counting algorithms exhibit improvements in terms of round complexity and approximation ratio, respectively, compared to two previous widely used algorithms for these problems.

Amartya Shankha Biswas, Talya Eden, Quanquan C. Liu, Ronitt Rubinfeld, and Slobodan Mitrović. Massively Parallel Algorithms for Small Subgraph Counting. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 245, pp. 39:1-39:28, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{biswas_et_al:LIPIcs.APPROX/RANDOM.2022.39, author = {Biswas, Amartya Shankha and Eden, Talya and Liu, Quanquan C. and Rubinfeld, Ronitt and Mitrovi\'{c}, Slobodan}, title = {{Massively Parallel Algorithms for Small Subgraph Counting}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)}, pages = {39:1--39:28}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-249-5}, ISSN = {1868-8969}, year = {2022}, volume = {245}, editor = {Chakrabarti, Amit and Swamy, Chaitanya}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2022.39}, URN = {urn:nbn:de:0030-drops-171619}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2022.39}, annote = {Keywords: triangle counting, massively parallel computation, clique counting, approximation algorithms, subgraph counting} }

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

For a graph G on n vertices, naively sampling the position of a random walk of at time t requires work Ω(t). We desire local access algorithms supporting position_G(t) queries, which return the position of a random walk from some fixed start vertex s at time t, where the joint distribution of returned positions is 1/poly(n) close to those of a uniformly random walk in 𝓁₁ distance.
We first give an algorithm for local access to random walks on a given undirected d-regular graph with Õ(1/(1-λ)√n) runtime per query, where λ is the second-largest eigenvalue of the random walk matrix of the graph in absolute value. Since random d-regular graphs G(n,d) are expanders with high probability, this gives an Õ(√n) algorithm for a graph drawn from G(n,d) whp, which improves on the naive method for small numbers of queries.
We then prove that no algorithm with subconstant error given probe access to an input d-regular graph can have runtime better than Ω(√n/log(n)) per query in expectation when the input graph is drawn from G(n,d), obtaining a nearly matching lower bound. We further show an Ω(n^{1/4}) runtime per query lower bound even with an oblivious adversary (i.e. when the query sequence is fixed in advance).
We then show that for families of graphs with additional group theoretic structure, dramatically better results can be achieved. We give local access to walks on small-degree abelian Cayley graphs, including cycles and hypercubes, with runtime polylog(n) per query. This also allows for efficient local access to walks on polylog degree expanders. We show that our techniques apply to graphs with high degree by extending or results to graphs constructed using the tensor product (giving fast local access to walks on degree n^ε graphs for any ε ∈ (0,1]) and Cartesian product.

Amartya Shankha Biswas, Edward Pyne, and Ronitt Rubinfeld. Local Access to Random Walks. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 24:1-24:22, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{biswas_et_al:LIPIcs.ITCS.2022.24, author = {Biswas, Amartya Shankha and Pyne, Edward and Rubinfeld, Ronitt}, title = {{Local Access to Random Walks}}, booktitle = {13th Innovations in Theoretical Computer Science Conference (ITCS 2022)}, pages = {24:1--24:22}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.24}, URN = {urn:nbn:de:0030-drops-156209}, doi = {10.4230/LIPIcs.ITCS.2022.24}, annote = {Keywords: sublinear time algorithms, random generation, local computation} }

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RANDOM

**Published in:** LIPIcs, Volume 207, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)

We present a sublinear time algorithm that allows one to sample multiple edges from a distribution that is pointwise ε-close to the uniform distribution, in an amortized-efficient fashion. We consider the adjacency list query model, where access to a graph G is given via degree and neighbor queries.
The problem of sampling a single edge in this model has been raised by Eden and Rosenbaum (SOSA 18). Let n and m denote the number of vertices and edges of G, respectively. Eden and Rosenbaum provided upper and lower bounds of Θ^*(n/√ m) for sampling a single edge in general graphs (where O^*(⋅) suppresses poly(1/ε) and poly(log n) dependencies). We ask whether the query complexity lower bound for sampling a single edge can be circumvented when multiple samples are required. That is, can we get an improved amortized per-sample cost if we allow a preprocessing phase? We answer in the affirmative.
We present an algorithm that, if one knows the number of required samples q in advance, has an overall cost that is sublinear in q, namely, O^*(√ q ⋅(n/√ m)), which is strictly preferable to O^*(q⋅ (n/√ m)) cost resulting from q invocations of the algorithm by Eden and Rosenbaum.
Subsequent to a preliminary version of this work, Tětek and Thorup (arXiv, preprint) proved that this bound is essentially optimal.

Talya Eden, Saleet Mossel, and Ronitt Rubinfeld. Sampling Multiple Edges Efficiently. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 51:1-51:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{eden_et_al:LIPIcs.APPROX/RANDOM.2021.51, author = {Eden, Talya and Mossel, Saleet and Rubinfeld, Ronitt}, title = {{Sampling Multiple Edges Efficiently}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)}, pages = {51:1--51:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-207-5}, ISSN = {1868-8969}, year = {2021}, volume = {207}, editor = {Wootters, Mary and Sanit\`{a}, Laura}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2021.51}, URN = {urn:nbn:de:0030-drops-147441}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2021.51}, annote = {Keywords: Sampling edges, graph algorithm, sublinear algorithms} }

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RANDOM

**Published in:** LIPIcs, Volume 207, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)

We consider the problem of sampling and approximately counting an arbitrary given motif H in a graph G, where access to G is given via queries: degree, neighbor, and pair, as well as uniform edge sample queries. Previous algorithms for these tasks were based on a decomposition of H into a collection of odd cycles and stars, denoted D^*(H) = {O_{k₁},...,O_{k_q}, S_{p₁},...,S_{p_𝓁}}. These algorithms were shown to be optimal for the case where H is a clique or an odd-length cycle, but no other lower bounds were known.
We present a new algorithm for sampling arbitrary motifs which, up to poly(log n) factors, is always at least as good, and for most graphs G is strictly better. The main ingredient leading to this improvement is an improved uniform algorithm for sampling stars, which might be of independent interest, as it allows to sample vertices according to the p-th moment of the degree distribution.
Finally, we prove that this algorithm is decomposition-optimal for decompositions that contain at least one odd cycle. These are the first lower bounds for motifs H with a nontrivial decomposition, i.e., motifs that have more than a single component in their decomposition.

Amartya Shankha Biswas, Talya Eden, and Ronitt Rubinfeld. Towards a Decomposition-Optimal Algorithm for Counting and Sampling Arbitrary Motifs in Sublinear Time. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 55:1-55:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{biswas_et_al:LIPIcs.APPROX/RANDOM.2021.55, author = {Biswas, Amartya Shankha and Eden, Talya and Rubinfeld, Ronitt}, title = {{Towards a Decomposition-Optimal Algorithm for Counting and Sampling Arbitrary Motifs in Sublinear Time}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)}, pages = {55:1--55:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-207-5}, ISSN = {1868-8969}, year = {2021}, volume = {207}, editor = {Wootters, Mary and Sanit\`{a}, Laura}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2021.55}, URN = {urn:nbn:de:0030-drops-147480}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2021.55}, annote = {Keywords: Sublinear time algorithms, Graph algorithms, Sampling subgraphs, Approximate counting} }

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

Consider an algorithm performing a computation on a huge random object (for example a random graph or a "long" random walk). Is it necessary to generate the entire object prior to the computation, or is it possible to provide query access to the object and sample it incrementally "on-the-fly" (as requested by the algorithm)? Such an implementation should emulate the random object by answering queries in a manner consistent with an instance of the random object sampled from the true distribution (or close to it). This paradigm is useful when the algorithm is sub-linear and thus, sampling the entire object up front would ruin its efficiency.
Our first set of results focus on undirected graphs with independent edge probabilities, i.e. each edge is chosen as an independent Bernoulli random variable. We provide a general implementation for this model under certain assumptions. Then, we use this to obtain the first efficient local implementations for the Erdös-Rényi G(n,p) model for all values of p, and the Stochastic Block model. As in previous local-access implementations for random graphs, we support Vertex-Pair and Next-Neighbor queries. In addition, we introduce a new Random-Neighbor query. Next, we give the first local-access implementation for All-Neighbors queries in the (sparse and directed) Kleinberg’s Small-World model. Our implementations require no pre-processing time, and answer each query using O(poly(log n)) time, random bits, and additional space.
Next, we show how to implement random Catalan objects, specifically focusing on Dyck paths (balanced random walks on the integer line that are always non-negative). Here, we support Height queries to find the location of the walk, and First-Return queries to find the time when the walk returns to a specified location. This in turn can be used to implement Next-Neighbor queries on random rooted ordered trees, and Matching-Bracket queries on random well bracketed expressions (the Dyck language).
Finally, we introduce two features to define a new model that: (1) allows multiple independent (and even simultaneous) instantiations of the same implementation, to be consistent with each other without the need for communication, (2) allows us to generate a richer class of random objects that do not have a succinct description. Specifically, we study uniformly random valid q-colorings of an input graph G with maximum degree Δ. This is in contrast to prior work in the area, where the relevant random objects are defined as a distribution with O(1) parameters (for example, n and p in the G(n,p) model). The distribution over valid colorings is instead specified via a "huge" input (the underlying graph G), that is far too large to be read by a sub-linear time algorithm. Instead, our implementation accesses G through local neighborhood probes, and is able to answer queries to the color of any given vertex in sub-linear time for q ≥ 9Δ, in a manner that is consistent with a specific random valid coloring of G. Furthermore, the implementation is memory-less, and can maintain consistency with non-communicating copies of itself.

Amartya Shankha Biswas, Ronitt Rubinfeld, and Anak Yodpinyanee. Local Access to Huge Random Objects Through Partial Sampling. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 27:1-27:65, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{biswas_et_al:LIPIcs.ITCS.2020.27, author = {Biswas, Amartya Shankha and Rubinfeld, Ronitt and Yodpinyanee, Anak}, title = {{Local Access to Huge Random Objects Through Partial Sampling}}, booktitle = {11th Innovations in Theoretical Computer Science Conference (ITCS 2020)}, pages = {27:1--27:65}, 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.2020.27}, URN = {urn:nbn:de:0030-drops-117126}, doi = {10.4230/LIPIcs.ITCS.2020.27}, annote = {Keywords: sublinear time algorithms, random generation, local computation} }

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

A probability distribution over the Boolean cube is monotone if flipping the value of a coordinate from zero to one can only increase the probability of an element. Given samples of an unknown monotone distribution over the Boolean cube, we give (to our knowledge) the first algorithm that learns an approximation of the distribution in statistical distance using a number of samples that is sublinear in the domain.
To do this, we develop a structural lemma describing monotone probability distributions. The structural lemma has further implications to the sample complexity of basic testing tasks for analyzing monotone probability distributions over the Boolean cube: We use it to give nontrivial upper bounds on the tasks of estimating the distance of a monotone distribution to uniform and of estimating the support size of a monotone distribution. In the setting of monotone probability distributions over the Boolean cube, our algorithms are the first to have sample complexity lower than known lower bounds for the same testing tasks on arbitrary (not necessarily monotone) probability distributions.
One further consequence of our learning algorithm is an improved sample complexity for the task of testing whether a distribution on the Boolean cube is monotone.

Ronitt Rubinfeld and Arsen Vasilyan. Monotone Probability Distributions over the Boolean Cube Can Be Learned with Sublinear Samples. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 28:1-28:34, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{rubinfeld_et_al:LIPIcs.ITCS.2020.28, author = {Rubinfeld, Ronitt and Vasilyan, Arsen}, title = {{Monotone Probability Distributions over the Boolean Cube Can Be Learned with Sublinear Samples}}, booktitle = {11th Innovations in Theoretical Computer Science Conference (ITCS 2020)}, pages = {28:1--28:34}, 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.2020.28}, URN = {urn:nbn:de:0030-drops-117138}, doi = {10.4230/LIPIcs.ITCS.2020.28}, annote = {Keywords: Learning distributions, monotone probability distributions, estimating support size} }

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RANDOM

**Published in:** LIPIcs, Volume 145, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)

The noise sensitivity of a Boolean function f: {0,1}^n - > {0,1} is one of its fundamental properties. For noise parameter delta, the noise sensitivity is denoted as NS_{delta}[f]. This quantity is defined as follows: First, pick x = (x_1,...,x_n) uniformly at random from {0,1}^n, then pick z by flipping each x_i independently with probability delta. NS_{delta}[f] is defined to equal Pr [f(x) != f(z)]. Much of the existing literature on noise sensitivity explores the following two directions: (1) Showing that functions with low noise-sensitivity are structured in certain ways. (2) Mathematically showing that certain classes of functions have low noise sensitivity. Combined, these two research directions show that certain classes of functions have low noise sensitivity and therefore have useful structure.
The fundamental importance of noise sensitivity, together with this wealth of structural results, motivates the algorithmic question of approximating NS_{delta}[f] given an oracle access to the function f. We show that the standard sampling approach is essentially optimal for general Boolean functions. Therefore, we focus on estimating the noise sensitivity of monotone functions, which form an important subclass of Boolean functions, since many functions of interest are either monotone or can be simply transformed into a monotone function (for example the class of unate functions consists of all the functions that can be made monotone by reorienting some of their coordinates [O'Donnell, 2014]).
Specifically, we study the algorithmic problem of approximating NS_{delta}[f] for monotone f, given the promise that NS_{delta}[f] >= 1/n^{C} for constant C, and for delta in the range 1/n <= delta <= 1/2. For such f and delta, we give a randomized algorithm performing O((min(1,sqrt{n} delta log^{1.5} n))/(NS_{delta}[f]) poly (1/epsilon)) queries and approximating NS_{delta}[f] to within a multiplicative factor of (1 +/- epsilon). Given the same constraints on f and delta, we also prove a lower bound of Omega((min(1,sqrt{n} delta))/(NS_{delta}[f] * n^{xi})) on the query complexity of any algorithm that approximates NS_{delta}[f] to within any constant factor, where xi can be any positive constant. Thus, our algorithm’s query complexity is close to optimal in terms of its dependence on n.
We introduce a novel descending-ascending view of noise sensitivity, and use it as a central tool for the analysis of our algorithm. To prove lower bounds on query complexity, we develop a technique that reduces computational questions about query complexity to combinatorial questions about the existence of "thin" functions with certain properties. The existence of such "thin" functions is proved using the probabilistic method. These techniques also yield new lower bounds on the query complexity of approximating other fundamental properties of Boolean functions: the total influence and the bias.

Ronitt Rubinfeld and Arsen Vasilyan. Approximating the Noise Sensitivity of a Monotone Boolean Function. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 52:1-52:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{rubinfeld_et_al:LIPIcs.APPROX-RANDOM.2019.52, author = {Rubinfeld, Ronitt and Vasilyan, Arsen}, title = {{Approximating the Noise Sensitivity of a Monotone Boolean Function}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)}, pages = {52:1--52:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-125-2}, ISSN = {1868-8969}, year = {2019}, volume = {145}, editor = {Achlioptas, Dimitris and V\'{e}gh, L\'{a}szl\'{o} A.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2019.52}, URN = {urn:nbn:de:0030-drops-112672}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2019.52}, annote = {Keywords: Monotone Boolean functions, noise sensitivity, influence} }

Document

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

A graph spanner is a fundamental graph structure that faithfully preserves the pairwise distances in the input graph up to a small multiplicative stretch. The common objective in the computation of spanners is to achieve the best-known existential size-stretch trade-off efficiently.
Classical models and algorithmic analysis of graph spanners essentially assume that the algorithm can read the input graph, construct the desired spanner, and write the answer to the output tape. However, when considering massive graphs containing millions or even billions of nodes not only the input graph, but also the output spanner might be too large for a single processor to store.
To tackle this challenge, we initiate the study of local computation algorithms (LCAs) for graph spanners in general graphs, where the algorithm should locally decide whether a given edge (u,v) in E belongs to the output (sparse) spanner or not. Such LCAs give the user the "illusion" that a specific sparse spanner for the graph is maintained, without ever fully computing it. We present several results for this setting, including:
- For general n-vertex graphs and for parameter r in {2,3}, there exists an LCA for (2r-1)-spanners with O~(n^{1+1/r}) edges and sublinear probe complexity of O~(n^{1-1/2r}). These size/stretch trade-offs are best possible (up to polylogarithmic factors).
- For every k >= 1 and n-vertex graph with maximum degree Delta, there exists an LCA for O(k^2) spanners with O~(n^{1+1/k}) edges, probe complexity of O~(Delta^4 n^{2/3}), and random seed of size polylog(n). This improves upon, and extends the work of [Lenzen-Levi, ICALP'18].
We also complement these constructions by providing a polynomial lower bound on the probe complexity of LCAs for graph spanners that holds even for the simpler task of computing a sparse connected subgraph with o(m) edges.
To the best of our knowledge, our results on 3 and 5-spanners are the first LCAs with sublinear (in Delta) probe-complexity for Delta = n^{Omega(1)}.

Merav Parter, Ronitt Rubinfeld, Ali Vakilian, and Anak Yodpinyanee. Local Computation Algorithms for Spanners. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 58:1-58:21, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{parter_et_al:LIPIcs.ITCS.2019.58, author = {Parter, Merav and Rubinfeld, Ronitt and Vakilian, Ali and Yodpinyanee, Anak}, title = {{Local Computation Algorithms for Spanners}}, booktitle = {10th Innovations in Theoretical Computer Science Conference (ITCS 2019)}, pages = {58:1--58:21}, 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.2019.58}, URN = {urn:nbn:de:0030-drops-101510}, doi = {10.4230/LIPIcs.ITCS.2019.58}, annote = {Keywords: Local Computation Algorithms, Sub-linear Algorithms, Graph Spanners} }

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**Published in:** LIPIcs, Volume 81, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)

We study the Fractional Set Cover problem in the streaming model. That is, we consider the relaxation of the set cover problem over a universe of n elements and a collection of m sets, where each set can be picked fractionally, with a value in [0,1]. We present a randomized (1+a)-approximation algorithm that makes p passes over the data, and uses O(polylog(m,n,1/a) (mn^(O(1/(pa)))+n)) memory space. The algorithm works in both the set arrival and the edge arrival models. To the best of our knowledge, this is the first streaming result for the fractional set cover problem. We obtain our results by employing the multiplicative weights update framework in the streaming settings.

Piotr Indyk, Sepideh Mahabadi, Ronitt Rubinfeld, Jonathan Ullman, Ali Vakilian, and Anak Yodpinyanee. Fractional Set Cover in the Streaming Model. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 12:1-12:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{indyk_et_al:LIPIcs.APPROX-RANDOM.2017.12, author = {Indyk, Piotr and Mahabadi, Sepideh and Rubinfeld, Ronitt and Ullman, Jonathan and Vakilian, Ali and Yodpinyanee, Anak}, title = {{Fractional Set Cover in the Streaming Model}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)}, pages = {12:1--12:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-044-6}, ISSN = {1868-8969}, year = {2017}, volume = {81}, editor = {Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.12}, URN = {urn:nbn:de:0030-drops-75613}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2017.12}, annote = {Keywords: Streaming Algorithms, Fractional Set Cover, LP relaxation, Multiplicative Weight Update} }

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Invited Talk

**Published in:** LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)

Consider a setting in which inputs to and outputs from a computational problem are so large, that there is not time to read them in their entirety. However, if one is only interested in small parts of the output at any given time, is it really necessary to solve the entire computational problem? Is it even necessary to view the whole input? We survey recent work in the model of local computation algorithms which for a given input, supports queries by a user to values of specified bits of a legal output. The goal is to design local computation algorithms in such a way that very little of the input needs to be seen in order to determine the value of any single bit of the output. In this talk, we describe results on a variety of problems for which sublinear time and space local computation algorithms have been developed - we will give special focus to finding maximal independent sets and sparse spanning graphs.

Ronitt Rubinfeld. Local Computation Algorithms (Invited Talk). In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, p. 3:1, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{rubinfeld:LIPIcs.ICALP.2017.3, author = {Rubinfeld, Ronitt}, title = {{Local Computation Algorithms}}, booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)}, pages = {3:1--3:1}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-041-5}, ISSN = {1868-8969}, year = {2017}, volume = {80}, editor = {Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.3}, URN = {urn:nbn:de:0030-drops-75068}, doi = {10.4230/LIPIcs.ICALP.2017.3}, annote = {Keywords: Massive Data Sets, Approximate Solutions, Maximal Independent Set, Sparse Spanning Graphs} }

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**Published in:** LIPIcs, Volume 60, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)

Constructing a spanning tree of a graph is one of the most basic tasks in graph theory. We consider this problem in the setting of local algorithms: one wants to quickly determine whether a given edge e is in a specific spanning tree, without computing the whole spanning tree, but rather by inspecting the local neighborhood of e. The challenge is to maintain consistency. That is, to answer queries about different edges according to the same spanning tree. Since it is known that this problem cannot be solved without essentially viewing all the graph, we consider the relaxed version of finding a spanning subgraph with (1+c)n edges instead of n-1 edges (where n is the number of vertices and c is a given approximation/sparsity parameter).
It is known that this relaxed problem requires inspecting order of n^{1/2} edges in general graphs (for any constant c), which motivates the study of natural restricted families of graphs. One such family is the family of graphs with an excluded minor (which in particular includes planar graphs). For this family there is an algorithm that achieves constant success probability, and inspects (d/c)^{poly(h)log(1/c)} edges (for each edge it is queried on), where d is the maximum degree in the graph and h is the size of the excluded minor. The distances between pairs of vertices in the spanning subgraph G' are at most a factor of poly(d, 1/c, h) larger than in G.
In this work, we show that for an input graph that is H-minor free for any H of size h, this task can be performed by inspecting only poly(d, 1/c, h) edges in G.
The distances between pairs of vertices in the spanning subgraph G' are at most a factor of h log(d)/c (up to poly-logarithmic factors) larger than in G. Furthermore, the error probability of the new algorithm is significantly improved to order of 1/n. This algorithm can also be easily adapted to yield an efficient algorithm for the distributed (message passing) setting.

Reut Levi, Dana Ron, and Ronitt Rubinfeld. A Local Algorithm for Constructing Spanners in Minor-Free Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 60, pp. 38:1-38:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{levi_et_al:LIPIcs.APPROX-RANDOM.2016.38, author = {Levi, Reut and Ron, Dana and Rubinfeld, Ronitt}, title = {{A Local Algorithm for Constructing Spanners in Minor-Free Graphs}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)}, pages = {38:1--38:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-018-7}, ISSN = {1868-8969}, year = {2016}, volume = {60}, editor = {Jansen, Klaus and Mathieu, Claire and Rolim, Jos\'{e} D. P. and Umans, Chris}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2016.38}, URN = {urn:nbn:de:0030-drops-66613}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2016.38}, annote = {Keywords: spanners, sparse subgraphs, local algorithms, excluded-minor} }

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**Published in:** LIPIcs, Volume 47, 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)

We study the question of testing structured properties (classes) of discrete distributions. Specifically, given sample access to an arbitrary distribution D over [n] and a property P, the goal is to distinguish between D in P and l_{1}(D,P)>epsilon. We develop a general algorithm for this question, which applies to a large range of "shape-constrained" properties, including monotone, log-concave, t-modal, piecewise-polynomial, and Poisson Binomial distributions. Moreover, for all cases considered, our algorithm has near-optimal sample complexity with regard to the domain size and is computationally efficient. For most of these classes, we provide the first non-trivial tester in the literature. In addition, we also describe a generic method to prove lower bounds for this problem, and use it to show our upper bounds are nearly tight. Finally, we extend some of our techniques to tolerant testing, deriving nearly-tight upper and lower bounds for the corresponding questions.

Clément L. Canonne, Ilias Diakonikolas, Themis Gouleakis, and Ronitt Rubinfeld. Testing Shape Restrictions of Discrete Distributions. In 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 47, pp. 25:1-25:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{canonne_et_al:LIPIcs.STACS.2016.25, author = {Canonne, Cl\'{e}ment L. and Diakonikolas, Ilias and Gouleakis, Themis and Rubinfeld, Ronitt}, title = {{Testing Shape Restrictions of Discrete Distributions}}, booktitle = {33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)}, pages = {25:1--25:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-001-9}, ISSN = {1868-8969}, year = {2016}, volume = {47}, editor = {Ollinger, Nicolas and Vollmer, Heribert}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2016.25}, URN = {urn:nbn:de:0030-drops-57260}, doi = {10.4230/LIPIcs.STACS.2016.25}, annote = {Keywords: property testing, probability distributions, statistics, lower bounds} }

Document

**Published in:** LIPIcs, Volume 28, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)

We initiate the study of the problem of designing sublinear-time (local) algorithms that, given an edge (u,v) in a connected graph G=(V,E), decide whether (u,v) belongs to a sparse spanning graph G' = (V,E') of G. Namely, G' should be connected and |E'| should be upper bounded by (1+epsilon)|V| for a given parameter epsilon > 0. To this end the algorithms may query the incidence relation of the graph G, and we seek algorithms whose query complexity and running time (per given edge (u,v)) is as small as possible. Such an algorithm may be randomized but (for a fixed choice of its random coins) its decision on different edges in the graph should be consistent with the same spanning graph G' and independent of the order of queries.
We first show that for general (bounded-degree) graphs, the query complexity of any such algorithm must be Omega(sqrt{|V|}). This lower bound holds for graphs that have high expansion. We then turn to design and analyze algorithms both for graphs with high expansion (obtaining a result that roughly matches the lower bound) and for graphs that are (strongly) non-expanding (obtaining results in which the complexity does not depend on |V|). The complexity of the problem for graphs that do not fall into these two categories is left as an open question.

Reut Levi, Dana Ron, and Ronitt Rubinfeld. Local Algorithms for Sparse Spanning Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 28, pp. 826-842, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2014)

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@InProceedings{levi_et_al:LIPIcs.APPROX-RANDOM.2014.826, author = {Levi, Reut and Ron, Dana and Rubinfeld, Ronitt}, title = {{Local Algorithms for Sparse Spanning Graphs}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)}, pages = {826--842}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-74-3}, ISSN = {1868-8969}, year = {2014}, volume = {28}, editor = {Jansen, Klaus and Rolim, Jos\'{e} and Devanur, Nikhil R. and Moore, Cristopher}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2014.826}, URN = {urn:nbn:de:0030-drops-47410}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2014.826}, annote = {Keywords: local, spanning graph, sparse} }

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**Published in:** Dagstuhl Seminar Proceedings, Volume 8341, Sublinear Algorithms (2008)

From August 17 to August 22, 2008, the Dagstuhl Seminar 08341 ``Sublinear Algorithms'' was held in the International Conference and Research Center (IBFI),
Schloss Dagstuhl.
During the seminar, several participants presented their current
research, and ongoing work and open problems were discussed. Abstracts of
the presentations given during the seminar as well as abstracts of
seminar results and ideas are put together in this paper. The first section
describes the seminar topics and goals in general.
Links to extended abstracts or full papers are provided, if available.

Artur Czumaj, S. Muthu Muthukrishnan, Ronitt Rubinfeld, and Christian Sohler. 08341 Abstracts Collection – Sublinear Algorithms. In Sublinear Algorithms. Dagstuhl Seminar Proceedings, Volume 8341, pp. 1-19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2008)

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@InProceedings{czumaj_et_al:DagSemProc.08341.1, author = {Czumaj, Artur and Muthukrishnan, S. Muthu and Rubinfeld, Ronitt and Sohler, Christian}, title = {{08341 Abstracts Collection – Sublinear Algorithms}}, booktitle = {Sublinear Algorithms}, pages = {1--19}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2008}, volume = {8341}, editor = {Artur Czumaj and S. Muthu Muthukrishnan and Ronitt Rubinfeld and Christian Sohler}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.08341.1}, URN = {urn:nbn:de:0030-drops-16981}, doi = {10.4230/DagSemProc.08341.1}, annote = {Keywords: Sublinear algorithms, property testing, data streaming, graph algorithms, approximation algorithms} }

Document

**Published in:** Dagstuhl Seminar Proceedings, Volume 8341, Sublinear Algorithms (2008)

This report summarizes the content and structure of the Dagstuhl seminar `Sublinear Algorithms', which was held from 17.8.2008 to 22.8.2008 in Schloss Dagstuhl, Germany.

Artur Czumaj, S. Muthu Muthukrishnan, Ronitt Rubinfeld, and Christian Sohler. 08341 Executive Summary – Sublinear Algorithms. In Sublinear Algorithms. Dagstuhl Seminar Proceedings, Volume 8341, pp. 1-2, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2008)

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@InProceedings{czumaj_et_al:DagSemProc.08341.2, author = {Czumaj, Artur and Muthukrishnan, S. Muthu and Rubinfeld, Ronitt and Sohler, Christian}, title = {{08341 Executive Summary – Sublinear Algorithms}}, booktitle = {Sublinear Algorithms}, pages = {1--2}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2008}, volume = {8341}, editor = {Artur Czumaj and S. Muthu Muthukrishnan and Ronitt Rubinfeld and Christian Sohler}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.08341.2}, URN = {urn:nbn:de:0030-drops-16964}, doi = {10.4230/DagSemProc.08341.2}, annote = {Keywords: Sublinear algorithms, property testing, data streaming, graph algorithms, approximation algorithms} }

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**Published in:** Dagstuhl Seminar Proceedings, Volume 5291, Sublinear Algorithms (2006)

From 17.07.05 to 22.07.05, the Dagstuhl Seminar
05291 ``Sublinear Algorithms'' was held
in the International Conference and Research Center (IBFI),
Schloss Dagstuhl.
During the seminar, several participants presented their current
research, and ongoing work and open problems were discussed. Abstracts of
the presentations given during the seminar as well as abstracts of
seminar results and ideas are put together in this paper. The first section
describes the seminar topics and goals in general.
Links to extended abstracts or full papers are provided, if available.

Artur Czumaj, S. Muthu Muthukrishnan, Ronitt Rubinfeld, and Christian Sohler. 05291 Abstracts Collection – Sublinear Algorithms. In Sublinear Algorithms. Dagstuhl Seminar Proceedings, Volume 5291, pp. 1-18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2006)

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@InProceedings{czumaj_et_al:DagSemProc.05291.1, author = {Czumaj, Artur and Muthukrishnan, S. Muthu and Rubinfeld, Ronitt and Sohler, Christian}, title = {{05291 Abstracts Collection – Sublinear Algorithms}}, booktitle = {Sublinear Algorithms}, pages = {1--18}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2006}, volume = {5291}, editor = {Artur Czumaj and S. Muthu Muthukrishnan and Ronitt Rubinfeld and Christian Sohler}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.05291.1}, URN = {urn:nbn:de:0030-drops-6814}, doi = {10.4230/DagSemProc.05291.1}, annote = {Keywords: Property testing, sublinear time approximation algorithms, data streaming algorithms} }

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