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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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**Published in:** LIPIcs, Volume 87, 25th Annual European Symposium on Algorithms (ESA 2017)

Estimating the size of the maximum matching is a canonical problem in graph analysis, and one that has attracted extensive study over a range of different computational models. We present improved streaming algorithms for approximating the size of maximum matching with sparse (bounded arboricity) graphs.
* (Insert-Only Streams) We present a one-pass algorithm that takes O(alpha log n) space and approximates the size of the maximum matching in graphs with arboricity alpha within a factor of O(alpha). This improves significantly upon the state-of-the-art tilde{O}(alpha n^{2/3})-space streaming algorithms, and is the first poly-logarithmic space algorithm for this problem.
* (Dynamic Streams) Given a dynamic graph stream (i.e., inserts and deletes) of edges of an underlying alpha-bounded arboricity graph, we present an one-pass algorithm that uses space tilde{O}(alpha^{10/3}n^{2/3}) and returns an O(alpha)-estimator for the size of the maximum matching on the condition that the number edge deletions in the stream is bounded by O(alpha n). For this class of inputs, our algorithm improves the state-of-the-art tilde{O}(\alpha n^{4/5})-space algorithms, where the \tilde{O}(.) notation hides logarithmic in n dependencies.
In contrast to prior work, our results take more advantage of the streaming access to the input and characterize the matching size based on the ordering of the edges in the stream in addition to the degree distributions and structural properties of the sparse graphs.

Graham Cormode, Hossein Jowhari, Morteza Monemizadeh, and S. Muthukrishnan. The Sparse Awakens: Streaming Algorithms for Matching Size Estimation in Sparse Graphs. In 25th Annual European Symposium on Algorithms (ESA 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 87, pp. 29:1-29:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{cormode_et_al:LIPIcs.ESA.2017.29, author = {Cormode, Graham and Jowhari, Hossein and Monemizadeh, Morteza and Muthukrishnan, S.}, title = {{The Sparse Awakens: Streaming Algorithms for Matching Size Estimation in Sparse Graphs}}, booktitle = {25th Annual European Symposium on Algorithms (ESA 2017)}, pages = {29:1--29:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-049-1}, ISSN = {1868-8969}, year = {2017}, volume = {87}, editor = {Pruhs, Kirk and Sohler, Christian}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2017.29}, URN = {urn:nbn:de:0030-drops-78499}, doi = {10.4230/LIPIcs.ESA.2017.29}, annote = {Keywords: streaming algorithms, matching size} }

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

We study which property testing and sublinear time algorithms can be transformed into graph streaming algorithms for random order streams. Our main result is that for bounded degree graphs, any property that is constant-query testable in the adjacency list model can be tested with constant space in a single-pass in random order streams. Our result is obtained by estimating the distribution of local neighborhoods of the vertices on a random order graph stream using constant space.
We then show that our approach can also be applied to constant time approximation algorithms for bounded degree graphs in the adjacency list model: As an example, we obtain a constant-space single-pass random order streaming algorithms for approximating the size of a maximum matching with additive error epsilon n (n is the number of nodes).
Our result establishes for the first time that a large class of sublinear algorithms can be simulated in random order streams, while Omega(n) space is needed for many graph streaming problems for adversarial orders.

Morteza Monemizadeh, S. Muthukrishnan, Pan Peng, and Christian Sohler. Testable Bounded Degree Graph Properties Are Random Order Streamable. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 131:1-131:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{monemizadeh_et_al:LIPIcs.ICALP.2017.131, author = {Monemizadeh, Morteza and Muthukrishnan, S. and Peng, Pan and Sohler, Christian}, title = {{Testable Bounded Degree Graph Properties Are Random Order Streamable}}, booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)}, pages = {131:1--131:14}, 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.2017.131}, URN = {urn:nbn:de:0030-drops-74782}, doi = {10.4230/LIPIcs.ICALP.2017.131}, annote = {Keywords: Graph streaming algorithms, graph property testing, constant-time approximation algorithms} }

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**Published in:** LIPIcs, Volume 4, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (2009)

We study functionally private approximations.
An approximation function $g$ is {\em functionally private} with respect to
$f$ if, for any input $x$, $g(x)$ reveals no more information about $x$ than
$f(x)$.
Our main result states that a function $f$ admits an efficiently-computable
functionally private approximation $g$ if there exists an efficiently-computable
and negligibly-biased estimator for $f$.
Contrary to previous generic results, our theorem is more general and
has a wider application reach.We provide two distinct applications of the above result to demonstrate its flexibility.
In the data stream model, we provide a functionally private approximation to the
$L_p$-norm estimation problem, a quintessential application in streaming, using only
polylogarithmic space in the input size.
The privacy guarantees rely on the use of pseudo-random {\em
functions} (PRF) (a stronger cryptographic notion than pseudo-random
generators) of which can be based on common cryptographic assumptions.The application of PRFs in this context appears to be novel and we expect other results to follow suit.Moreover, this is the first known functionally private streaming result for {\em any} problem.
Our second application result states that every problem in some subclasses of \SP of
hard counting problems admit efficient and functionally private approximation protocols.
This result is based on a functionally private approximation for the \SDNF
problem (or estimating the number of satisfiable truth assignments to a
Boolean formula in disjunctive normal form), which is an application of our
main theorem and previously known results.

André Madeira and S. Muthukrishnan. Functionally Private Approximations of Negligibly-Biased Estimators. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 4, pp. 323-334, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{madeira_et_al:LIPIcs.FSTTCS.2009.2329, author = {Madeira, Andr\'{e} and Muthukrishnan, S.}, title = {{Functionally Private Approximations of Negligibly-Biased Estimators}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science}, pages = {323--334}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-13-2}, ISSN = {1868-8969}, year = {2009}, volume = {4}, editor = {Kannan, Ravi and Narayan Kumar, K.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2009.2329}, URN = {urn:nbn:de:0030-drops-23298}, doi = {10.4230/LIPIcs.FSTTCS.2009.2329}, annote = {Keywords: Functional privacy, privacy, data streams, #P-complete} }

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**Published in:** LIPIcs, Volume 3, 26th International Symposium on Theoretical Aspects of Computer Science (2009)

Given string $S[1..N]$ and integer $k$, the {\em suffix selection} problem is to determine the $k$th lexicographically smallest amongst the suffixes $S[i\ldots N]$, $1 \leq i \leq N$. We study the suffix selection problem in the cache-aware model that captures two-level memory inherent in computing systems, for a \emph{cache} of limited size $M$ and block size $B$. The complexity of interest is the number of block transfers. We present an optimal suffix selection algorithm in the cache-aware model, requiring $\Theta\left(N/B\right)$ block transfers, for any string $S$ over an unbounded alphabet (where characters can only be compared), under the common tall-cache assumption (i.e. $M=\Omega\left(B^{1+\epsilon}\right)$, where $\epsilon<1$). Our algorithm beats the bottleneck bound for permuting an input array to the desired output array, which holds for nearly any nontrivial problem in hierarchical memory models.

Gianni Franceschini, Roberto Grossi, and S. Muthukrishnan. Optimal Cache-Aware Suffix Selection. In 26th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 3, pp. 457-468, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{franceschini_et_al:LIPIcs.STACS.2009.1845, author = {Franceschini, Gianni and Grossi, Roberto and Muthukrishnan, S.}, title = {{Optimal Cache-Aware Suffix Selection}}, booktitle = {26th International Symposium on Theoretical Aspects of Computer Science}, pages = {457--468}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-09-5}, ISSN = {1868-8969}, year = {2009}, volume = {3}, editor = {Albers, Susanne and Marion, Jean-Yves}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2009.1845}, URN = {urn:nbn:de:0030-drops-18452}, doi = {10.4230/LIPIcs.STACS.2009.1845}, annote = {Keywords: } }

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