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**Published in:** LIPIcs, Volume 148, 14th International Symposium on Parameterized and Exact Computation (IPEC 2019)

Parameterized complexity attempts to give a more fine-grained analysis of the complexity of problems: instead of measuring the running time as a function of only the input size, we analyze the running time with respect to additional parameters. This approach has proven to be highly successful in delineating our understanding of NP-hard problems. Given this success with the TIME resource, it seems but natural to use this approach for dealing with the SPACE resource. First attempts in this direction have considered a few individual problems, with some success: Fafianie and Kratsch [MFCS'14] and Chitnis et al. [SODA'15] introduced the notions of streaming kernels and parameterized streaming algorithms respectively. For example, the latter shows how to refine the Omega(n^2) bit lower bound for finding a minimum Vertex Cover (VC) in the streaming setting by designing an algorithm for the parameterized k-VC problem which uses O(k^{2}log n) bits.
In this paper, we initiate a systematic study of graph problems from the paradigm of parameterized streaming algorithms. We first define a natural hierarchy of space complexity classes of FPS, SubPS, SemiPS, SupPS and BrutePS, and then obtain tight classifications for several well-studied graph problems such as Longest Path, Feedback Vertex Set, Dominating Set, Girth, Treewidth, etc. into this hierarchy (see Figure 1 and Table 1). On the algorithmic side, our parameterized streaming algorithms use techniques from the FPT world such as bidimensionality, iterative compression and bounded-depth search trees. On the hardness side, we obtain lower bounds for the parameterized streaming complexity of various problems via novel reductions from problems in communication complexity. We also show a general (unconditional) lower bound for space complexity of parameterized streaming algorithms for a large class of problems inspired by the recently developed frameworks for showing (conditional) kernelization lower bounds.
Parameterized algorithms and streaming algorithms are approaches to cope with TIME and SPACE intractability respectively. It is our hope that this work on parameterized streaming algorithms leads to two-way flow of ideas between these two previously separated areas of theoretical computer science.

Rajesh Chitnis and Graham Cormode. Towards a Theory of Parameterized Streaming Algorithms. In 14th International Symposium on Parameterized and Exact Computation (IPEC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 148, pp. 7:1-7:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{chitnis_et_al:LIPIcs.IPEC.2019.7, author = {Chitnis, Rajesh and Cormode, Graham}, title = {{Towards a Theory of Parameterized Streaming Algorithms}}, booktitle = {14th International Symposium on Parameterized and Exact Computation (IPEC 2019)}, pages = {7:1--7:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-129-0}, ISSN = {1868-8969}, year = {2019}, volume = {148}, editor = {Jansen, Bart M. P. and Telle, Jan Arne}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2019.7}, URN = {urn:nbn:de:0030-drops-114682}, doi = {10.4230/LIPIcs.IPEC.2019.7}, annote = {Keywords: Parameterized Algorithms, Streaming Algorithms, Kernels} }

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**Published in:** LIPIcs, Volume 149, 30th International Symposium on Algorithms and Computation (ISAAC 2019)

Motivated by the growth in outsourced data analysis, we describe methods for verifying basic linear algebra operations performed by a cloud service without having to recalculate the entire result. We provide novel protocols in the streaming setting for inner product, matrix multiplication and vector-matrix-vector multiplication where the number of rounds of interaction can be adjusted to tradeoff space, communication, and duration of the protocol. Previous work suggests that the costs of these interactive protocols are optimized by choosing O(log n) rounds. However, we argue that we can reduce the number of rounds without incurring a significant time penalty by considering the total end-to-end time, so fewer rounds and larger messages are preferable. We confirm this claim with an experimental study that shows that a constant number of rounds gives the fastest protocol.

Graham Cormode and Chris Hickey. Efficient Interactive Proofs for Linear Algebra. In 30th International Symposium on Algorithms and Computation (ISAAC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 149, pp. 48:1-48:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{cormode_et_al:LIPIcs.ISAAC.2019.48, author = {Cormode, Graham and Hickey, Chris}, title = {{Efficient Interactive Proofs for Linear Algebra}}, booktitle = {30th International Symposium on Algorithms and Computation (ISAAC 2019)}, pages = {48:1--48:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-130-6}, ISSN = {1868-8969}, year = {2019}, volume = {149}, editor = {Lu, Pinyan and Zhang, Guochuan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2019.48}, URN = {urn:nbn:de:0030-drops-115449}, doi = {10.4230/LIPIcs.ISAAC.2019.48}, annote = {Keywords: Streaming Interactive Proofs, Linear Algebra} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

We consider the maximal and maximum independent set problems in three models of graph streams:
- In the edge model we see a stream of edges which collectively define a graph; this model is well-studied for a variety of problems. We show that the space complexity for a one-pass streaming algorithm to find a maximal independent set is quadratic (i.e. we must store all edges). We further show that it is not much easier if we only require approximate maximality. This contrasts strongly with the other two vertex-based models, where one can greedily find an exact solution in only the space needed to store the independent set.
- In the "explicit" vertex model, the input stream is a sequence of vertices making up the graph. Every vertex arrives along with its incident edges that connect to previously arrived vertices. Various graph problems require substantially less space to solve in this setting than in edge-arrival streams. We show that every one-pass c-approximation streaming algorithm for maximum independent set (MIS) on explicit vertex streams requires Omega({n^2}/{c^6}) bits of space, where n is the number of vertices of the input graph. It is already known that Theta~({n^2}/{c^2}) bits of space are necessary and sufficient in the edge arrival model (Halldórsson et al. 2012), thus the MIS problem is not significantly easier to solve under the explicit vertex arrival order assumption. Our result is proved via a reduction from a new multi-party communication problem closely related to pointer jumping.
- In the "implicit" vertex model, the input stream consists of a sequence of objects, one per vertex. The algorithm is equipped with a function that maps pairs of objects to the presence or absence of edges, thus defining the graph. This model captures, for example, geometric intersection graphs such as unit disc graphs. Our final set of results consists of several improved upper and lower bounds for interval and square intersection graphs, in both explicit and implicit streams. In particular, we show a gap between the hardness of the explicit and implicit vertex models for interval graphs.

Graham Cormode, Jacques Dark, and Christian Konrad. Independent Sets in Vertex-Arrival Streams. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 45:1-45:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{cormode_et_al:LIPIcs.ICALP.2019.45, author = {Cormode, Graham and Dark, Jacques and Konrad, Christian}, title = {{Independent Sets in Vertex-Arrival Streams}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {45:1--45:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-109-2}, ISSN = {1868-8969}, year = {2019}, volume = {132}, editor = {Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.45}, URN = {urn:nbn:de:0030-drops-106212}, doi = {10.4230/LIPIcs.ICALP.2019.45}, annote = {Keywords: streaming algorithms, independent set size, lower bounds} }

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**Published in:** LIPIcs, Volume 98, 21st International Conference on Database Theory (ICDT 2018)

Many data sources can be interpreted as time-series, and a key problem is to identify which pairs out of a large collection of signals are highly correlated. We expect that there will be few, large, interesting correlations, while most signal pairs do not have any strong correlation. We abstract this as the problem of identifying the highly correlated pairs in a collection of n mostly pairwise uncorrelated random variables, where observations of the variables arrives as a stream. Dimensionality reduction can remove dependence on the number of observations, but further techniques are required to tame the quadratic (in n) cost of a search through all possible pairs.
We develop a new algorithm for rapidly finding large correlations based on sketch techniques with an added twist: we quickly generate sketches of random combinations of signals, and use these in concert with ideas from coding theory to decode the identity of correlated pairs. We prove correctness and compare performance and effectiveness with the best LSH (locality sensitive hashing) based approach.

Graham Cormode and Jacques Dark. Fast Sketch-based Recovery of Correlation Outliers. In 21st International Conference on Database Theory (ICDT 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 98, pp. 13:1-13:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{cormode_et_al:LIPIcs.ICDT.2018.13, author = {Cormode, Graham and Dark, Jacques}, title = {{Fast Sketch-based Recovery of Correlation Outliers}}, booktitle = {21st International Conference on Database Theory (ICDT 2018)}, pages = {13:1--13:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-063-7}, ISSN = {1868-8969}, year = {2018}, volume = {98}, editor = {Kimelfeld, Benny and Amsterdamer, Yael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICDT.2018.13}, URN = {urn:nbn:de:0030-drops-86094}, doi = {10.4230/LIPIcs.ICDT.2018.13}, annote = {Keywords: correlation, sketching, streaming, dimensionality reduction} }

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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 75, 16th International Symposium on Experimental Algorithms (SEA 2017)

Streaming algorithms must process a large quantity of small updates quickly to allow queries about the input to be answered from a small summary. Initial work on streaming algorithms laid out theoretical results, and subsequent efforts have involved engineering these for practical use. Informed by experiments, streaming algorithms have been widely implemented and used in practice. This talk will survey this line of work, and identify some lessons learned.

Graham Cormode. Engineering Streaming Algorithms. In 16th International Symposium on Experimental Algorithms (SEA 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 75, p. 3:1, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{cormode:LIPIcs.SEA.2017.3, author = {Cormode, Graham}, title = {{Engineering Streaming Algorithms}}, booktitle = {16th International Symposium on Experimental Algorithms (SEA 2017)}, pages = {3:1--3:1}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-036-1}, ISSN = {1868-8969}, year = {2017}, volume = {75}, editor = {Iliopoulos, Costas S. and Pissis, Solon P. and Puglisi, Simon J. and Raman, Rajeev}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2017.3}, URN = {urn:nbn:de:0030-drops-76270}, doi = {10.4230/LIPIcs.SEA.2017.3}, annote = {Keywords: Data stream algorithms} }

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

In the setting of streaming interactive proofs (SIPs), a client (verifier) needs to compute a given function on a massive stream of data, arriving online, but is unable to store even a small fraction of the data. It outsources the processing to a third party service (prover), but is unwilling to blindly trust answers returned by this service. Thus, the service cannot simply supply the desired answer; it must convince the verifier of its correctness via a short interaction after the stream has been seen.
In this work we study "barely interactive" SIPs. Specifically, we show that two or three rounds of interaction suffice to solve several query problems - including Index, Median, Nearest Neighbor Search, Pattern Matching, and Range Counting - with polylogarithmic space and communication costs. Such efficiency with O(1) rounds of interaction was thought to be impossible based on previous work.
On the other hand, we initiate a formal study of the limitations of constant-round SIPs by introducing a new hierarchy of communication models called Online Interactive Proofs (OIPs). The online nature of these models is analogous to the streaming restriction placed upon the verifier in an SIP. We give upper and lower bounds that (1) characterize, up to quadratic blowups, every finite level of the OIP hierarchy in terms of other well-known communication complexity classes, (2) separate the first four levels of the hierarchy, and (3) reveal that the hierarchy collapses to the fourth level. Our study of OIPs reveals marked contrasts and some parallels with the classic Turing Machine theory of interactive proofs, establishes limits on the power of existing techniques for developing constant-round SIPs, and provides a new characterization of (non-online) Arthur-Merlin communication in terms of an online model.

Amit Chakrabarti, Graham Cormode, Andrew McGregor, Justin Thaler, and Suresh Venkatasubramanian. Verifiable Stream Computation and Arthur–Merlin Communication. In 30th Conference on Computational Complexity (CCC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 33, pp. 217-243, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{chakrabarti_et_al:LIPIcs.CCC.2015.217, author = {Chakrabarti, Amit and Cormode, Graham and McGregor, Andrew and Thaler, Justin and Venkatasubramanian, Suresh}, title = {{Verifiable Stream Computation and Arthur–Merlin Communication}}, booktitle = {30th Conference on Computational Complexity (CCC 2015)}, pages = {217--243}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-81-1}, ISSN = {1868-8969}, year = {2015}, volume = {33}, editor = {Zuckerman, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2015.217}, URN = {urn:nbn:de:0030-drops-50680}, doi = {10.4230/LIPIcs.CCC.2015.217}, annote = {Keywords: Arthur-Merlin communication complexity, streaming interactive proofs} }

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Invited Talk

**Published in:** LIPIcs, Volume 31, 18th International Conference on Database Theory (ICDT 2015)

The demands to make data available are growing ever louder, including open data initiatives and "data monetization". But the problem of doing so without disclosing confidential information is a subtle and difficult one. Is "private data release" an oxymoron? This paper (accompanying an invited talk) aims to delve into the motivations of data release, explore the challenges, and outline some of the current statistical approaches developed in response to this confounding problem.

Graham Cormode. The Confounding Problem of Private Data Release (Invited Talk). In 18th International Conference on Database Theory (ICDT 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 31, pp. 1-12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{cormode:LIPIcs.ICDT.2015.1, author = {Cormode, Graham}, title = {{The Confounding Problem of Private Data Release}}, booktitle = {18th International Conference on Database Theory (ICDT 2015)}, pages = {1--12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-79-8}, ISSN = {1868-8969}, year = {2015}, volume = {31}, editor = {Arenas, Marcelo and Ugarte, Mart{\'\i}n}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICDT.2015.1}, URN = {urn:nbn:de:0030-drops-49977}, doi = {10.4230/LIPIcs.ICDT.2015.1}, annote = {Keywords: privacy, anonymization, data release} }

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