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**Published in:** LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)

We present novel lower bounds in the Merlin-Arthur (MA) communication model and the related annotated streaming or stream verification model. The MA communication model extends the classical communication model by introducing an all-powerful but untrusted player, Merlin, who knows the inputs of the usual players, Alice and Bob, and attempts to convince them about the output. We focus on the online MA (OMA) model where Alice and Merlin each send a single message to Bob, who needs to catch Merlin if he is dishonest and announce the correct output otherwise. Most known functions have OMA protocols with total communication significantly smaller than what would be needed without Merlin. In this work, we introduce the notion of non-trivial-OMA complexity of a function. This is the minimum total communication required when we restrict ourselves to only non-trivial protocols where Alice sends Bob fewer bits than what she would have sent without Merlin. We exhibit the first explicit functions that have this complexity superlinear - even exponential - in their classical one-way complexity: this means the trivial protocol, where Merlin communicates nothing and Alice and Bob compute the function on their own, is exponentially better than any non-trivial protocol in terms of total communication. These OMA lower bounds also translate to the annotated streaming model, the MA analogue of single-pass data streaming. We show large separations between the classical streaming complexity and the non-trivial annotated streaming complexity (for the analogous notion in this setting) of fundamental problems such as counting distinct items, as well as of graph problems such as connectivity and k-connectivity in a certain edge update model called the support graph turnstile model that we introduce here.

Prantar Ghosh and Vihan Shah. New Lower Bounds in Merlin-Arthur Communication and Graph Streaming Verification. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 53:1-53:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{ghosh_et_al:LIPIcs.ITCS.2024.53, author = {Ghosh, Prantar and Shah, Vihan}, title = {{New Lower Bounds in Merlin-Arthur Communication and Graph Streaming Verification}}, booktitle = {15th Innovations in Theoretical Computer Science Conference (ITCS 2024)}, pages = {53:1--53:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-309-6}, ISSN = {1868-8969}, year = {2024}, volume = {287}, editor = {Guruswami, Venkatesan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.53}, URN = {urn:nbn:de:0030-drops-195815}, doi = {10.4230/LIPIcs.ITCS.2024.53}, annote = {Keywords: Graph Algorithms, Streaming, Communication Complexity, Stream Verification, Merlin-Arthur Communication, Lower Bounds} }

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**Published in:** LIPIcs, Volume 255, 26th International Conference on Database Theory (ICDT 2023)

Estimating quantiles, like the median or percentiles, is a fundamental task in data mining and data science. A (streaming) quantile summary is a data structure that can process a set S of n elements in a streaming fashion and at the end, for any ϕ ∈ (0,1], return a ϕ-quantile of S up to an ε error, i.e., return a ϕ'-quantile with ϕ' = ϕ ± ε. We are particularly interested in comparison-based summaries that only compare elements of the universe under a total ordering and are otherwise completely oblivious of the universe. The best known deterministic quantile summary is the 20-year old Greenwald-Khanna (GK) summary that uses O((1/ε) log{(ε n)}) space [SIGMOD'01]. This bound was recently proved to be optimal for all deterministic comparison-based summaries by Cormode and Vesleý [PODS'20].
In this paper, we study weighted quantiles, a generalization of the quantiles problem, where each element arrives with a positive integer weight which denotes the number of copies of that element being inserted. The only known method of handling weighted inputs via GK summaries is the naive approach of breaking each weighted element into multiple unweighted items, and feeding them one by one to the summary, which results in a prohibitively large update time (proportional to the maximum weight of input elements).
We give the first non-trivial extension of GK summaries for weighted inputs and show that it takes O((1/ε) log(εn)) space and O(log(1/ε)+log log(εn)) update time per element to process a stream of length n (under some quite mild assumptions on the range of weights and ε). En route to this, we also simplify the original GK summaries for unweighted quantiles.

Sepehr Assadi, Nirmit Joshi, Milind Prabhu, and Vihan Shah. Generalizing Greenwald-Khanna Streaming Quantile Summaries for Weighted Inputs. In 26th International Conference on Database Theory (ICDT 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 255, pp. 19:1-19:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{assadi_et_al:LIPIcs.ICDT.2023.19, author = {Assadi, Sepehr and Joshi, Nirmit and Prabhu, Milind and Shah, Vihan}, title = {{Generalizing Greenwald-Khanna Streaming Quantile Summaries for Weighted Inputs}}, booktitle = {26th International Conference on Database Theory (ICDT 2023)}, pages = {19:1--19:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-270-9}, ISSN = {1868-8969}, year = {2023}, volume = {255}, editor = {Geerts, Floris and Vandevoort, Brecht}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICDT.2023.19}, URN = {urn:nbn:de:0030-drops-177618}, doi = {10.4230/LIPIcs.ICDT.2023.19}, annote = {Keywords: Streaming algorithms, Quantile summaries, Rank estimation} }

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APPROX

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

We optimally resolve the space complexity for the problem of finding an α-approximate minimum vertex cover (αMVC) in dynamic graph streams. We give a randomised algorithm for αMVC which uses O(n²/α²) bits of space matching Dark and Konrad’s lower bound [CCC 2020] up to constant factors. By computing a random greedy matching, we identify "easy" instances of the problem which can trivially be solved by returning the entire vertex set. The remaining "hard" instances, then have sparse induced subgraphs which we exploit to get our space savings and solve αMVC.
Achieving this type of optimality result is crucial for providing a complete understanding of a problem, and it has been gaining interest within the dynamic graph streaming community. For connectivity, Nelson and Yu [SODA 2019] improved the lower bound showing that Ω(n log³ n) bits of space is necessary while Ahn, Guha, and McGregor [SODA 2012] have shown that O(n log³ n) bits is sufficient. For finding an α-approximate maximum matching, the upper bound was improved by Assadi and Shah [ITCS 2022] showing that O(n²/α³) bits is sufficient while Dark and Konrad [CCC 2020] have shown that Ω(n²/α³) bits is necessary. The space complexity, however, remains unresolved for many other dynamic graph streaming problems where further improvements can still be made.

Kheeran K. Naidu and Vihan Shah. Space Optimal Vertex Cover in Dynamic Streams. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 245, pp. 53:1-53:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{naidu_et_al:LIPIcs.APPROX/RANDOM.2022.53, author = {Naidu, Kheeran K. and Shah, Vihan}, title = {{Space Optimal Vertex Cover in Dynamic Streams}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)}, pages = {53:1--53:15}, 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.2022.53}, URN = {urn:nbn:de:0030-drops-171753}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2022.53}, annote = {Keywords: Graph Streaming Algorithms, Vertex Cover, Dynamic Streams, Approximation Algorithm} }

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

We present an algorithm for the maximum matching problem in dynamic (insertion-deletions) streams with asymptotically optimal space: for any n-vertex graph, our algorithm with high probability outputs an α-approximate matching in a single pass using O(n²/α³) bits of space.
A long line of work on the dynamic streaming matching problem has reduced the gap between space upper and lower bounds first to n^{o(1)} factors [Assadi-Khanna-Li-Yaroslavtsev; SODA 2016] and subsequently to polylog factors [Dark-Konrad; CCC 2020]. Our upper bound now matches the Dark-Konrad lower bound up to O(1) factors, thus completing this research direction.
Our approach consists of two main steps: we first (provably) identify a family of graphs, similar to the instances used in prior work to establish the lower bounds for this problem, as the only "hard" instances to focus on. These graphs include an induced subgraph which is both sparse and contains a large matching. We then design a dynamic streaming algorithm for this family of graphs which is more efficient than prior work. The key to this efficiency is a novel sketching method, which bypasses the typical loss of polylog(n)-factors in space compared to standard L₀-sampling primitives, and can be of independent interest in designing optimal algorithms for other streaming problems.

Sepehr Assadi and Vihan Shah. An Asymptotically Optimal Algorithm for Maximum Matching in Dynamic Streams. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 9:1-9:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{assadi_et_al:LIPIcs.ITCS.2022.9, author = {Assadi, Sepehr and Shah, Vihan}, title = {{An Asymptotically Optimal Algorithm for Maximum Matching in Dynamic Streams}}, booktitle = {13th Innovations in Theoretical Computer Science Conference (ITCS 2022)}, pages = {9:1--9:23}, 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.2022.9}, URN = {urn:nbn:de:0030-drops-156054}, doi = {10.4230/LIPIcs.ITCS.2022.9}, annote = {Keywords: Graph streaming algorithms, Sketching, Maximum matching} }

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