Document

Complete Volume

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

LIPIcs, Volume 245, APPROX/RANDOM 2022, Complete Volume

Amit Chakrabarti and Chaitanya Swamy. LIPIcs, Volume 245, APPROX/RANDOM 2022, Complete Volume. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 245, pp. 1-1064, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@Proceedings{chakrabarti_et_al:LIPIcs.APPROX/RANDOM.2022, title = {{LIPIcs, Volume 245, APPROX/RANDOM 2022, Complete Volume}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)}, pages = {1--1064}, 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}, URN = {urn:nbn:de:0030-drops-171211}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2022}, annote = {Keywords: LIPIcs, Volume 245, APPROX/RANDOM 2022, Complete Volume} }

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Front Matter

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

Front Matter, Table of Contents, Preface, Conference Organization

Amit Chakrabarti and Chaitanya Swamy. Front Matter, Table of Contents, Preface, Conference Organization. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 245, pp. 0:i-0:xx, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{chakrabarti_et_al:LIPIcs.APPROX/RANDOM.2022.0, author = {Chakrabarti, Amit and Swamy, Chaitanya}, title = {{Front Matter, Table of Contents, Preface, Conference Organization}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)}, pages = {0:i--0:xx}, 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.2022.0}, URN = {urn:nbn:de:0030-drops-171229}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2022.0}, annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization} }

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**Published in:** LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

We consider the problem of space-efficiently estimating the number of simplices in a hypergraph stream. This is the most natural hypergraph generalization of the highly-studied problem of estimating the number of triangles in a graph stream. Our input is a k-uniform hypergraph H with n vertices and m hyperedges, each hyperedge being a k-sized subset of vertices. A k-simplex in H is a subhypergraph on k+1 vertices X such that all k+1 possible hyperedges among X exist in H. The goal is to process the hyperedges of H, which arrive in an arbitrary order as a data stream, and compute a good estimate of T_k(H), the number of k-simplices in H.
We design a suite of algorithms for this problem. As with triangle-counting in graphs (which is the special case k = 2), sublinear space is achievable but only under a promise of the form T_k(H) ≥ T. Under such a promise, our algorithms use at most four passes and together imply a space bound of O(ε^{-2} log δ^{-1} polylog n ⋅ min{(m^{1+1/k})/T, m/(T^{2/(k+1)})}) for each fixed k ≥ 3, in order to guarantee an estimate within (1±ε)T_k(H) with probability ≥ 1-δ. We also give a simpler 1-pass algorithm that achieves O(ε^{-2} log δ^{-1} log n⋅ (m/T) (Δ_E + Δ_V^{1-1/k})) space, where Δ_E (respectively, Δ_V) denotes the maximum number of k-simplices that share a hyperedge (respectively, a vertex), which generalizes a previous result for the k = 2 case. We complement these algorithmic results with space lower bounds of the form Ω(ε^{-2}), Ω(m^{1+1/k}/T), Ω(m/T^{1-1/k}) and Ω(mΔ_V^{1/k}/T) for multi-pass algorithms and Ω(mΔ_E/T) for 1-pass algorithms, which show that some of the dependencies on parameters in our upper bounds are nearly tight. Our techniques extend and generalize several different ideas previously developed for triangle counting in graphs, using appropriate innovations to handle the more complicated combinatorics of hypergraphs.

Amit Chakrabarti and Themistoklis Haris. Counting Simplices in Hypergraph Streams. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 32:1-32:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{chakrabarti_et_al:LIPIcs.ESA.2022.32, author = {Chakrabarti, Amit and Haris, Themistoklis}, title = {{Counting Simplices in Hypergraph Streams}}, booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)}, pages = {32:1--32:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-247-1}, ISSN = {1868-8969}, year = {2022}, volume = {244}, editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.32}, URN = {urn:nbn:de:0030-drops-169705}, doi = {10.4230/LIPIcs.ESA.2022.32}, annote = {Keywords: data streaming, graph algorithms, hypergraphs, sub-linear algorithms, triangle counting} }

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

A streaming algorithm is considered to be adversarially robust if it provides correct outputs with high probability even when the stream updates are chosen by an adversary who may observe and react to the past outputs of the algorithm. We grow the burgeoning body of work on such algorithms in a new direction by studying robust algorithms for the problem of maintaining a valid vertex coloring of an n-vertex graph given as a stream of edges. Following standard practice, we focus on graphs with maximum degree at most Δ and aim for colorings using a small number f(Δ) of colors.
A recent breakthrough (Assadi, Chen, and Khanna; SODA 2019) shows that in the standard, non-robust, streaming setting, (Δ+1)-colorings can be obtained while using only Õ(n) space. Here, we prove that an adversarially robust algorithm running under a similar space bound must spend almost Ω(Δ²) colors and that robust O(Δ)-coloring requires a linear amount of space, namely Ω(nΔ). We in fact obtain a more general lower bound, trading off the space usage against the number of colors used. From a complexity-theoretic standpoint, these lower bounds provide (i) the first significant separation between adversarially robust algorithms and ordinary randomized algorithms for a natural problem on insertion-only streams and (ii) the first significant separation between randomized and deterministic coloring algorithms for graph streams, since deterministic streaming algorithms are automatically robust.
We complement our lower bounds with a suite of positive results, giving adversarially robust coloring algorithms using sublinear space. In particular, we can maintain an O(Δ²)-coloring using Õ(n √Δ) space and an O(Δ³)-coloring using Õ(n) space.

Amit Chakrabarti, Prantar Ghosh, and Manuel Stoeckl. Adversarially Robust Coloring for Graph Streams. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 37:1-37:23, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{chakrabarti_et_al:LIPIcs.ITCS.2022.37, author = {Chakrabarti, Amit and Ghosh, Prantar and Stoeckl, Manuel}, title = {{Adversarially Robust Coloring for Graph Streams}}, booktitle = {13th Innovations in Theoretical Computer Science Conference (ITCS 2022)}, pages = {37:1--37:23}, 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.2022.37}, URN = {urn:nbn:de:0030-drops-156332}, doi = {10.4230/LIPIcs.ITCS.2022.37}, annote = {Keywords: Data streaming, graph algorithms, graph coloring, lower bounds, online algorithms} }

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RANDOM

**Published in:** LIPIcs, Volume 176, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)

We study graph computations in an enhanced data streaming setting, where a space-bounded client reading the edge stream of a massive graph may delegate some of its work to a cloud service. We seek algorithms that allow the client to verify a purported proof sent by the cloud service that the work done in the cloud is correct. A line of work starting with Chakrabarti et al. (ICALP 2009) has provided such algorithms, which we call schemes, for several statistical and graph-theoretic problems, many of which exhibit a tradeoff between the length of the proof and the space used by the streaming verifier.
This work designs new schemes for a number of basic graph problems - including triangle counting, maximum matching, topological sorting, and single-source shortest paths - where past work had either failed to obtain smooth tradeoffs between these two key complexity measures or only obtained suboptimal tradeoffs. Our key innovation is having the verifier compute certain nonlinear sketches of the input stream, leading to either new or improved tradeoffs. In many cases, our schemes in fact provide optimal tradeoffs up to logarithmic factors.
Specifically, for most graph problems that we study, it is known that the product of the verifier’s space cost v and the proof length h must be at least Ω(n²) for n-vertex graphs. However, matching upper bounds are only known for a handful of settings of h and v on the curve h ⋅ v = Θ̃(n²). For example, for counting triangles and maximum matching, schemes with costs lying on this curve are only known for (h = Õ(n²), v = Õ(1)), (h = Õ(n), v = Õ(n)), and the trivial (h = Õ(1), v = Õ(n²)). A major message of this work is that by exploiting nonlinear sketches, a significant "portion" of costs on the tradeoff curve h ⋅ v = n² can be achieved.

Amit Chakrabarti, Prantar Ghosh, and Justin Thaler. Streaming Verification for Graph Problems: Optimal Tradeoffs and Nonlinear Sketches. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 22:1-22:23, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{chakrabarti_et_al:LIPIcs.APPROX/RANDOM.2020.22, author = {Chakrabarti, Amit and Ghosh, Prantar and Thaler, Justin}, title = {{Streaming Verification for Graph Problems: Optimal Tradeoffs and Nonlinear Sketches}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)}, pages = {22:1--22:23}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-164-1}, ISSN = {1868-8969}, year = {2020}, volume = {176}, editor = {Byrka, Jaros{\l}aw and Meka, Raghu}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2020.22}, URN = {urn:nbn:de:0030-drops-126258}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2020.22}, annote = {Keywords: data streams, interactive proofs, Arthur-Merlin, graph algorithms} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

We study the problem of coloring a given graph using a small number of colors in several well-established models of computation for big data. These include the data streaming model, the general graph query model, the massively parallel communication (MPC) model, and the CONGESTED-CLIQUE and the LOCAL models of distributed computation. On the one hand, we give algorithms with sublinear complexity, for the appropriate notion of complexity in each of these models. Our algorithms color a graph G using κ(G)⋅(1+o(1)) colors, where κ(G) is the degeneracy of G: this parameter is closely related to the arboricity α(G). As a function of κ(G) alone, our results are close to best possible, since the optimal number of colors is κ(G)+1. For several classes of graphs, including real-world "big graphs," our results improve upon the number of colors used by the various (Δ(G)+1)-coloring algorithms known for these models, where Δ(G) is the maximum degree in G, since Δ(G) ⩾ κ(G) and can in fact be arbitrarily larger than κ(G).
On the other hand, we establish certain lower bounds indicating that sublinear algorithms probably cannot go much further. In particular, we prove that any randomized coloring algorithm that uses at most κ(G)+O(1) colors would require Ω(n²) storage in the one pass streaming model, and Ω(n²) many queries in the general graph query model, where n is the number of vertices in the graph. These lower bounds hold even when the value of κ(G) is known in advance; at the same time, our upper bounds do not require κ(G) to be given in advance.

Suman K. Bera, Amit Chakrabarti, and Prantar Ghosh. Graph Coloring via Degeneracy in Streaming and Other Space-Conscious Models. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 11:1-11:21, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bera_et_al:LIPIcs.ICALP.2020.11, author = {Bera, Suman K. and Chakrabarti, Amit and Ghosh, Prantar}, title = {{Graph Coloring via Degeneracy in Streaming and Other Space-Conscious Models}}, booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)}, pages = {11:1--11:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-138-2}, ISSN = {1868-8969}, year = {2020}, volume = {168}, editor = {Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.11}, URN = {urn:nbn:de:0030-drops-124182}, doi = {10.4230/LIPIcs.ICALP.2020.11}, annote = {Keywords: Data streaming, Graph coloring, Sublinear algorithms, Massively parallel communication, Distributed algorithms} }

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RANDOM

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

We give new algorithms in the annotated data streaming setting - also known as verifiable data stream computation - for certain graph problems. This setting is meant to model outsourced computation, where a space-bounded verifier limited to sequential data access seeks to overcome its computational limitations by engaging a powerful prover, without needing to trust the prover. As is well established, several problems that admit no sublinear-space algorithms under traditional streaming do allow protocols using a sublinear amount of prover/verifier communication and sublinear-space verification. We give algorithms for many well-studied graph problems including triangle counting, its generalization to subgraph counting, maximum matching, problems about the existence (or not) of short paths, finding the shortest path between two vertices, and testing for an independent set. While some of these problems have been studied before, our results achieve new tradeoffs between space and communication costs that were hitherto unknown. In particular, two of our results disprove explicit conjectures of Thaler (ICALP, 2016) by giving triangle counting and maximum matching algorithms for n-vertex graphs, using o(n) space and o(n^2) communication.

Amit Chakrabarti and Prantar Ghosh. Streaming Verification of Graph Computations via Graph Structure. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 70:1-70:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{chakrabarti_et_al:LIPIcs.APPROX-RANDOM.2019.70, author = {Chakrabarti, Amit and Ghosh, Prantar}, title = {{Streaming Verification of Graph Computations via Graph Structure}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)}, pages = {70:1--70:20}, 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.2019.70}, URN = {urn:nbn:de:0030-drops-112856}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2019.70}, annote = {Keywords: data streams, interactive proofs, Arthur-Merlin, graph algorithms} }

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**Published in:** LIPIcs, Volume 66, 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)

We revisit the much-studied problem of space-efficiently estimating the number of triangles in a graph stream, and extensions of this problem to counting fixed-sized cliques and cycles. For the important special case of counting triangles, we give a 4-pass, (1 +/- epsilon)-approximate, randomized algorithm using O-tilde(epsilon^(-2) m^(3/2) / T) space, where m is the number of edges and T is a promised lower bound on the number of triangles. This matches the space bound of a recent algorithm (McGregor et al., PODS 2016), with an arguably simpler and more general technique. We give an improved multi-pass lower bound of Omega(min{m^(3/2)/T , m/sqrt(T)}), applicable at essentially all densities Omega(n) <= m <= O(n^2). We prove other multi-pass lower bounds in terms of various structural parameters of the input graph. Together, our results resolve a couple of open questions raised in recent work (Braverman et al., ICALP 2013).
Our presentation emphasizes more general frameworks, for both upper and lower bounds. We give a sampling algorithm for counting arbitrary subgraphs and then improve it via combinatorial means in the special cases of counting odd cliques and odd cycles. Our results show that these problems are considerably easier in the cash-register streaming model than in the turnstile model, where previous work had focused. We use Turán graphs and related gadgets to derive lower bounds for counting cliques and cycles, with triangle-counting lower bounds following as a corollary.

Suman K. Bera and Amit Chakrabarti. Towards Tighter Space Bounds for Counting Triangles and Other Substructures in Graph Streams. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 11:1-11:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{bera_et_al:LIPIcs.STACS.2017.11, author = {Bera, Suman K. and Chakrabarti, Amit}, title = {{Towards Tighter Space Bounds for Counting Triangles and Other Substructures in Graph Streams}}, booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)}, pages = {11:1--11:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-028-6}, ISSN = {1868-8969}, year = {2017}, volume = {66}, editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.11}, URN = {urn:nbn:de:0030-drops-70222}, doi = {10.4230/LIPIcs.STACS.2017.11}, annote = {Keywords: data streaming, graph algorithms, triangles, subgraph counting, lower bounds} }

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**Published in:** LIPIcs, Volume 45, 35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015)

In this paper, we study the maximum density, threshold and emptiness
queries for intervals in the streaming model. The input is a stream S
of n points in the real line R and a floating closed interval W of width alpha. The specific problems we consider in this paper are as follows.
- Maximum density: find a placement of W in R containing the
maximum number of points of S.
- Threshold query: find a placement of W in R, if it exists,
that contains at least Delta elements of S.
- Emptiness query: find, if possible, a placement of W within the extent of S so that the interior of W does not contain any element of S.
The stream S, being huge, does not fit into main memory and can be read sequentially at most a constant number of times, usually once.
The problems studied here in the geometric setting have relations to frequency estimation and heavy hitter identification in a stream of data. We provide lower bounds and results on trade-off between extra space and quality of solution. We also discuss generalizations for the higher dimensional variants for a few cases.

Arijit Bishnu, Amit Chakrabarti, Subhas C. Nandy, and Sandeep Sen. On Density, Threshold and Emptiness Queries for Intervals in the Streaming Model. In 35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 45, pp. 336-349, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{bishnu_et_al:LIPIcs.FSTTCS.2015.336, author = {Bishnu, Arijit and Chakrabarti, Amit and Nandy, Subhas C. and Sen, Sandeep}, title = {{On Density, Threshold and Emptiness Queries for Intervals in the Streaming Model}}, booktitle = {35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015)}, pages = {336--349}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-97-2}, ISSN = {1868-8969}, year = {2015}, volume = {45}, editor = {Harsha, Prahladh and Ramalingam, G.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2015.336}, URN = {urn:nbn:de:0030-drops-56488}, doi = {10.4230/LIPIcs.FSTTCS.2015.336}, annote = {Keywords: Density, threshold, emptiness queries, interval queries, streaming model, heavy hitter, frequency estimation} }

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

We prove that certain instances of the iterated matrix multiplication (IMM) family of polynomials with N variables and degree n require N^(Omega(sqrt(n))) gates when expressed as a homogeneous depth-five Sigma Pi Sigma Pi Sigma arithmetic circuit with the bottom fan-in bounded by N^(1/2-epsilon). By a depth-reduction result of Tavenas, this size lower bound is optimal and can be achieved by the weaker class of homogeneous depth-four Sigma Pi Sigma Pi circuits.
Our result extends a recent result of Kumar and Saraf, who gave the same N^(Omega(sqrt(n))) lower bound for homogeneous depth-four Sigma Pi Sigma Pi circuits computing IMM. It is analogous to a recent result of Kayal and Saha, who gave the same lower bound for homogeneous Sigma Pi Sigma Pi Sigma circuits (over characteristic zero) with bottom fan-in at most N^(1-epsilon), for the harder problem of computing certain polynomials defined by Nisan-Wigderson designs.

Suman K. Bera and Amit Chakrabarti. A Depth-Five Lower Bound for Iterated Matrix Multiplication. In 30th Conference on Computational Complexity (CCC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 33, pp. 183-197, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{bera_et_al:LIPIcs.CCC.2015.183, author = {Bera, Suman K. and Chakrabarti, Amit}, title = {{A Depth-Five Lower Bound for Iterated Matrix Multiplication}}, booktitle = {30th Conference on Computational Complexity (CCC 2015)}, pages = {183--197}, 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.2015.183}, URN = {urn:nbn:de:0030-drops-50622}, doi = {10.4230/LIPIcs.CCC.2015.183}, annote = {Keywords: arithmetic circuits, iterated matrix multiplication, depth five circuits, lower bound} }

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

The EQUALITY problem is usually one’s first encounter with communication complexity and is one of the most fundamental problems in the field. Although its deterministic and randomized communication complexity were settled decades ago, we find several new things to say about the problem by focusing on three subtle aspects. The first is to consider the expected communication cost (at a worst-case input) for a protocol that uses limited interaction—i.e., a bounded number of rounds of communication—and whose error probability is zero or close to it. The second is to treat the false negative error rate separately from the false positive error rate. The third is to consider the information cost of such protocols. We obtain asymptotically optimal rounds-versus-cost tradeoffs for EQUALITY: both expected communication cost and information cost scale as Theta(log log ... log n), with r-1 logs, where r is the number of rounds. These bounds hold even when the false negative rate approaches 1. For the case of zero-error communication cost, we obtain essentially matching bounds, up to a tiny additive constant. We also provide some applications.

Joshua Brody, Amit Chakrabarti, Ranganath Kondapally, David P. Woodruff, and Grigory Yaroslavtsev. Certifying Equality With Limited Interaction. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 28, pp. 545-581, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{brody_et_al:LIPIcs.APPROX-RANDOM.2014.545, author = {Brody, Joshua and Chakrabarti, Amit and Kondapally, Ranganath and Woodruff, David P. and Yaroslavtsev, Grigory}, title = {{Certifying Equality With Limited Interaction}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)}, pages = {545--581}, 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.2014.545}, URN = {urn:nbn:de:0030-drops-47229}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2014.545}, annote = {Keywords: equality, communication complexity, information complexity} }

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**Published in:** LIPIcs, Volume 1, 25th International Symposium on Theoretical Aspects of Computer Science (2008)

We study the one-way number-on-the-forehead (NOF) communication
complexity of the $k$-layer pointer jumping problem with $n$
vertices per layer. This classic problem, which has connections to
many aspects of complexity theory, has seen a recent burst of
research activity, seemingly preparing the ground for an
$Omega(n)$ lower bound, for constant $k$. Our first result is a
surprising sublinear --- i.e., $o(n)$ --- upper bound for the
problem that holds for $k ge 3$, dashing hopes for such a lower
bound.
A closer look at the protocol achieving the upper bound shows that
all but one of the players involved are collapsing, i.e., their
messages depend only on the composition of the layers ahead of
them. We consider protocols for the pointer jumping problem where
all players are collapsing. Our second result shows that a strong
$n - O(log n)$ lower bound does hold in this case. Our third
result is another upper bound showing that nontrivial protocols for
(a non-Boolean version of) pointer jumping are possible even when
all players are collapsing.
Our lower bound result uses a novel proof technique, different from
those of earlier lower bounds that had an information-theoretic
flavor. We hope this is useful in further study of the problem.

Joshua Brody and Amit Chakrabarti. Sublinear Communication Protocols for Multi-Party Pointer Jumping and a Related Lower Bound. In 25th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 1, pp. 145-156, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2008)

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@InProceedings{brody_et_al:LIPIcs.STACS.2008.1341, author = {Brody, Joshua and Chakrabarti, Amit}, title = {{Sublinear Communication Protocols for Multi-Party Pointer Jumping and a Related Lower Bound}}, booktitle = {25th International Symposium on Theoretical Aspects of Computer Science}, pages = {145--156}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-06-4}, ISSN = {1868-8969}, year = {2008}, volume = {1}, editor = {Albers, Susanne and Weil, Pascal}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2008.1341}, URN = {urn:nbn:de:0030-drops-13415}, doi = {10.4230/LIPIcs.STACS.2008.1341}, annote = {Keywords: Communication complexity, pointer jumping, number on the forehead} }

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