Document

RANDOM

**Published in:** LIPIcs, Volume 275, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)

Beame et al. [ITCS'18 & TALG'20] introduced and used the Bipartite Independent Set (BIS) and Independent Set (IS) oracle access to an unknown, simple, unweighted and undirected graph and solved the edge estimation problem. The introduction of this oracle set forth a series of works in a short time that either solved open questions mentioned by Beame et al. or were generalizations of their work as in Dell and Lapinskas [STOC'18 and TOCT'21], Dell, Lapinskas, and Meeks [SODA'20 and SICOMP'22], Bhattacharya et al. [ISAAC'19 & TOCS'21], and Chen et al. [SODA'20]. Edge estimation using BIS can be done using polylogarithmic queries, while IS queries need sub-linear but more than polylogarithmic queries. Chen et al. improved Beame et al.’s upper bound result for edge estimation using IS and also showed an almost matching lower bound. Beame et al. in their introductory work asked a few open questions out of which one was on estimating structures of higher order than edges, like triangles and cliques, using BIS queries.
In this work, we almost resolve the query complexity of estimating triangles using BIS oracle. While doing so, we prove a lower bound for an even stronger query oracle called Edge Emptiness (EE) oracle, recently introduced by Assadi, Chakrabarty, and Khanna [ESA'21] to test graph connectivity.

Arijit Bishnu, Arijit Ghosh, and Gopinath Mishra. On the Complexity of Triangle Counting Using Emptiness Queries. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 48:1-48:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{bishnu_et_al:LIPIcs.APPROX/RANDOM.2023.48, author = {Bishnu, Arijit and Ghosh, Arijit and Mishra, Gopinath}, title = {{On the Complexity of Triangle Counting Using Emptiness Queries}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)}, pages = {48:1--48:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-296-9}, ISSN = {1868-8969}, year = {2023}, volume = {275}, editor = {Megow, Nicole and Smith, Adam}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2023.48}, URN = {urn:nbn:de:0030-drops-188739}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2023.48}, annote = {Keywords: Triangle Counting, Emptiness Queries, Bipartite Independent Set Query} }

Document

**Published in:** LIPIcs, Volume 250, 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)

For an unknown n × n matrix A having non-negative entries, the inner product (IP) oracle takes as inputs a specified row (or a column) of A and a vector 𝐯 ∈ ℝⁿ with non-negative entries, and returns their inner product. Given two input vectors x and y in ℝⁿ with non-negative entries, and an unknown matrix A with non-negative entries with IP oracle access, we design almost optimal sublinear time algorithms for the following two fundamental matrix problems:
- Find an estimate 𝒳 for the bilinear form x^T A y such that 𝒳 ≈ x^T A y.
- Designing a sampler 𝒵 for the entries of the matrix A such that ℙ(𝒵 = (i,j)) ≈ x_i A_{ij} y_j /(x^T A y), where x_i and y_j are i-th and j-th coordinate of 𝐱 and 𝐲 respectively. As special cases of the above results, for any submatrix of an unknown matrix with non-negative entries and IP oracle access, we can efficiently estimate the sum of the entries of any submatrix, and also sample a random entry from the submatrix with probability proportional to its weight. We will show that the above results imply that if we are given IP oracle access to the adjacency matrix of a graph, with non-negative weights on the edges, then we can design sublinear time algorithms for the following two fundamental graph problems:
- Estimating the sum of the weights of the edges of an induced subgraph, and
- Sampling edges proportional to their weights from an induced subgraph. We show that compared to the classical local queries (degree, adjacency, and neighbor queries) on graphs, we can get a quadratic speedup if we use IP oracle access for the above two problems.
Apart from the above, we study several matrix problems through the lens of IP oracle, like testing if the matrix is diagonal, symmetric, doubly stochastic, etc. Note that IP oracle is in the class of linear algebraic queries used lately in a series of works by Ben-Eliezer et al. [SODA'08], Nisan [SODA'21], Rashtchian et al. [RANDOM'20], Sun et al. [ICALP'19], and Shi and Woodruff [AAAI'19]. Recently, IP oracle was used by Bishnu et al. [RANDOM'21] to estimate dissimilarities between two matrices.

Arijit Bishnu, Arijit Ghosh, Gopinath Mishra, and Manaswi Paraashar. Counting and Sampling from Substructures Using Linear Algebraic Queries. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 8:1-8:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bishnu_et_al:LIPIcs.FSTTCS.2022.8, author = {Bishnu, Arijit and Ghosh, Arijit and Mishra, Gopinath and Paraashar, Manaswi}, title = {{Counting and Sampling from Substructures Using Linear Algebraic Queries}}, booktitle = {42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)}, pages = {8:1--8:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-261-7}, ISSN = {1868-8969}, year = {2022}, volume = {250}, editor = {Dawar, Anuj and Guruswami, Venkatesan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.8}, URN = {urn:nbn:de:0030-drops-174009}, doi = {10.4230/LIPIcs.FSTTCS.2022.8}, annote = {Keywords: Query complexity, Bilinear form, Uniform sampling, Weighted graphs} }

Document

**Published in:** LIPIcs, Volume 219, 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)

In this work, we consider d-Hyperedge Estimation and d-Hyperedge Sample problem in a hypergraph H(U(H),F(H)) in the query complexity framework, where U(H) denotes the set of vertices and F(H) denotes the set of hyperedges. The oracle access to the hypergraph is called Colorful Independence Oracle (CID), which takes d (non-empty) pairwise disjoint subsets of vertices A₁,…, A_d ⊆ U(ℋ) as input, and answers whether there exists a hyperedge in H having (exactly) one vertex in each A_i, i ∈ {1,2,…,d}. The problem of d-Hyperedge Estimation and d-Hyperedge Sample with CID oracle access is important in its own right as a combinatorial problem. Also, Dell et al. [SODA '20] established that decision vs counting complexities of a number of combinatorial optimization problems can be abstracted out as d-Hyperedge Estimation problems with a CID oracle access.
The main technical contribution of the paper is an algorithm that estimates m = |F(H)| with m̂ such that
1/(C_{d)log^{d-1} n) ≤ m̂/m ≤ C_{d} log ^{d-1} n.
by using at most C_{d}log ^{d+2} n many CID queries, where n denotes the number of vertices in the hypergraph H and C_d is a constant that depends only on d}. Our result coupled with the framework of Dell et al. [SODA '21] implies improved bounds for the following fundamental problems:
Edge Estimation using the Bipartite Independent Set (BIS). We improve the bound obtained by Beame et al. [ITCS '18, TALG '20].
Triangle Estimation using the Tripartite Independent Set (TIS). The previous best bound for the case of graphs with low co-degree (Co-degree for an edge in the graph is the number of triangles incident to that edge in the graph) was due to Bhattacharya et al. [ISAAC '19, TOCS '21], and Dell {et al.}’s result gives the best bound for the case of general graphs [SODA '21]. We improve both of these bounds.
Hyperedge Estimation & Sampling using Colorful Independence Oracle (CID). We give an improvement over the bounds obtained by Dell et al. [SODA '21].

Anup Bhattacharya, Arijit Bishnu, Arijit Ghosh, and Gopinath Mishra. Faster Counting and Sampling Algorithms Using Colorful Decision Oracle. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 10:1-10:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bhattacharya_et_al:LIPIcs.STACS.2022.10, author = {Bhattacharya, Anup and Bishnu, Arijit and Ghosh, Arijit and Mishra, Gopinath}, title = {{Faster Counting and Sampling Algorithms Using Colorful Decision Oracle}}, booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)}, pages = {10:1--10:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-222-8}, ISSN = {1868-8969}, year = {2022}, volume = {219}, editor = {Berenbrink, Petra and Monmege, Benjamin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.10}, URN = {urn:nbn:de:0030-drops-158205}, doi = {10.4230/LIPIcs.STACS.2022.10}, annote = {Keywords: Query Complexity, Subset Query, Hyperedge Estimation, and Colorful Independent Set oracle} }

Document

APPROX

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

In this work, we resolve the query complexity of global minimum cut problem for a graph by designing a randomized algorithm for approximating the size of minimum cut in a graph, where the graph can be accessed through local queries like Degree, Neighbor, and Adjacency queries.
Given ε ∈ (0,1), the algorithm with high probability outputs an estimate t̂ satisfying the following (1-ε) t ≤ t̂ ≤ (1+ε) t, where t is the size of minimum cut in the graph. The expected number of local queries used by our algorithm is min{m+n,m/t}poly(log n,1/(ε)) where n and m are the number of vertices and edges in the graph, respectively. Eden and Rosenbaum showed that Ω(m/t) local queries are required for approximating the size of minimum cut in graphs, {but no local query based algorithm was known. Our algorithmic result coupled with the lower bound of Eden and Rosenbaum [APPROX 2018] resolve the query complexity of the problem of estimating the size of minimum cut in graphs using local queries.}
Building on the lower bound of Eden and Rosenbaum, we show that, for all t ∈ ℕ, Ω(m) local queries are required to decide if the size of the minimum cut in the graph is t or t-2. Also, we show that, for any t ∈ ℕ, Ω(m) local queries are required to find all the minimum cut edges even if it is promised that the input graph has a minimum cut of size t. Both of our lower bound results are randomized, and hold even if we can make Random Edge queries in addition to local queries.

Arijit Bishnu, Arijit Ghosh, Gopinath Mishra, and Manaswi Paraashar. Query Complexity of Global Minimum Cut. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 6:1-6:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{bishnu_et_al:LIPIcs.APPROX/RANDOM.2021.6, author = {Bishnu, Arijit and Ghosh, Arijit and Mishra, Gopinath and Paraashar, Manaswi}, title = {{Query Complexity of Global Minimum Cut}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)}, pages = {6:1--6:15}, 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.2021.6}, URN = {urn:nbn:de:0030-drops-146992}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2021.6}, annote = {Keywords: Query complexity, Global mincut} }

Document

RANDOM

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

Using geometric techniques like projection and dimensionality reduction, we show that there exists a randomized sub-linear time algorithm that can estimate the Hamming distance between two matrices. Consider two matrices A and B of size n × n whose dimensions are known to the algorithm but the entries are not. The entries of the matrix are real numbers. The access to any matrix is through an oracle that computes the projection of a row (or a column) of the matrix on a vector in {0,1}ⁿ. We call this query oracle to be an Inner Product oracle (shortened as IP). We show that our algorithm returns a (1± ε) approximation to {D}_M (A,B) with high probability by making O(n/(√{{D)_M (A,B)}}poly(log n, 1/(ε))) oracle queries, where {D}_M (A,B) denotes the Hamming distance (the number of corresponding entries in which A and B differ) between two matrices A and B of size n × n. We also show a matching lower bound on the number of such IP queries needed. Though our main result is on estimating {D}_M (A,B) using IP, we also compare our results with other query models.

Arijit Bishnu, Arijit Ghosh, and Gopinath Mishra. Distance Estimation Between Unknown Matrices Using Sublinear Projections on Hamming Cube. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 44:1-44:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{bishnu_et_al:LIPIcs.APPROX/RANDOM.2021.44, author = {Bishnu, Arijit and Ghosh, Arijit and Mishra, Gopinath}, title = {{Distance Estimation Between Unknown Matrices Using Sublinear Projections on Hamming Cube}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)}, pages = {44:1--44:22}, 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.2021.44}, URN = {urn:nbn:de:0030-drops-147378}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2021.44}, annote = {Keywords: Distance estimation, Property testing, Dimensionality reduction, Sub-linear algorithms} }

Document

**Published in:** LIPIcs, Volume 185, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)

We study a graph coloring problem that is otherwise easy in the RAM model but becomes quite non-trivial in the one-pass streaming model. In contrast to previous graph coloring problems in streaming that try to find an assignment of colors to vertices, our main work is on estimating the number of conflicting or monochromatic edges given a coloring function that is streaming along with the graph; we call the problem Conflict-Est. The coloring function on a vertex can be read or accessed only when the vertex is revealed in the stream. If we need the color on a vertex that has streamed past, then that color, along with its vertex, has to be stored explicitly. We provide algorithms for a graph that is streaming in different variants of the vertex arrival in one-pass streaming model, viz. the Vertex Arrival (VA), Vertex Arrival With Degree Oracle (VAdeg), Vertex Arrival in Random Order (VArand) models, with special focus on the random order model. We also provide matching lower bounds for most of the cases. The mainstay of our work is in showing that the properties of a random order stream can be exploited to design efficient streaming algorithms for estimating the number of monochromatic edges. We have also obtained a lower bound, though not matching the upper bound, for the random order model. Among all the three models vis-a-vis this problem, we can show a clear separation of power in favor of the VArand model.

Anup Bhattacharya, Arijit Bishnu, Gopinath Mishra, and Anannya Upasana. Even the Easiest(?) Graph Coloring Problem Is Not Easy in Streaming!. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 15:1-15:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{bhattacharya_et_al:LIPIcs.ITCS.2021.15, author = {Bhattacharya, Anup and Bishnu, Arijit and Mishra, Gopinath and Upasana, Anannya}, title = {{Even the Easiest(?) Graph Coloring Problem Is Not Easy in Streaming!}}, booktitle = {12th Innovations in Theoretical Computer Science Conference (ITCS 2021)}, pages = {15:1--15:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-177-1}, ISSN = {1868-8969}, year = {2021}, volume = {185}, editor = {Lee, James R.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.15}, URN = {urn:nbn:de:0030-drops-135544}, doi = {10.4230/LIPIcs.ITCS.2021.15}, annote = {Keywords: Streaming, random ordering, graph coloring, estimation, lower bounds} }

Document

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

Estimating the number of triangles in a graph is one of the most fundamental problems in sublinear algorithms. In this work, we provide an approximate triangle counting algorithm using only polylogarithmic queries when the number of triangles on any edge in the graph is polylogarithmically bounded. Our query oracle Tripartite Independent Set (TIS) takes three disjoint sets of vertices A, B and C as input, and answers whether there exists a triangle having one endpoint in each of these three sets. Our query model generally belongs to the class of group queries (Ron and Tsur, ACM ToCT, 2016; Dell and Lapinskas, STOC 2018) and in particular is inspired by the Bipartite Independent Set (BIS) query oracle of Beame et al. (ITCS 2018). We extend the algorithmic framework of Beame et al., with TIS replacing BIS, for triangle counting using ideas from color coding due to Alon et al. (J. ACM, 1995) and a concentration inequality for sums of random variables with bounded dependency (Janson, Rand. Struct. Alg., 2004).

Anup Bhattacharya, Arijit Bishnu, Arijit Ghosh, and Gopinath Mishra. Triangle Estimation Using Tripartite Independent Set Queries. In 30th International Symposium on Algorithms and Computation (ISAAC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 149, pp. 19:1-19:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{bhattacharya_et_al:LIPIcs.ISAAC.2019.19, author = {Bhattacharya, Anup and Bishnu, Arijit and Ghosh, Arijit and Mishra, Gopinath}, title = {{Triangle Estimation Using Tripartite Independent Set Queries}}, booktitle = {30th International Symposium on Algorithms and Computation (ISAAC 2019)}, pages = {19:1--19:17}, 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.2019.19}, URN = {urn:nbn:de:0030-drops-115156}, doi = {10.4230/LIPIcs.ISAAC.2019.19}, annote = {Keywords: Triangle estimation, query complexity, sublinear algorithm} }

Document

**Published in:** LIPIcs, Volume 123, 29th International Symposium on Algorithms and Computation (ISAAC 2018)

In this paper, we study the query complexity of parameterized decision and optimization versions of Hitting-Set. We also investigate the query complexity of Packing. In doing so, we use generalizations to hypergraphs of an earlier query model, known as BIS introduced by Beame et al. in ITCS'18. The query models considered are the GPIS and GPISE oracles. The GPIS and GPISE oracles are used for the decision and optimization versions of the problems, respectively. We use color coding and queries to the oracles to generate subsamples from the hypergraph, that retain some structural properties of the original hypergraph. We use the stability of the sunflowers in a non-trivial way to do so.

Arijit Bishnu, Arijit Ghosh, Sudeshna Kolay, Gopinath Mishra, and Saket Saurabh. Parameterized Query Complexity of Hitting Set Using Stability of Sunflowers. In 29th International Symposium on Algorithms and Computation (ISAAC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 123, pp. 25:1-25:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{bishnu_et_al:LIPIcs.ISAAC.2018.25, author = {Bishnu, Arijit and Ghosh, Arijit and Kolay, Sudeshna and Mishra, Gopinath and Saurabh, Saket}, title = {{Parameterized Query Complexity of Hitting Set Using Stability of Sunflowers}}, booktitle = {29th International Symposium on Algorithms and Computation (ISAAC 2018)}, pages = {25:1--25:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-094-1}, ISSN = {1868-8969}, year = {2018}, volume = {123}, editor = {Hsu, Wen-Lian and Lee, Der-Tsai and Liao, Chung-Shou}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2018.25}, URN = {urn:nbn:de:0030-drops-99735}, doi = {10.4230/LIPIcs.ISAAC.2018.25}, annote = {Keywords: Query complexity, Hitting set, Parameterized complexity} }

Document

**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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