arXiv subjects

Arijit Bishnu

Publications and source records attributed to Arijit Bishnu.

At least 19 recordsLinked to original sources

arXiv2025

Optimal non-adaptive algorithm for edge estimation

We present a simple nonadaptive randomized algorithm that estimates the number of edges in a simple, unweighted, undirected graph, possibly containing isolated vertices, using only degree and random edge queries. For an $n$-vertex graph, our method requires only $\widetilde{O}(\sqrt{n})$ queries, achieving sublinear query complexity. The algorithm independently samples a set of vertices and queries their degrees, and also independently samples a set of edges, using the answers to these queries to estimate the total number of edges in the graph. We further prove a matching lower bound, establishing the optimality of our algorithm and resolving the non-adaptive query complexity of this problem with respect to degree and random-edge queries.

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arXiv2025

Curves, points, incidences and covering

Given a point set, mostly a grid in our case, we seek upper and lower bounds on the number of curves that are needed to cover the point set. We say a curve covers a point if the curve passes through the point. We consider such coverings by monotonic curves, lines, orthoconvex curves, circles, etc. We also study a problem that is converse of the covering problem -- if a set of $n^2$ points in the plane is covered by $n$ lines then can we say something about the configuration of the points?

math.CO↗

arXiv2025

Towards Tight Bounds for Estimating Degree Distribution in Streaming and Query Models

The degree distribution of a graph $G=(V,E)$, $|V|=n$, $|E|=m$ is one of the most fundamental objects of study in the analysis of graphs as it embodies relationship among entities. In particular, an important derived distribution from degree distribution is the complementary cumulative degree histogram (ccdh). The ccdh is a fundamental summary of graph structure, capturing, for each threshold $d$, the number of vertices with degree at least $d$. For approximating ccdh, we consider the $(\varepsilon_D,\varepsilon_R)$-BiCriteria Multiplicative Approximation, which allows for controlled multiplicative slack in both the domain and the range. The exact complexity of the problem was not known and had been posed as an open problem in WOLA 2019 [Sublinear.info, Problem 98]. In this work, we first design an algorithm that can approximate ccdh if a suitable vertex sample and an edge sample can be obtained and thus, the algorithm is independent of any sublinear model. Next, we show that in the streaming and query models, these samples can be obtained efficiently. On the other end, we establish the first lower bounds for this problem in both query and streaming models, and (almost) settle the complexity of the problem across both the sublinear models.

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arXiv2025

Arboricity and Random Edge Queries Matter for Triangle Counting using Sublinear Queries

Given a simple, unweighted, undirected graph $G=(V,E)$ with $|V|=n$ and $|E|=m$, and parameters $0 < \varepsilon, \delta <1$, along with \texttt{Degree}, \texttt{Neighbour}, \texttt{Edge} and \texttt{RandomEdge} query access to $G$, we provide a query based randomized algorithm to generate an estimate $\widehat{T}$ of the number of triangles $T$ in $G$, such that $\widehat{T} \in [(1-\varepsilon)T , (1+\varepsilon)T]$ with probability at least $1-\delta$. The query complexity of our algorithm is $\widetilde{O}\left({m \alpha \log(1/\delta)}/{\varepsilon^3 T}\right)$, where $\alpha$ is the arboricity of $G$. Our work can be seen as a continuation in the line of recent works [Eden et al., SIAM J Comp., 2017; Assadi et al., ITCS 2019; Eden et al. SODA 2020] that considered subgraph or triangle counting with or without the use of \texttt{RandomEdge} query. Of these works, Eden et al. [SODA 2020] considers the role of arboricity. Our work considers how \texttt{RandomEdge} query can leverage the notion of arboricity. Furthermore, continuing in the line of work of Assadi et al. [APPROX/RANDOM 2022], we also provide a lower bound of $\widetilde{\Omega}\left({m \alpha \log(1/\delta)}/{\varepsilon^2 T}\right)$ that matches the upper bound exactly on arboricity and the parameter $\delta$ and almost on $\varepsilon$.

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arXiv2024

Near Uniform Triangle Sampling Over Adjacency List Graph Streams

Triangle counting and sampling are two fundamental problems for streaming algorithms. Arguably, designing sampling algorithms is more challenging than their counting variants. It may be noted that triangle counting has received far greater attention in the literature than the sampling variant. In this work, we consider the problem of approximately sampling triangles in different models of streaming with the focus being on the adjacency list model. In this problem, the edges of a graph $G$ will arrive over a data stream. The goal is to design efficient streaming algorithms that can sample and output a triangle from a distribution, over the triangles in $G$, that is close to the uniform distribution over the triangles in $G$. The distance between distributions is measured in terms of $\ell_1$-distance. The main technical contribution of this paper is to design algorithms for this triangle sampling problem in the adjacency list model with the space complexities matching their counting variants. For the sake of completeness, we also show results on the vertex and edge arrival models.

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arXiv2022

Faster Counting and Sampling Algorithms using Colorful Decision Oracle

In this work, we consider $d$-{\sc Hyperedge Estimation} and $d$-{\sc Hyperedge Sample} problem in a hypergraph $\mathcal{H}(U(\mathcal{H}),\mathcal{F}(\mathcal{H}))$ in the query complexity framework, where $U(\mathcal{H})$ denotes the set of vertices and $\mathcal{F}(\mathcal{H})$ denotes the set of hyperedges. The oracle access to the hypergraph is called {\sc Colorful Independence Oracle} ({\sc CID}), which takes $d$ (non-empty) pairwise disjoint subsets of vertices $A_1,\ldots,A_d \subseteq U(\mathcal{H})$ as input, and answers whether there exists a hyperedge in $\mathcal{H}$ having (exactly) one vertex in each $A_i, i \in \{1,2,\ldots,d\}$. The problem of $d$-{\sc Hyperedge Estimation} and $d$-{\sc Hyperedge Sample} with {\sc CID} oracle access is important in its own right as a combinatorial problem. Also, Dell {\it{et al.}}~[SODA '20] established that {\em decision} vs {\em counting} complexities of a number of combinatorial optimization problems can be abstracted out as $d$-{\sc Hyperedge Estimation} problems with a {\sc CID} oracle access. The main technical contribution of the paper is an algorithm that estimates $m= \lvert {\mathcal{F}(\mathcal{H})}\rvert$ with $\widehat{m}$ such that { $$ \frac{1}{C_{d}\log^{d-1} n} \;\leq\; \frac{\widehat{m}}{m} \;\leq\; C_{d} \log ^{d-1} n . $$ by using at most $C_{d}\log ^{d+2} n$ many {\sc CID} queries, where $n$ denotes the number of vertices in the hypergraph $\mathcal{H}$ and $C_{d}$ is a constant that depends only on $d$}. Our result coupled with the framework of Dell {\it{et al.}}~[SODA '21] implies improved bounds for a number of fundamental problems.

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arXiv2021

On the Complexity of Triangle Counting using Emptiness Queries

Beame et al. [ITCS 2018 & TALG 2021] 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 span of time that either solved open questions mentioned by Beame et al. or were generalizations of their work as in Dell and Lapinskas [STOC 2018], Dell, Lapinskas and Meeks [SODA 2020], Bhattacharya et al. [ISAAC 2019 & Theory Comput. Syst. 2021], and Chen et al. [SODA 2020]. 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 completely 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 2021] to test graph connectivity.

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arXiv2021

Distance Estimation Between Unknown Matrices Using Sublinear Projections on Hamming Cube

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 ${\bf A}$ and ${\bf B}$ of size $n \times n$ 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 {\sc Inner Product} oracle (shortened as {\sc IP}). We show that our algorithm returns a $(1\pm \epsilon)$ approximation to ${{\bf D}}_{\bf M} ({\bf A},{\bf B})$ with high probability by making ${\cal O}\left(\frac{n}{\sqrt{{{\bf D}}_{\bf M} ({\bf A},{\bf B})}}\mbox{poly}\left(\log n, \frac{1}{\epsilon}\right)\right)$ oracle queries, where ${{\bf D}}_{\bf M} ({\bf A},{\bf B})$ denotes the Hamming distance (the number of corresponding entries in which ${\bf A}$ and ${\bf B}$ differ) between two matrices ${\bf A}$ and ${\bf B}$ of size $n \times n$. We also show a matching lower bound on the number of such {\sc IP} queries needed. Though our main result is on estimating ${{\bf D}}_{\bf M} ({\bf A},{\bf B})$ using {\sc IP}, we also compare our results with other query models.

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arXiv2020

Even the Easiest(?) Graph Coloring Problem is not Easy in Streaming!

We study a graph coloring problem that is otherwise easy 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 {\sc 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 one-pass vertex arrival streaming model, viz. the {\sc Vertex Arrival} ({\sc VA}), {Vertex Arrival With Degree Oracle} ({\sc VAdeg}), {\sc Vertex Arrival in Random Order} ({\sc 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 streaming algorithms for estimating the number of conflicting 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 {\sc VArand} model.

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arXiv2020

Query Complexity of Global Minimum Cut

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 {\sc Degree}, {\sc Neighbor}, and {\sc Adjacency} queries. Given $\epsilon \in (0,1)$, the algorithm with high probability outputs an estimate $\hat{t}$ satisfying the following $(1-\epsilon) t \leq \hat{t} \leq (1+\epsilon) t$, where $m$ is the number of edges in the graph and $t$ is the size of minimum cut in the graph. The expected number of local queries used by our algorithm is $\min\left\{m+n,\frac{m}{t}\right\}\mbox{poly}\left(\log n,\frac{1}{\epsilon}\right)$ where $n$ is the number of vertices in the graph. Eden and Rosenbaum showed that $\Omega(m/t)$ many local queries are required for approximating the size of minimum cut in graphs. These two results together 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 $t \in \mathbb{N}$, $\Omega(m)$ 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 $t \in \mathbb{N}$, $\Omega(m)$ 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 {\sc Random Edge} query apart from local queries.

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arXiv2020

The linear arboricity conjecture for graphs of low degeneracy

A linear forest is an acyclic graph whose each connected component is a path; or in other words, it is an acyclic graph whose maximum degree is at most 2. A linear coloring of a graph $G$ is an edge coloring of $G$ such that the edges in each color class form a linear forest. The linear arboricity of $G$, denoted as $\chi'_l(G)$, is the minimum number of colors required in any linear coloring of $G$. It is easy to see that for any graph $G$, $\chi'_l(G)\geq\left\lceil\frac{\Delta(G)}{2}\right\rceil$, where $\Delta(G)$ is the maximum degree of $G$. The Linear Arboricity Conjecture of Akiyama, Exoo and Harary from 1980 states that for every graph $G$, $\chi'_l(G)\leq \left \lceil \frac{\Delta(G)+1}{2}\right\rceil$. Basavaraju et al. showed that the conjecture is true for 3-degenerate graphs and provided a linear time algorithm for computing a linear coloring using at most $\left\lceil\frac{\Delta(G)+1}{2}\right\rceil$ colors for any input 3-degenerate graph $G$. Recently, Chen, Hao and Yu showed that $\chi'_l(G)=\left\lceil\frac{\Delta(G)}{2}\right\rceil$ for any $k$-degenerate graph $G$ having $\Delta(G)\geq 2k^2-k$. From this result, we have $\chi'_l(G)=\left\lceil\frac{\Delta(G)}{2}\right\rceil$ for every 3-degenerate graph $G$ having $\Delta(G)\geq 15$. We show that this equality holds for every 3-degenerate graph $G$ having $\Delta(G)\geq 9$. Moreover, by extending the techniques used, we show a different proof for the Linear Arboricity Conjecture on 3-degenerate graphs. Next, we prove that for every 2-degenerate graph $G$, $\chi'_l(G)=\left\lceil\frac{\Delta(G)}{2}\right\rceil$ if $\Delta(G)\geq 5$. We conjecture that this equality holds also when $\Delta(G)\in\{3,4\}$ and show that this is the case for some well-known subclasses of 2-degenerate graphs.

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arXiv2019

Hyperedge Estimation using Polylogarithmic Subset Queries

In this work, we estimate the number of hyperedges in a hypergraph ${\cal H}(U({\cal H}), {\cal F}({\cal H}))$, where $U({\cal H})$ denotes the set of vertices and ${\cal F}({\cal H}))$ denotes the set of hyperedges. We assume a query oracle access to the hypergraph ${\cal H}$. Estimating the number of edges, triangles or small subgraphs in a graph is a well studied problem. Beame \etal~and Bhattacharya \etal~gave algorithms to estimate the number of edges and triangles in a graph using queries to the {\sc Bipartite Independent Set} ({\sc BIS}) and the {\sc Tripartite Independent Set} ({\sc TIS}) oracles, respectively. We generalize the earlier works by estimating the number of hyperedges using a query oracle, known as the {\bf Generalized $d$-partite independent set oracle ({\sc GPIS})}, that takes $d$ (non-empty) pairwise disjoint subsets of vertices $A_1,\ldots,A_d \subseteq U({\cal H})$ as input, and answers whether there exists a hyperedge in ${\cal H}$ having (exactly) one vertex in each $A_i, i \in \{1,2,\ldots,d\}$. We give a randomized algorithm for the hyperedge estimation problem using the {\sc GPIS} query oracle to output $\widehat{m}$ for $m({\cal H})$ satisfying $(1-\epsilon) \cdot m({\cal H}) \leq \widehat{m} \leq (1+\epsilon) \cdot m({\cal H})$. The number of queries made by our algorithm, assuming $d$ to be a constant, is polylogarithmic in the number of vertices of the hypergraph.

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arXiv2019

Efficiently Sampling and Estimating from Substructures using Linear Algebraic Queries

Given an unknown $n \times n$ matrix $A$ having non-negative entries, the \emph{inner product} (IP) oracle takes as inputs a specified row (or a column) of $A$ and a vector $v \in \mathbb{R}^{n}$, and returns their inner product. A derivative of IP is the induced degree query in an unknown graph $G=(V(G), E(G))$ that takes a vertex $u \in V(G)$ and a subset $S \subseteq V(G)$ as input and reports the number of neighbors of $u$ that are present in $S$. The goal of this paper is to understand the strength of the inner product oracle. Our results in that direction are as follows: (I) IP oracle can solve bilinear form estimation, i.e., estimate the value of ${\bf x}^{T}A\bf{y}$ given two vectors ${\bf x},\, {\bf y} \in \mathbb{R}^{n}$ with non-negative entries and can sample almost uniformly entries of a matrix with non-negative entries; (ii) We tackle for the first time weighted edge estimation and weighted sampling of edges that follow as an application to the bilinear form estimation and almost uniform sampling problems, respectively; (iii) induced degree query, a derivative of IP can solve edge estimation and an almost uniform edge sampling in induced subgraphs. To the best of our knowledge, these are the first set of Oracle-based query complexity results for induced subgraphs. We show that IP/induced degree queries over the whole graph can simulate local queries in any induced subgraph; (iv) Apart from the above, we also show that IP can solve several problems related to matrix, like testing if the matrix is diagonal, symmetric, doubly stochastic, etc.

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arXiv2019

Structural Parameterization for Graph Deletion Problems over Data Streams

The study of parameterized streaming complexity on graph problems was initiated by Fafianie et al. (MFCS'14) and Chitnis et al. (SODA'15 and SODA'16). Simply put, the main goal is to design streaming algorithms for parameterized problems such that $O\left(f(k)\log^{O(1)}n\right)$ space is enough, where $f$ is an arbitrary computable function depending only on the parameter $k$. However, in the past few years, very few positive results have been established. Most of the graph problems that do have streaming algorithms of the above nature are ones where localized checking is required, like Vertex Cover or Maximum Matching parameterized by the size $k$ of the solution we are seeking. Many important parameterized problems that form the backbone of traditional parameterized complexity are known to require $\Omega(n)$ bits for any streaming algorithm; e.g., Feedback Vertex Set, Even/Odd Cycle Transversal, Triangle Deletion or the more general ${\cal F}$-Subgraph Deletion when parameterized by solution size $k$. Our main conceptual contribution is to overcome the obstacles to efficient parameterized streaming algorithms by utilizing the power of parameterization. To the best of our knowledge, this is the first work in parameterized streaming complexity that considers structural parameters instead of the solution size as a parameter. We focus on the vertex cover size $K$ as the parameter for the parameterized graph deletion problems we consider. At the same time, most of the previous work in parameterized streaming complexity was restricted to the EA (edge arrival) or DEA (dynamic edge arrival) models. In this work, we consider the above mentioned graph deletion problems in the four most well-studied streaming models, i.e., the EA, DEA, VA (vertex arrival) and AL (adjacency list) models.

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arXiv2018

On Triangle Estimation using Tripartite Independent Set Queries

Estimating the number of triangles in a graph is one of the most fundamental problems in sublinear algorithms. In this work, we provide an algorithm that approximately counts the number of triangles in a graph using only polylogarithmic queries when \emph{the number of triangles on any edge in the graph is polylogarithmically bounded}. Our query oracle {\em Tripartite Independent Set} (TIS) takes three disjoint sets of vertices $A$, $B$ and $C$ as inputs, 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 \emph{group queries} (Ron and Tsur, ACM ToCT, 2016; Dell and Lapinskas, STOC 2018) and in particular is inspired by the {\em Bipartite Independent Set} (BIS) query oracle of Beame {\em et al.} (ITCS 2018). We extend the algorithmic framework of Beame {\em et al.}, with \tis replacing \bis, for approximately counting triangles in graphs.

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arXiv2018

Almost optimal query algorithm for hitting set using a subset query

Given access to the hypergraph through a subset query oracle in the query model, we give sublinear time algorithms for Hitting-Set with almost tight parameterized query complexity. In parameterized query complexity, we estimate the number of queries to the oracle based on the parameter $k$, the size of the Hitting-Set. The subset query oracle we use in this paper is called Generalized $d$-partite Independent Set query oracle (GPIS) and it was introduced by Bishnu et al. (ISAAC'18). GPIS is a generalization to hypergraphs of the Bipartite Independent Set query oracle (BIS) introduced by Beame et al. (ITCS'18 and TALG'20) for estimating the number of edges in graphs. Formally, GPIS is defined as follows: GPIS oracle for a $d$-uniform hypergraph $\mathcal{H}$ takes as input $d$ pairwise disjoint non-empty subsets $A_1, \ldots, A_d$ of vertices in $\cal H$ and answers whether there is a hyperedge in $\mathcal{H}$ that intersects each set $A_i$, where $i \in \{1, \, 2, \, \ldots, d\}$. } For $d=2$, the GPIS oracle is nothing but BIS oracle. We show that $d$-Hitting-Set, the hitting set problem for $d$-uniform hypergraphs, can be solved using $\widetilde{\mathcal{O}}_d(k^{d} \log n)$ GPIS queries. Additionally, we also showed that $d$-Decesion-Hitting-Set, the decision version of $d$-Hitting-Set can be solved with $\widetilde{\mathcal{O}}_d\left( \min \left\{ k^d\log n, k^{2d^2} \right\} \right)$ {\sc GPIS} queries. We complement these parameterized upper bounds with an almost matching parameterized lower bound that states that any algorithm that solves $d$-Decesion-Hitting-Set requires $\Omega \left( \binom{k+d}{d} \right)$ GPIS queries.

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arXiv2018

On the streaming complexity of fundamental geometric problems

In this paper, we focus on lower bounds and algorithms for some basic geometric problems in the one-pass (insertion only) streaming model. The problems considered are grouped into three categories: (i) Klee's measure (ii) Convex body approximation, geometric query, and (iii) Discrepancy Klee's measure is the problem of finding the area of the union of hyperrectangles. Under convex body approximation, we consider the problems of convex hull, convex body approximation, linear programming in fixed dimensions. The results for convex body approximation implies a property testing type result to find if a query point lies inside a convex polyhedron. Under discrepancy, we consider both the geometric and combinatorial discrepancy. For all the problems considered, we present (randomized) lower bounds on space. Most of our lower bounds are in terms of approximating the solution with respect to an error parameter $\epsilon$. We provide approximation algorithms that closely match the lower bound on space for most of the problems.

cs.CG↗

arXiv2017

Grid Obstacle Representation of Graphs

The grid obstacle representation, or alternately, $\ell_1$-obstacle representation of a graph $G=(V,E)$ is an injective function $f:V \rightarrow \mathbb{Z}^2$ and a set of point obstacles $\mathcal{O}$ on the grid points of $\mathbb{Z}^2$ (where no vertex of $V$ has been mapped) such that $uv$ is an edge in $G$ if and only if there exists a Manhattan path between $f(u)$ and $f(v)$ in $\mathbb{Z}^2$ avoiding the obstacles of $\mathcal{O}$ and points in $f(V)$. This work shows that planar graphs admit such a representation while there exist some non-planar graphs that do not admit such a representation. Moreover, we show that every graph admits a grid obstacle representation in $\mathbb{Z}^3$. We also show NP-hardness result for the point set embeddability of an $\ell_1$-obstacle representation.

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