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Saswata Shannigrahi

Publications and source records attributed to Saswata Shannigrahi.

13 recordsLinked to original sources
arXiv2022

On the Rise and Fall of Online Social Networks

The rise and fall of online social networks recently generated an enormous amount of interest among people, both inside and outside of academia. Gillette [Businessweek magazine, 2011] did a detailed analysis of MySpace, which started losing its popularity since 2008. Cannarella and Spechler [ArXiv, 2014] used a model of disease spread to explain the rise and fall of MySpace. In this paper, we present a graph theoretical model that may be able to provide an alternative explanation for the rise and fall of online social networks. Our model is motivated by the well-known Barabasi-Albert model of generating random scale-free networks using preferential attachment or `rich-gets-richer' phenomenon. As shown by our empirical analysis, we conjecture that such an online social network growth model is inherently flawed as it fails to maintain the stability of such networks while ensuring their growth. In the process, we also conjecture that our model of preferential attachment also exhibits scale-free phenomenon.

cs.SI↗
arXiv2019

Rectilinear Crossings in Complete Balanced d-Partite d-Uniform Hypergraphs

In this paper, we study the embedding of a complete balanced $d$-partite $d$-uniform hypergraph with all its $nd$ vertices represented as points in general position in $\mathbb{R}^d$ and each hyperedge drawn as a convex hull of $d$ corresponding vertices. We assume that the set of vertices is partitioned into $d$ disjoint sets, each of size $n$, such that each of the vertices in a hyperedge is from a different set. Two hyperedges are said to be crossing if they are vertex disjoint and contain a common point in their relative interiors. Using the Generalized Colored Tverberg Theorem, we observe that such an embedding of a complete balanced $d$-partite $d$-uniform hypergraph with $nd$ vertices contains $Ω\left((8/3)^{d/2}\right){\left({n/2}\right)^d{\left((n-1)/2\right)}^d}$ crossing pairs of hyperedges for sufficiently large $n$ and $d$. Using the Gale Transform and the Ham-Sandwich Theorem, we improve this lower bound to $ Ω\left(2^{d}\right){\left({n/2}\right)^d{\left((n-1)/2\right)}^d}$ for sufficiently large $n$ and $d$.

math.CO↗
arXiv2019

$k$-Sets and Rectilinear Crossings in Complete Uniform Hypergraphs

In this paper, we study the $d$-dimensional rectilinear drawings of the complete $d$-uniform hypergraph $K_{2d}^d$. Anshu et al. [Computational Geometry: Theory and Applications, 2017] used Gale transform and Ham-Sandwich theorem to prove that there exist $Ω\left(2^d\right)$ crossing pairs of hyperedges in such a drawing of $K_{2d}^d$. We improve this lower bound by showing that there exist $Ω\left(2^d \sqrt{ d}\right)$ crossing pairs of hyperedges in a $d$-dimensional rectilinear drawing of $K_{2d}^d$. We also prove the following results. 1. There are $Ω\left(2^d {d^{3/2}}\right)$ crossing pairs of hyperedges in a $d$-dimensional rectilinear drawing of $K_{2d}^d$ when its $2d$ vertices are either not in convex position in $\mathbb{R}^d$ or form the vertices of a $d$-dimensional convex polytope that is $t$-neighborly but not $(t+1)$-neighborly for some constant $t\geq1$ independent of $d$. 2. There are $Ω\left(2^d {d^{5/2}}\right)$ crossing pairs of hyperedges in a $d$-dimensional rectilinear drawing of $K_{2d}^d$ when its $2d$ vertices form the vertices of a $d$-dimensional convex polytope that is $(\lfloor{d/2}\rfloor-t')$-neighborly for some constant $t' \geq 0$ independent of $d$.

math.CO↗
arXiv2019

Improved Bounds for Uniform Hypergraphs without Property B

A hypergraph is said to be properly 2-colorable if there exists a 2-coloring of its vertices such that no hyperedge is monochromatic. On the other hand, a hypergraph is called non-2-colorable if there exists at least one monochromatic hyperedge in each of the possible 2-colorings of its vertex set. Let m(n) denote the minimum number of hyperedges in a non-2-colorable n-uniform hypergraph. Establishing the lower and upper bounds on m(n) is a well-studied research direction over several decades. In this paper, we present new constructions for non-2-colorable n-uniform hypergraphs. These constructions improve the upper bounds for m(8), m(13), m(14), m(16) and m(17). We also improve the lower bound for m(5).

math.CO↗
arXiv2018

Improved Encoding and Counting of Uniform Hypertrees

We consider labeled $r$-uniform hypertrees having $n \ge r \ge 2$ vertices. The number of hyperedges in such a hypertree is $m = (n - 1)/(r - 1)$. We show that there are exactly $f(n, r) = \frac{(n-1)! n^{m-1}}{(r-1)!^m m!}$ $r$-uniform hypertrees with $n$ vertices labeled with distinct integers. We also give an encoding scheme that encodes such hypertrees using, on an average, at most $1 + \log_2 e$ bits more than $\log_2(f(n, r))$.

math.CO↗
arXiv2018

Hypergraph Two-Coloring in the Streaming Model

We consider space-efficient algorithms for two-coloring $n$-uniform hypergraphs $H=(V,E)$ in the streaming model, when the hyperedges arrive one at a time. It is known that any such hypergraph with at most $0.7 \sqrt{\frac{n}{\ln n}} 2^n$ hyperedges has a two-coloring [Radhakrishnan & Srinivasan, RSA, 2000], which can be found deterministically in polynomial time, if allowed full access to the input. 1. Let $s^D(v, q, n)$ be the minimum space used by a deterministic one-pass streaming algorithm that on receiving an $n$-uniform hypergraph $H$ on $v$ vertices and $q$ hyperedges produces a proper two-coloring of $H$. We show that $s^D(n^2, q, n) = Ω(q/n)$ when $q \leq 0.7 \sqrt{\frac{n}{\ln n}} 2^n$, and $s^D(n^2, q, n) = Ω(\sqrt{\frac{1}{n\ln n}} 2^n)$ otherwise. 2. Let $s^R(v, q,n)$ be the minimum space used by a randomized one-pass streaming algorithm that on receiving an $n$-uniform hypergraph $H$ on $v$ vertices and $q$ hyperedges with high probability produces a proper two-coloring of $H$ (or declares failure). We show that $s^R(v, \frac{1}{10}\sqrt{\frac{n}{\ln n}} 2^n, n) = O(v \log v)$ by giving an efficient randomized streaming algorithm. The above results are inspired by the study of the number $q(n)$, the minimum possible number of hyperedges in a $n$-uniform hypergraph that is not two-colorable. It is known that $q(n) = Ω(\sqrt{\frac{n}{\ln n}})$ [Radhakrishnan-Srinivasan] and $ q(n)= O(n^2 2^n)$ [Erdos, 1963]. Our first result shows that no space-efficient deterministic streaming algorithm can match the performance of the offline algorithm of Radhakrishnan and Srinivasan; the second result shows that there is, however, a space-efficient randomized streaming algorithm for the task.

cs.DS↗
arXiv2016

On the Rectilinear Crossing Number of Complete Uniform Hypergraphs

In this paper, we consider a generalized version of the rectilinear crossing number problem of drawing complete graphs on a plane. The minimum number of crossing pairs of hyperedges in the $d$-dimensional rectilinear drawing of a $d$-uniform hypergraph is known as the $d$-dimensional rectilinear crossing number of the hypergraph. The currently best-known lower bound on the $d$-dimensional rectilinear crossing number of a complete $d$-uniform hypergraph with $n$ vertices in general position in $\mathbb{R}^d$ is $Ω(\frac{2^d}{\sqrt{d}} \log d) {n \choose 2d}$. In this paper, we improve this lower bound to $Ω(2^d) {n \choose 2d}$. We also consider the special case when all the vertices of a $d$-uniform hypergraph are placed on the $d$-dimensional moment curve. For such complete $d$-uniform hypergraphs with $n$ vertices, we show that the number of pairwise crossing hyperedges is $Θ(\frac{4^d}{\sqrt{d}}) {n \choose 2d}$.

math.CO↗
arXiv2016

A new hierarchical clustering algorithm to identify non-overlapping like-minded communities

A network has a non-overlapping community structure if the nodes of the network can be partitioned into disjoint sets such that each node in a set is densely connected to other nodes inside the set and sparsely connected to the nodes out- side it. There are many metrics to validate the efficacy of such a structure, such as clustering coefficient, betweenness, centrality, modularity and like-mindedness. Many methods have been proposed to optimize some of these metrics, but none of these works well on the recently introduced metric like-mindedness. To solve this problem, we propose a be- havioral property based algorithm to identify communities that optimize the like-mindedness metric and compare its performance on this metric with other behavioral data based methodologies as well as community detection methods that rely only on structural data. We execute these algorithms on real-life datasets of Filmtipset and Twitter and show that our algorithm performs better than the existing algorithms with respect to the like-mindedness metric.

cs.SI↗
arXiv2014

On the construction of non-2-colorable uniform hypergraphs

The problem of 2-coloring uniform hypergraphs has been extensively studied over the last few decades. An n-uniform hypergraph is not 2-colorable if its vertices can't be colored with two colors, Red and Blue, such that every hyperedge contains Red as well as Blue vertices. The least possible number of hyperedges in an n-uniform hypergraph which is not 2-colorable is denoted by m(n). In this paper, we consider the problem of finding an upper bound on m(n) for small values of n. We provide constructions which improve the existing results for some such values of n. We obtain the first improvement in the case of n=8.

math.CO↗
arXiv2014

A Lower Bound on the Crossing Number of Uniform Hypergraphs

In this paper, we consider the embedding of a complete $d$-uniform geometric hypergraph with $n$ vertices in general position in $\mathbb{R}^d$, where each hyperedge is represented as a $(d-1)$-simplex, and a pair of hyperedges is defined to cross if they are vertex-disjoint and contains a common point in the relative interior of the simplices corresponding to them. As a corollary of the Van Kampen-Flores Theorem, it can be seen that such a hypergraph contains $Ω(\frac{2^d}{\sqrt{d}})$ $n\choose 2d$ crossing pairs of hyperedges. Using Gale Transform and Ham Sandwich Theorem, we improve this lower bound to $Ω(\frac{2^d \log d}{\sqrt{d}})$ $n\choose 2d$.

math.CO↗
arXiv2013

NP-Hardness of Speed Scaling with a Sleep State

A modern processor can dynamically set it's speed while it's active, and can make a transition to sleep state when required. When the processor is operating at a speed $s$, the energy consumed per unit time is given by a convex power function $P(s)$ having the property that $P(0) > 0$ and $P"(s) > 0$ for all values of $s$. Moreover, $C > 0$ units of energy is required to make a transition from the sleep state to the active state. The jobs are specified by their arrival time, deadline and the processing volume. We consider a scheduling problem, called speed scaling with sleep state, where each job has to be scheduled within their arrival time and deadline, and the goal is to minimize the total energy consumption required to process these jobs. Albers et. al. proved the NP-hardness of this problem by reducing an instance of an NP-hard partition problem to an instance of this scheduling problem. The instance of this scheduling problem consists of the arrival time, the deadline and the processing volume for each of the jobs, in addition to $P$ and $C$. Since $P$ and $C$ depend on the instance of the partition problem, this proof of the NP-hardness of the speed scaling with sleep state problem doesn't remain valid when $P$ and $C$ are fixed. In this paper, we prove that the speed scaling with sleep state problem remains NP-hard for any fixed positive number $C$ and convex $P$ satisfying $P(0) > 0$ and $P"(s) > 0$ for all values of $s$.

cs.DS↗
arXiv2013

New Online Algorithm for Dynamic Speed Scaling with Sleep State

In this paper, we consider an energy-efficient scheduling problem where $n$ jobs $J_1, J_2, ..., J_n$ need to be executed such that the total energy usage of these jobs is minimized while ensuring that each job is finished within it's deadline. We work in an online setting where a job is known only at it's arrival time, along with it's processing volume and deadline. In such a setting, the currently best-known algorithm by Han et al. \cite{han} provides a competitive ratio max $\{4, 2 + α^α\}$ of energy usage. In this paper, we present a new online algorithm SqOA which provides a competitive ratio max $\{4, 2 + (2-1/α)^α2^{α-1}\}$ of energy usage. For $α\geq 3$, the competitive ratio of our algorithm is better than that of any other existing algorithms for this problem.

cs.DS↗