arXiv subjects

Subhas C. Nandy

Publications and source records attributed to Subhas C. Nandy.

At least 19 recordsLinked to original sources

arXiv2025

Partial Domination in Some Geometric Intersection Graphs and Some Complexity Results

{\em Partial domination problem} is a generalization of the {\em minimum dominating set problem} on graphs. Here, instead of dominating all the nodes, one asks to dominate at least a fraction of the nodes of the given graph by choosing a minimum number of nodes. For any real number $α\in(0,1]$, $α$-partial domination problem can be proved to be NP-complete for general graphs. In this paper, we define the {\em maximum dominating $k$-set} of a graph, which is polynomially transformable to the partial domination problem. The existence of a graph class for which the minimum dominating set problem is polynomial-time solvable, whereas the partial dominating set problem is NP-hard, is shown. We also propose polynomial-time algorithms for the maximum dominating $k$-set problem for the unit and arbitrary interval graphs. The problem can also be solved in polynomial time for the intersection graphs of a set of 2D objects intersected by a straight line, where each object is an axis-parallel unit square, as well as in the case where each object is a unit disk. Our technique also works for axis-parallel unit-height rectangle intersection graphs, where a straight line intersects all the rectangles. Finally, a parametrized algorithm for the maximum dominating $k$-set problem in a disk graph where the input disks are intersected by a straight line is proposed; here the parameter is the ratio of the diameters of the largest and smallest input disks.

cs.CG↗

arXiv2021

On the minimum spanning tree problem in imprecise set-up

In this article, we study the Euclidean minimum spanning tree problem in an imprecise setup. The problem is known as the \emph{Minimum Spanning Tree Problem with Neighborhoods} in the literature. We study the problem where the neighborhoods are represented as non-crossing line segments. Given a set ${\cal S}$ of $n$ disjoint line segments in $I\!\!R^2$, the objective is to find a minimum spanning tree (MST) that contains exactly one end-point from each segment in $\cal S$ and the cost of the MST is minimum among $2^n$ possible MSTs. We show that finding such an MST is NP-hard in general, and propose a $2α$-factor approximation algorithm for the same, where $α$ is the approximation factor of the best-known approximation algorithm to compute a minimum cost Steiner tree in an undirected graph with non-negative edge weights. As an implication of our reduction, we can show that the unrestricted version of the problem (i.e., one point must be chosen from each segment such that the cost of MST is as minimum as possible) is also NP-hard. We also propose a parameterized algorithm for the problem based on the "separability" parameter defined for segments.

cs.CG↗

arXiv2021

Sparsity of weighted networks: measures and applications

A majority of real life networks are weighted and sparse. The present article aims at characterization of weighted networks based on sparsity, as a measure of inherent diversity, of different network parameters. It utilizes sparsity index defined on ordered degree sequence of simple networks and derives further properties of this index. The range of possible values of sparsity index of any connected network, with edge-count in specific intervals, is worked out analytically in terms of node-count; a pattern is uncovered in corresponding degree sequences to produce highest sparsities. Given the edge-weight frequency distribution of a network, we have formulated an expression of the sparsity index of edge-weights. Its properties are analyzed under different distributions of edge-weights. For example, the upper and lower bounds of sparsity index of edge-weights of a network, having all distinct edge-weights, is determined in terms of its node-count and edge density. The article highlights that this summary index with low computational cost, computed on different network parameters, is useful to reveal different structural and organizational aspects of networks for performing analysis. An application of this index has been demonstrated through overlapping community detection of networks. The results validated on artificial and real-world networks show its efficacy.

cs.DM↗

arXiv2020

The Generalized Independent and Dominating Set Problems on Unit Disk Graphs

In this article, we study a generalized version of the maximum independent set and minimum dominating set problems, namely, the maximum $d$-distance independent set problem and the minimum $d$-distance dominating set problem on unit disk graphs for a positive integer $d>0$. We first show that the maximum $d$-distance independent set problem and the minimum $d$-distance dominating set problem belongs to NP-hard class. Next, we propose a simple polynomial-time constant-factor approximation algorithms and PTAS for both the problems.

cs.DS↗

arXiv2019

Corrigendum to: "Linear time algorithm to cover and hit a set of line segments optimally by two axis-parallel squares", Theoretical Computer Science 769 (2019) 63--74

In the paper "Linear time algorithm to cover and hit a set of line segments optimally by two axis-parallel squares", TCS Volume 769 (2019), pages 63--74, the LHIT problem is proposed as follows: For a given set of non-intersecting line segments ${\cal L} = \{\ell_1, \ell_2, \ldots, \ell_n\}$ in $I\!\!R^2$, compute two axis-parallel congruent squares ${\cal S}_1$ and ${\cal S}_2$ of minimum size whose union hits all the line segments in $\cal L$, and a linear time algorithm was proposed. Later it was observed that the algorithm has a bug. In this corrigendum, we corrected the algorithm. The time complexity of the corrected algorithm is $O(n^2)$.

cs.CG↗

arXiv2019

Variations of largest rectangle recognition amidst a bichromatic point set

Classical separability problem involving multi-color point sets is an important area of study in computational geometry. In this paper, we study different separability problems for bichromatic point set P=P_r\cup P_b on a plane, where $P_r$ and $P_b$ represent the set of n red points and m blue points respectively, and the objective is to compute a monochromatic object of the desired type and of maximum size. We propose in-place algorithms for computing (i) an arbitrarily oriented monochromatic rectangle of maximum size in R^2, (ii) an axis-parallel monochromatic cuboid of maximum size in R^3. The time complexities of the algorithms for problems (i) and (ii) are O(m(m+n)(m\sqrt{n}+m\log m+n \log n)) and O(m^3\sqrt{n}+m^2n\log n), respectively. As a prerequisite, we propose an in-place construction of the classic data structure the k-d tree, which was originally invented by J. L. Bentley in 1975. Our in-place variant of the $k$-d tree for a set of n points in R^k supports both orthogonal range reporting and counting query using O(1) extra workspace, and these query time complexities are the same as the classical complexities, i.e., O(n^{1-1/k}+μ) and O(n^{1-1/k}), respectively, where μis the output size of the reporting query. The construction time of this data structure is O(n\log n). Both the construction and query algorithms are non-recursive in nature that do not need O(\log n) size recursion stack compared to the previously known construction algorithm for in-place k-d tree and query in it. We believe that this result is of independent interest. We also propose an algorithm for the problem of computing an arbitrarily oriented rectangle of maximum weight among a point set P=P_r \cup P_b, where each point in P_b (resp. P_r) is associated with a negative (resp. positive) real-valued weight that runs in O(m^2(n+m)\log(n+m)) time using O(n) extra space.

cs.CG↗

arXiv2019

Color spanning Localized query

Let P be a set of n points and each of the points is colored with one of the k possible colors. We present efficient algorithms to pre-process P such that for a given query point q, we can quickly identify the smallest color spanning object of the desired type containing q. In this paper, we focus on (i) intervals, (ii) axis-parallel square, (iii) axis-parallel rectangle, (iv) equilateral triangle of fixed orientation and (v) circle, as our desired type of objects.

cs.CG↗

arXiv2017

A linear time algorithm to cover and hit a set of line segments optimally by two axis-parallel squares

This paper discusses the problem of covering and hitting a set of line segments $\cal L$ in ${\mathbb R}^2$ by a pair of axis-parallel squares such that the side length of the larger of the two squares is minimized. We also discuss the restricted version of covering, where each line segment in $\cal L$ is to be covered completely by at least one square. The proposed algorithm for the covering problem reports the optimum result by executing only two passes of reading the input data sequentially. The algorithm proposed for the hitting and restricted covering problems produces optimum result in $O(n)$ time. All the proposed algorithms are in-place, and they use only $O(1)$ extra space. The solution of these problems also give a $\sqrt{2}$ approximation for covering and hitting those line segments $\cal L$ by two congruent disks of minimum radius with same computational complexity.

cs.CG↗

arXiv2017

Range Assignment of Base-Stations Maximizing Coverage Area without Interference

We study the problem of assigning non-overlapping geometric objects centered at a given set of points such that the sum of area covered by them is maximized. If the points are placed on a straight-line and the objects are disks, then the problem is solvable in polynomial time. However, we show that the problem is NP-hard even for simplest objects like disks or squares in ${\mathbb{R}}^2$. Eppstein [CCCG, pages 260--265, 2016] proposed a polynomial time algorithm for maximizing the sum of radii (or perimeter) of non-overlapping balls or disks when the points are arbitrarily placed on a plane. We show that Eppstein's algorithm for maximizing sum of perimeter of the disks in ${\mathbb{R}}^2$ gives a $2$-approximation solution for the sum of area maximization problem. We propose a PTAS for our problem. These approximation results are extendible to higher dimensions. All these approximation results hold for the area maximization problem by regular convex polygons with even number of edges centered at the given points.

cs.CG↗

arXiv2016

Faster Approximation for Maximum Independent Set on Unit Disk Graph

Maximum independent set from a given set $D$ of unit disks intersecting a horizontal line can be solved in $O(n^2)$ time and $O(n^2)$ space. As a corollary, we design a factor 2 approximation algorithm for the maximum independent set problem on unit disk graph which takes both time and space of $O(n^2)$. The best known factor 2 approximation algorithm for this problem runs in $O(n^2 \log n)$ time and takes $O(n^2)$ space [Jallu and Das 2016, Das et al. 2016].

cs.CG↗

arXiv2016

Covering segments with unit squares

We study several variations of line segment covering problem with axis-parallel unit squares in $I\!\!R^2$. A set $S$ of $n$ line segments is given. The objective is to find the minimum number of axis-parallel unit squares which cover at least one end-point of each segment. The variations depend on the orientation and length of the input segments. We prove some of these problems to be NP-complete, and give constant factor approximation algorithms for those problems. For some variations, we have polynomial time exact algorithms. For the general version of the problem, where the segments are of arbitrary length and orientation, and the squares are given as input, we propose a factor 16 approximation result based on multilevel linear programming relaxation technique, which may be useful for solving some other problems. Further, we show that our problems have connections with the problems studied by Arkin et al. 2015 on conflict-free covering problem. Our NP-completeness results hold for more simplified types of objects than those of Arkin et al. 2015.

cs.CG↗

arXiv2016

Color Spanning Annulus: Square, Rectangle and Equilateral Triangle

In this paper, we study different variations of minimum width color-spanning annulus problem among a set of points $P=\{p_1,p_2,\ldots,p_n\}$ in $I\!\!R^2$, where each point is assigned with a color in $\{1, 2, \ldots, k\}$. We present algorithms for finding a minimum width color-spanning axis parallel square annulus $(CSSA)$, minimum width color spanning axis parallel rectangular annulus $(CSRA)$, and minimum width color-spanning equilateral triangular annulus of fixed orientation $(CSETA)$. The time complexities of computing (i) a $CSSA$ is $O(n^3+n^2k\log k)$ which is an improvement by a factor $n$ over the existing result on this problem, (ii) that for a $CSRA$ is $O(n^4\log n)$, and for (iii) a $CSETA$ is $O(n^3k)$. The space complexity of all the algorithms is $O(k)$.

cs.CG↗

arXiv2015

Approximation algorithms for the two-center problem of convex polygon

Given a convex polygon $P$ with $n$ vertices, the two-center problem is to find two congruent closed disks of minimum radius such that they completely cover $P$. We propose an algorithm for this problem in the streaming setup, where the input stream is the vertices of the polygon in clockwise order. It produces a radius $r$ satisfying $r\leq2r_{opt}$ using $O(1)$ space, where $r_{opt}$ is the optimum solution. Next, we show that in non-streaming setup, we can improve the approximation factor by $r\leq 1.84 r_{opt}$, maintaining the time complexity of the algorithm to $O(n)$, and using $O(1)$ extra space in addition to the space required for storing the input.

cs.CG↗

arXiv2014

Facility location problems in the constant work-space read-only memory model

Facility location problems are captivating both from theoretical and practical point of view. In this paper, we study some fundamental facility location problems from the space-efficient perspective. Here the input is considered to be given in a read-only memory and only constant amount of work-space is available during the computation. This {\em constant-work-space model} is well-motivated for handling big-data as well as for computing in smart portable devices with small amount of extra-space. First, we propose a strategy to implement prune-and-search in this model. As a warm up, we illustrate this technique for finding the Euclidean 1-center constrained on a line for a set of points in $\IR^2$. This method works even if the input is given in a sequential access read-only memory. Using this we show how to compute (i) the Euclidean 1-center of a set of points in $\IR^2$, and (ii) the weighted 1-center and weighted 2-center of a tree network. The running time of all these algorithms are $O(n~poly(\log n))$. While the result of (i) gives a positive answer to an open question asked by Asano, Mulzer, Rote and Wang in 2011, the technique used can be applied to other problems which admit solutions by prune-and-search paradigm. For example, we can apply the technique to solve two and three dimensional linear programming in $O(n~poly(\log n))$ time in this model. To the best of our knowledge, these are the first sub-quadratic time algorithms for all the above mentioned problems in the constant-work-space model. We also present optimal linear time algorithms for finding the centroid and weighted median of a tree in this model.

cs.DS↗

arXiv2012

Convex Hull and Linear Programming in Read-only Setup with Limited Work-space

Prune-and-search is an important paradigm for solving many important geometric problems. We show that the general prune-and-search technique can be implemented where the objects are given in read-only memory. As examples we consider convex-hull in 2D, and linear programming in 2D and 3D. For the convex-hull problem, designing sub-quadratic algorithm in a read-only setup with sub-linear space is an open problem for a long time. We first propose a simple algorithm for this problem that runs in $O(n^{3/2+ε)}$ time and $O(n^(1/2))$ space. Next, we consider a restricted version of the problem where the points in $P$ are given in sorted order with respect to their $x$-coordinates in a read-only array. For the linear programming problems, the constraints are given in the read-only array. The last three algorithms use {\it prune-and-search}, and their time and extra work-space complexities are $O(n^{1 + ε})$ and $O(\log n)$ respectively, where $ε$ is a small constant satisfying $\sqrt{\frac{\log\log n}{\log n}} < ε< 1$.

cs.CG↗

arXiv2012

Greedy is good: An experimental study on minimum clique cover and maximum independent set problems for randomly generated rectangles

Given a set ${\cal R}=\{R_1,R_2,..., R_n\}$ of $n$ randomly positioned axis parallel rectangles in 2D, the problem of computing the minimum clique cover (MCC) and maximum independent set (MIS) for the intersection graph $G({\cal R})$ of the members in $\cal R$ are both computationally hard \cite{CC05}. For the MCC problem, it is proved that polynomial time constant factor approximation is impossible to obtain \cite{PT11}. Though such a result is not proved yet for the MIS problem, no polynomial time constant factor approximation algorithm exists in the literature. We study the performance of greedy algorithms for computing these two parameters of $G({\cal R})$. Experimental results shows that for each of the MCC and MIS problems, the corresponding greedy algorithm produces a solution that is very close to its optimum solution. Scheinerman \cite{Scheinerman80} showed that the size of MIS is tightly bounded by $\sqrt{n}$ for a random instance of the 1D version of the problem, (i.e., for the interval graph). Our experiment shows that the size of independent set and the clique cover produced by the greedy algorithm is at least $2\sqrt{n}$ and at most $3\sqrt{n}$, respectively. Thus the experimentally obtained approximation ratio of the greedy algorithm for MIS problem is at most 3/2 and the same for the MCC problem is at least 2/3. Finally we will provide refined greedy algorithms based on a concept of {\it simplicial rectangle}. The characteristics of this algorithm may be of interest in getting a provably constant factor approximation algorithm for random instance of both the problems. We believe that the result also holds true for any finite dimension.

cs.CG↗

arXiv2012

Localized Geometric Query Problems

A new class of geometric query problems are studied in this paper. We are required to preprocess a set of geometric objects $P$ in the plane, so that for any arbitrary query point $q$, the largest circle that contains $q$ but does not contain any member of $P$, can be reported efficiently. The geometric sets that we consider are point sets and boundaries of simple polygons.

cs.CG↗

arXiv2012

Space-efficient Algorithms for Visibility Problems in Simple Polygon

Given a simple polygon $P$ consisting of $n$ vertices, we study the problem of designing space-efficient algorithms for computing (i) the visibility polygon of a point inside $P$, (ii) the weak visibility polygon of a line segment inside $P$ and (iii) the minimum link path between a pair of points inside $P$. For problem (i) two algorithms are proposed. The first one is an in-place algorithm where the input array may be lost. It uses only O(1) extra space apart from the input array. The second one assumes that the input is given in a read-only array, and it needs $O(\sqrt{n})$ extra space. The time complexity of both the algorithms are O(n). For problem (ii), we have assumed that the input polygon is given in a read-only array. Our proposed algorithm runs in $O(n^2)$ time using O(1) extra space. For problem (iii) the time and space complexities of our proposed algorithm are $O(kn)$ and O(1) respectively; $k$ is the length (number of links) in a minimum link path between the given pair of points.

cs.CG↗