arXiv subjects

Sharareh Alipour

Publications and source records attributed to Sharareh Alipour.

24 records · Page 2Linked to original sources

arXiv2018

On Triangluar Separation of Bichromatic Point Sets in Polygonal Environment

Let $\mathcal P$ be a simple polygonal environment with $k$ vertices in the plane. Assume that a set $B$ of $b$ blue points and a set $R$ of $r$ red points are distributed in $\mathcal P$. We study the problem of computing triangles that separate the sets $B$ and $R$, and fall in $\mathcal P$. We call these triangles \emph{inscribed triangular separators}. We propose an output-sensitive algorithm to solve this problem in $O(r \cdot (r+c_B+k)+h_\triangle)$ time, where $c_B$ is the size of convex hull of $B$, and $h_\triangle$ is the number of inscribed triangular separators. We also study the case where there does not exist any inscribed triangular separators. This may happen due to the tight distribution of red points around convex hull of $B$ while no red points are inside this hull. In this case we focus to compute a triangle that separates most of the blue points from the red points. We refer to these triangles as \emph{maximum triangular separators}. Assuming $n=r+b$, we design a constant-factor approximation algorithm to compute such a separator in $O(n^{4/3} \log^3 n)$ time. "Eligible for best student paper"

cs.CG↗

arXiv2018

Upper bounds for domination numbers of graphs using Turán's Theorem and Lovász local lemma

Let $G$ be a connected graph of order $n$ with vertex set $V(G)$. A subset $S\subseteq V(G)$ is an $(a,b)$-dominating set if every vertex $v\in S$ is adjacent to at least $a$ vertices in $S$ and every $v\in V\setminus S$ is adjacent to at least $b$ vertices in $S$. The minimum cardinality of an $(a,b)$-dominating set of $G$ is the $(a,b)$-domination number of $G$, denoted by $γ_{a,b}(G)$. There are various results about upper bounds for $γ_{a,b}(G)$ when $G$ is regular or $a$ and $b$ are small numbers. In the first part of this paper, for a given graph $G$ with the minimum degree of $\max\{a,b\}$, we define a new graph $G'$ associated to $G$ and show that the independence number of this graph is related to $γ_{a,b}(G)$. In the next part, using Lovász local lemma, we give a randomized approach to improve previous results in some special cases.

math.CO↗

arXiv2018

On $k$-tuple and $k$-tuple total domination numbers of regular graphs

Let $G$ be a connected graph of order $n$, whose minimum vertex degree is at least $k$. A subset $S$ of vertices in $G$ is a $k$-tuple total dominating set if every vertex of $G$ is adjacent to at least $k$ vertices in $S$. The minimum cardinality of a $k$-tuple total dominating set of $G$ is the $k$-tuple total domination number of $G$, denoted by $γ_{\times k,t}(G)$. Henning and Yeo in \cite{hen} proved that if $G$ is a cubic graph different from the Heawood graph, $γ_{\times 2, t}(G) \leq \frac{5}{6}n$, and this bound is sharp. Similarly, a $k$-tuple dominating set is a subset $S$ of vertices of $G$, $V (G)$ such that $|N[v] \cap S| \geq k$ for every vertex $v$, where $N[v] = \{v\}\cup \{u \in V(G) : uv \in E(G)\}$. The $k$-tuple domination number of $G$, denoted by $γ_{\times k}(G)$, is the minimum cardinality of a $k$-tuple dominating set of $G$. In this paper, we give a simple approach to compute an upper bound for $(r-1)$-tuple total domination number of $r$-regular graphs. Also, we give an upper bound for the $r$-tuple dominating number of $r$-regular graphs. In addition, our method gives algorithms to compute dominating sets with the given bounds, while the previous methods are existential.

math.CO↗

arXiv2017

Improvements on the k-center problem for uncertain data

In real applications, there are situations where we need to model some problems based on uncertain data. This leads us to define an uncertain model for some classical geometric optimization problems and propose algorithms to solve them. In this paper, we study the $k$-center problem, for uncertain input. In our setting, each uncertain point $P_i$ is located independently from other points in one of several possible locations $\{P_{i,1},\dots, P_{i,z_i}\}$ in a metric space with metric $d$, with specified probabilities and the goal is to compute $k$-centers $\{c_1,\dots, c_k\}$ that minimize the following expected cost $$Ecost(c_1,\dots, c_k)=\sum_{R\in Ω} prob(R)\max_{i=1,\dots, n}\min_{j=1,\dots k} d(\hat{P}_i,c_j)$$ here $Ω$ is the probability space of all realizations $$R=\{\hat{P}_1,\dots, \hat{P}_n\}$$ of given uncertain points and $$prob(R)=\prod_{i=1}^n prob(\hat{P}_i).$$ In restricted assigned version of this problem, an assignment $A:\{P_1,\dots, P_n\}\rightarrow \{c_1,\dots, c_k\}$ is given for any choice of centers and the goal is to minimize $$Ecost_A(c_1,\dots, c_k)=\sum_{R\in Ω} prob(R)\max_{i=1,\dots, n} d(\hat{P}_i,A(P_i)).$$ In unrestricted version, the assignment is not specified and the goal is to compute $k$ centers $\{c_1,\dots, c_k\}$ and an assignment $A$ that minimize the above expected cost. We give several improved constant approximation factor algorithms for the assigned versions of this problem in a Euclidean space and in a general metric space. Our results significantly improve the results of \cite{guh} and generalize the results of \cite{wang} to any dimension. Our approach is to replace a certain center point for each uncertain point and study the properties of these certain points. The proposed algorithms are efficient and simple to implement.

cs.CG↗

arXiv2016

An improved Constant-Factor Approximation Algorithm for Planar Visibility Counting Problem

Given a set $S$ of $n$ disjoint line segments in $\mathbb{R}^{2}$, the visibility counting problem (VCP) is to preprocess $S$ such that the number of segments in $S$ visible from any query point $p$ can be computed quickly. This problem can trivially be solved in logarithmic query time using $O(n^{4})$ preprocessing time and space. Gudmundsson and Morin proposed a 2-approximation algorithm for this problem with a tradeoff between the space and the query time. They answer any query in $O_ε(n^{1-α})$ with $O_ε(n^{2+2α})$ of preprocessing time and space, where $α$ is a constant $0\leq α\leq 1$, $ε> 0$ is another constant that can be made arbitrarily small, and $O_ε(f(n))=O(f(n)n^ε)$. In this paper, we propose a randomized approximation algorithm for VCP with a tradeoff between the space and the query time. We will show that for an arbitrary constants $0\leq β\leq \frac{2}{3}$ and $0<δ<1$, the expected preprocessing time, the expected space, and the query time of our algorithm are $O(n^{4-3β}\log n)$, $O(n^{4-3β})$, and $O(\frac{1}{δ^3}n^β\log n)$, respectively. The algorithm computes the number of visible segments from $p$, or $m_p$, exactly if $m_p\leq \frac{1}{δ^3}n^β\log n$. Otherwise, it computes a $(1+δ)$-approximation $m'_p$ with the probability of at least $1-\frac{1}{\log n}$, where $m_p\leq m'_p\leq (1+δ)m_p$.

cs.CG↗

arXiv2014

Fast Approximation and Randomized Algorithms for Diameter

We consider approximation of diameter of a set $S$ of $n$ points in dimension $m$. E$\tilde{g}$ecio$\tilde{g}$lu and Kalantari \cite{kal} have shown that given any $p \in S$, by computing its farthest in $S$, say $q$, and in turn the farthest point of $q$, say $q'$, we have ${\rm diam}(S) \leq \sqrt{3} d(q,q')$. Furthermore, iteratively replacing $p$ with an appropriately selected point on the line segment $pq$, in at most $t \leq n$ additional iterations, the constant bound factor is improved to $c_*=\sqrt{5-2\sqrt{3}} \approx 1.24$. Here we prove when $m=2$, $t=1$. This suggests in practice a few iterations may produce good solutions in any dimension. Here we also propose a randomized version and present large scale computational results with these algorithm for arbitrary $m$. The algorithms outperform many existing algorithms. On sets of data as large as $1,000,000$ points, the proposed algorithms compute solutions to within an absolute error of $10^{-4}$.

cs.CG↗