Sharif Digital Repository

We revisit the problem of finding (Formula presented.) paths with a minimum number of shared edges between two vertices of a graph. An edge is called shared if it is used in more than one of the (Formula presented.) paths. We provide a (Formula presented.)-approximation algorithm for this problem, improving the best previous approximation factor of (Formula presented.). We also provide the first approximation algorithm for the problem with a sublinear approximation factor of (Formula presented.), where (Formula presented.) is the number of vertices in the input graph. For sparse graphs, such as bounded-degree and planar graphs, we show that the approximation factor of our algorithm can be... 

Suppose n colored points with k colors in Rd are given. The Smallest Color-Spanning Ball (SCSB) is the smallest ball containing at least one point of each color. As the computation of the SCSB in Lp metric (p≥1) is time-consuming, we focus on approximately computing the SCSB in near-linear time. Initially, we propose a 3-approximation algorithm running in O(n logn) time. This algorithm is then utilized to present a (1+ε)-approximation algorithm with the running time of O((1/ε)dn logn). We improve the running time to O((1/ε)dn) using randomized techniques. Afterward, spanning colors with two balls is studied. For a special case where d=1, there is an algorithm with O(n2) time. We demonstrate... 

In this paper, we consider path simplification problem under difference area (diff-area) measure. Diff-area measure is defined as AA(Q) AB(Q) , where AA(Q) is the area under Q and above P and AB(Q) is the area above Q and under P (sec Figure 1). Bose et al. [1] presented an approximation algorithm for finding a simplificd path with at most k verticcs that minimizes the cliff-area measure which only works on i-monotone paths. The constraint of being i-monotone is restrictive in some applications like tracking bird migration paths or map boundary simplification. Here, we extend the method of Rosc et al. [1] and present algorithms with the same time complexities as theirs for gcneral paths.... 

The lazy bureaucrat scheduling is a new class of scheduling problems that was introduced in [1]. In these problems, there is one employee (or more) who should perform the assigned jobs. The objective of the employee is to minimize the amount of work he performs and to be as inefficient as possible. He is subject to a constraint, however, that he should be busy when there is some work to do. In this paper, we focus on the cases of this problem where all jobs have the same common deadline. We show that with this constraint, the problem is still NP-hard, and prove some hardness results. We then present a tight 2-approximation algorithm for this problem under one of the defined objective... 

Given a set S of n disjoint line segments in ℝ2, 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(n4) 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 Oε(n1−α) with Oε(n2+2α) of preprocessing time and space, where α is a constant 0 ≤ α ≤ 1, ε > 0 is another constant that can be made arbitrarily small, and Oε(f(n)) = O(f(n)nε). In this paper, we propose a randomized approximation algorithm... 

Motivated by a security problem in geographic information systems, we study the following graph theoretical problem: given a graph G, two special nodes s and t in G, and a number k, find k paths from s to t in G so as to minimize the number of edges shared among the paths. This is a generalization of the well-known disjoint paths problem. While disjoint paths can be computed efficiently, we show that finding paths with minimum shared edges is NP-hard. Moreover, we show that it is even hard to approximate the minimum number of shared edges within a factor of 2log1-εn, for any constant ε>0. On the positive side, we show that there exists a (k-1)-approximation algorithm for the problem, using... 

The edit distance between two strings is defined as the smallest number of insertions, deletions, and substitutions that need to be made to transform one of the strings to another one. Approximating edit distance in subquadratic time is one of the biggest unsolved problems in the field of combinatorial pattern matching" [21]. Our main result is a quantum constant approximation algorithm for computing the edit distance in truly subquadratic time. More precisely, we give an O(n1:858) quantum algorithm that approximates the edit distance within a factor of 7. We further extend this result to an O(n1:781) quantum algorithm that approximates the edit distance within a larger constant factor. Our... 

“How many guards are required to cover an art gallery?” asked Victor Klee in 1973, initiated a deep and interesting research area in computational geometry. This problem, referred to as the Art Gallery Problem, has been considered thoroughly in the literature. A recent version of this problem, introduced by Sadhu et al. in CCCG'10, is related to the connectivity of the guards. In this version, for a given set of initial guards inside a given simple polygon, the goal is to obtain a minimum set of new guards, such that the new guards alongside the initial ones have a connected visibility graph. The visibility graph of a set of points inside a simple polygon is a graph whose vertices correspond... 

This paper is aimed at investigating some computational aspects of different isoperimetric problems on weighted trees. In this regard, we consider different connectivity parameters called minimum normalized cuts/isoperimetric numbers defined through taking the minimum of the maximum or the mean of the normalized outgoing flows from a set of subdomains of vertices, where these subdomains constitute a partition/subpartition. We show that the decision problem for the case of taking k-partitions and the maximum (called the max normalized cut problem NCP^M), and the other two decision problems for the mean version (referred to as IPP^m and NCP^m) are NP-complete problems for weighted trees. On... 

We consider a class of optimization problems called movement minimization on euclidean plane. Given a set of nodes on the plane, the aim is to achieve some spe- cific property by minimum movement of the nodes. We consider two specific properties, namely the connectiv- ity (Con) and realization of a given topology (Topol). By minimum movement, we mean either the sum of all movements (Sum) or the maximum movement (Max). We obtain several approximation algorithms and some hardness results for these four problems. We obtain an O(m)-factor approximation for ConMax and ConSum and an O( p m=OPT)-factor approximation for Con- Max. We also extend some known result on graphical grounds in [1, 2] and... 

Given a 2.5D terrain and a query point p on or above it, we want to find the triangles of terrain that are visible from p. We present an approximation algorithm to solve this problem. We implement the algorithm and test it on real data sets. The experimental results show that our approximate solution is very close to the exact solution and compared to the other similar works, the computational cost of our algorithm is lower. We analyze the computational complexity of the algorithm. We consider the visibility testing problem where the goal is to test whether a given triangle of the terrain is visible or not with respect to p. We present an algorithm for this problem and show that the average... 

Let S be a set of n points on a polyhedral terrain T in ℝ3, and let ϵ > 0 be a fixed constant. We prove that S admits a (2 + ϵ )-spanner with O(n log n) edges with respect to the geodesic distance. This is the first spanner with constant spanning ratio and a near-linear number of edges for points on a terrain. On our way to this result, we prove that any set of n weighted points in Rd admits an additively weighted (2 + ϵ )-spanner with O(n) edges; this improves the previously best known bound on the spanning ratio (which was 5 + ϵ ) and almost matches the lower bound. © 2019 Society for Industrial and Applied Mathematics Publications. All rights reserved  

Let S be a set of n points on a polyhedral terrain T in ℝ3, and let ϵ > 0 be a fixed constant. We prove that S admits a (2 + ϵ )-spanner with O(n log n) edges with respect to the geodesic distance. This is the first spanner with constant spanning ratio and a near-linear number of edges for points on a terrain. On our way to this result, we prove that any set of n weighted points in Rd admits an additively weighted (2 + ϵ )-spanner with O(n) edges; this improves the previously best known bound on the spanning ratio (which was 5 + ϵ ) and almost matches the lower bound. © 2019 Society for Industrial and Applied Mathematics Publications. All rights reserved  

Given a metric graph G, we are concerned with finding a spanning tree of G where the maximum weighted degree of its vertices is minimum. In a metric graph (or its spanning tree), the weighted degree of a vertex is defined as the sum of the weights of its incident edges. In this paper, we propose a 4.5-approximation algorithm for this problem. We also prove it is NP-hard to approximate this problem within a 2 - ε factor. © 2007 Elsevier B.V. All rights reserved  

The use of classical Bossiness approximation is a straightforward strategy to take into account the buoyancy effect in incompressible solvers. This strategy is highly effective if the density variation is low. However, ignoring the importance of density variation in high thermo buoyant flows can cause considerable deviation in predicting the correct fluid flow behavior and heat transfer phenomenon. Indeed, there are many technological and environmental problems where the Bossiness approximation is not valid. In this study, an incompressible algorithm is suitably extended in order to solve compressible flow problems with natural-convection heat transfer. In this regard, the density field is... 

The spread of influence in social networks is studied in two main categories: progressive models and non-progressive models (see, e.g., the seminal work of Kempe et al. [8]). While the progressive models are suitable for modeling the spread of influence in monopolistic settings, non-progressive models are more appropriate for non-monopolistic settings, e.g., modeling diffusion of two competing technologies over a social network. Despite the extensive work on progressive models, non-progressive models have not been considered as much. In this paper, we study the spread of influence in the non-progressive model under the strict majority threshold: given a graph G with a set of initially... 

In this paper, we introduce a new version of the shortest path problem appeared in many applications. In this problem, there is a start point s, an end point t, an ordered sequence =(S0 = s, S1,...,S k, Sk+1 = t) of sets of polygons, and an ordered sequence =(F0,...,Fk) of simple polygons named fences in such that each fence Fi contains polygons of Si and Si+1. The goal is to find a path of minimum possible length from s to t which orderly touches the sets of polygons of in at least one point supporting the fences constraints. This is the general version of the previously answered Touring Polygons Problem (TPP). We prove that this problem is NP-Hard and propose a precision sensitive FPTAS...