Sharif Digital Repository

We revisit the problem of finding k 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 k paths. We provide a ⌊k/2⌋-approximation algorithm for this problem, improving the best previous approximation factor of k - 1. We also provide the first approximation algorithm for the problem with a sublinear approximation factor of O(n3/4), where n 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 improved to O(√n). While the problem is NP-hard, and even hard to approximate to within an O(log n)... 

Motivated by the bus escape routing problem in printed circuit boards, we study the following rectangle escape problem: given a set S of n axis-aligned rectangles inside an axis-aligned rectangular region R, extend each rectangle in S toward one of the four borders of R so that the maximum density over the region R is minimized. The density of each point p∈R is defined as the number of extended rectangles containing p. We show that the problem is hard to approximate to within a factor better than 3/2 in general. When the optimal density is sufficiently large, we provide a randomized algorithm that achieves an approximation factor of 1+ε with high probability improving over the current best... 

Given a set of points in the plane, the unit clustering problem asks for finding a minimum-size set of unit disks that cover the whole input set. We study the unit clustering problem in a distributed setting, where input data is partitioned among several machines. We present a (3 + ϵ)-approximation algorithm for the problem in the Euclidean plane, and a (4 + ϵ)-approximation algorithm for the problem under general Lp metric (p1). We also study the capacitated version of the problem, where each cluster has a limited capacity for covering the points. We present a distributed algorithm for the capacitated version of the problem that achieves an approximation factor of 4+" in the L2 plane, and a... 

We focus on two metric clusterings namely r-gather and (r, ?)-gather. The objective of r-gather is to minimize the radius of clustering, such that each cluster has at least r points. (r, ?)-gather is a version of r-gather with the extra condition that at most n? points can be left unclustered (outliers). MapReduce is a model used for processing big data. In each round, it distributes data to multiple servers, then simultaneously processes each server's data. We prove a lower bound 2 on the approximation factor of metric r-gather in the MapReduce model, even if an optimal algorithm for r-gather exists. Then, we give a (4+ δ)-approximation algorithm for r-gather in MapReduce which runs in O(... 

Given a set of points in the plane, the unit clustering problem asks for finding a minimum-size set of unit disks that cover the whole input set. We study the unit clustering problem in a distributed setting, where input data is partitioned among several machines. We present a (3 + ϵ)-approximation algorithm for the problem in the Euclidean plane, and a (4 + ϵ)-approximation algorithm for the problem under general Lp metric (p1). We also study the capacitated version of the problem, where each cluster has a limited capacity for covering the points. We present a distributed algorithm for the capacitated version of the problem that achieves an approximation factor of 4+" in the L2 plane, and a... 

We focus on two metric clusterings namely r-gather and (r, ?)-gather. The objective of r-gather is to minimize the radius of clustering, such that each cluster has at least r points. (r, ?)-gather is a version of r-gather with the extra condition that at most n? points can be left unclustered (outliers). MapReduce is a model used for processing big data. In each round, it distributes data to multiple servers, then simultaneously processes each server's data. We prove a lower bound 2 on the approximation factor of metric r-gather in the MapReduce model, even if an optimal algorithm for r-gather exists. Then, we give a (4+ δ)-approximation algorithm for r-gather in MapReduce which runs in O(... 

Edit distance is one of the most fundamental problems in combinatorial optimization to measure the similarity between strings. Ulam distance is a special case of edit distance where no character is allowed to appear more than once in a string. Recent developments have been very fruitful for obtaining fast and parallel algorithms for both edit distance and Ulam distance. In this work, we present an almost optimal MPC (massively parallel computation) algorithm for Ulam distance and improve MPC algorithms for edit distance. Our algorithm for Ulam distance is almost optimal in the sense that (1) the approximation factor of our algorithm is $1+epsilon$1+ϵ, (2) the round complexity of our... 

For a set of n disjoint line segments S in R2, the visibility counting problem (VCP) is to preprocess S such that the number of visible segments in S from a query point p can be computed quickly. For this configuration, the visibility testing problem (VTP) is to test whether p sees a fixed segment s. These problems can be solved in logarithmic query time by using O(n4) preprocessing time and space. In this paper, we approximately solve this problem using quadratic preprocessing time and space. Our methods are superior to current approximation algorithms in terms of both approximation factor and preprocessing cost. In this paper, we propose a 2-approximation algorithm for the VCP using at... 

We study a weighted balanced version of the k-center problem, where each center has a fixed capacity, and each element has an arbitrary demand. The objective is to assign demands of the elements to the centers, so as the total demand assigned to each center does not exceed its capacity, while the maximum distance between centers and their assigned elements is minimized. We present a deterministic O(1)-approximation algorithm for this generalized version of the k-center problem in the distributed setting, where data is partitioned among a number of machines. Our algorithm substantially improves the approximation factor of the current best randomized algorithm available for the problem. We... 

We study the problem of maximizing constrained non-monotone submodular functions and provide approximation algorithms that improve existing algorithms in terms of either the approximation factor or simplicity. Different constraints that we study are exact cardinality and multiple knapsack constraints for which we achieve (0.25-)-factor algorithms. We also show, as our main contribution, how to use the continuous greedy process for non-monotone functions and, as a result, obtain a 0.13-factor approximation algorithm for maximization over any solvable down-monotone polytope  

Given a graph G with a set of terminals, two weight functions c and d defined on the edge set of G, and a bound D, a popular NP-hard problem in designing networks is to find the minimum cost Steiner tree (under function c) in G, to connect all terminals in such a way that its diameter (under function d) is bounded by D. Marathe et al. (J. Algoritm. 28(1), 142–171, 1998) proposed an (O(lnn),O(lnn)) approximation algorithm for this bicriteria problem, where n is the number of terminals. The first factor reflects the approximation ratio on the diameter bound D, and the second factor indicates the cost-approximation ratio. Later, Kapoor and Sarwat (Theory Comput. Syst. 41(4), 779–794, 2007)... 

Dynamic Flows were introduced by Ford and Fulkerson in 1958 to model flows over time. They define edge capacities to be the total amount of flow that can enter an edge in one time unit. Each edge also has a length, representing the time needed to traverse it. Dynamic Flows have been used to model many problems including traffic congestion, hop-routing of packets and evacuation protocols in buildings. While the basic problem of moving the maximal amount of supplies from sources to sinks is polynomial time solvable, natural minor modifications can make it NP-hard. One such modification is that flows be confluent, i.e., all flows leaving a vertex must leave along the same edge. This corresponds... 

Feature selection is a fundamental problem in machine learning and data mining. The majority of feature selection algorithms are designed for running on a single machine (centralized setting) and they are less applicable to very large datasets. Although there are some distributed methods to tackle this problem, most of them are distributing the data horizontally which are not suitable for datasets with a large number of features and few number of instances. Thus, in this paper, we introduce a novel vertically distributable feature selection method in order to speed up this process and be able to handle very large datasets in a scalable manner. In general, feature selection methods aim at... 

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. The assigned version of the k-center problem for n uncertain points in a metric space is studied in this paper. The main approach is to replace each uncertain point with a clever choice of a certain point. We argue that the k-center solution for these certain replacements of our uncertain points, is a good constant approximation factor for the original uncertain k-center problem. This approach enables us to present fast and simple algorithms that give... 

We study two well-known planar visibility problems, namely visibility testing and visibility counting, in a model where there is uncertainty about the input data. The standard versions of these problems are defined as follows: we are given a set S of n segments in R 2 , and we would like to preprocess S so that we can quickly answer queries of the form: is the given query segment s∈S visible from the given query point q∈R 2 (for visibility testing) and how many segments in S are visible from the given query point q∈R 2 (for visibility counting). In our model of uncertainty, each segment may or may not exist, and if it does, it is located in one of finitely many possible locations, given by a... 

Edit distance is one of the most fundamental problems in combinatorial optimization. Ulam distance is a special case of edit distance where no character is allowed to appear more than once in a string. Recent developments have been very fruitful for obtaining fast and parallel algorithms for both edit distance and Ulam distance. In this work, we present an almost optimal MPC algorithm for Ulam distance and improve MPC algorithms for edit distance. Our algorithm for Ulam distance is optimal in the sense that (1) the approximation factor of our algorithm is 1 + ϵ, (2) the round complexity of our algorithm is constant, (3) the total memory of our algorithm is almost linear (OH(n)), and (4)] the... 

In this paper, we present a randomized LOCAL constant approximation factor algorithm for minimum dominating set (MDS) problem and minimum total dominating set (MTDS) problem in graphs. The approximation factor of this algorithm for planar graphs with no 4-cycles is 18 and 9 for MDS and MTDS problems, respectively. © 2020 Owner/Author  

Motivated by the bus escape routing problem in printed circuit boards (PCBs), we study the following optimization problem: given a set of rectangles attached to the boundary of a rectangular region, find a subset of nonoverlapping rectangles with maximum total weight. We present an efficient algorithm that solves this problem optimally in O(n4) time, where n is the number of rectangles in the input instance. This improves over the best previous O(n6) -time algorithm available for the problem. We also present two efficient approximation algorithms for the problem that find near-optimal solutions with guaranteed approximation factors. The first algorithm finds a 2-approximate solution in O(n2)... 

Clustering is the problem of finding relations in a data set in an supervised manner. These relations can be extracted using the density of a data set, where density of a data point is defined as the number of data points around it. To find the number of data points around another point, region queries are adopted. Region queries are the most expensive construct in density based algorithm, so it should be optimized to enhance the performance of density based clustering algorithms specially on large data sets. Finding the optimum set of region queries to cover all the data points has been proven to be NP-complete. This optimum set is called the skeletal points of a data set. In this paper, we... 

This paper studies how to optimally embed a general metric, represented by a graph, into a target space while preserving the relative magnitudes of most distances. More precisely, in an ordinal embedding, we must preserve the relative order between pairs of distances (which pairs are larger or smaller), and not necessarily the values of the distances themselves. The relaxation of an ordinal embedding is the maximum ratio between two distances whose relative order is inverted by the embedding. We develop polynomial-time constant-factor approximation algorithms for minimizing the relaxation in an embedding of an unweighted graph into a line metric and into a tree metric. These two basic target...