Sharif Digital Repository

The Frechet distance is a well-known geometric measure for evaluating the similarity of two polygonal curves. This distance is also known as dog-leash distance, as one can intuitively imagine the distance as the minimum length of a leash needed for a person to walk his/her dog, each walking along one of the curves from beginning to the end, without backtracking. The Frechet distance has several applications e.g., in computer graphics, computer vision, handwriting recognition and GIS.
In the standard denition of the Frechet distance, there is no limit on the speed of the motion on each of the curves. In 2009, Maheshwari et al. introduced a new variant of the Frechet distance in which each... 

Power dominating set is a concept in graph theory that was first defined as a result of studying the controllability of electric power systems. Assume that a graph G and a subset S of vertices of G are given. First, we color all vertices in S black, and all other vertices of G white. Then we color all vertices that have a neighbor in S black (Domination step). After that, for each black vertex v, if all neighbors of v except one (the vertex u) are black, then we also color the vertex u black (Propagation step). If after a number of Propagation steps all vertices of G are black, then we call S a power dominating set of G. The minimum cardinality of a power dominating set of G is called the... 

Game theory is widely used to model diverse phenomena in the real world such as people’s behavior in elections and auctions. It also has natural applications to some other areas such as computer networks, cryptography, and security. In this thesis, we present a general approach to model two important classes of optimization problems, namely, covering and packing problems, using game theory concepts. This model provides an integrated language to explain the problems, and enables us to use game-theoretic tools to further explore and analyze the problems. In our proposed model, the optimum solutions of the modeled problem are always one of the equilibria of the game. Therefore, one can find... 

The problem of finding the near neighbours is as follows: given a set of npoints, build a data structure that for any query point, can quickly find all points in distancer from the query point. The problem finds applications in various areas of computer science such as data mining, pattern recognition, databases, and search engines. An important factor here is to determine the number of points to be reported. If this number is too small, the answers may be too homogeneous (similar to the query point), and therefore, convey no useful information.On the ther hand, if the number of reported points is too high, again the informativeness decreases because of the large output size. Therefore, in... 

Clustering is one of the most well-known fundamental problems in computer science. In this thesis we have focused on a particular version of such problem, called capacititated k-center, and we analyze and survey them in the distributed model, when massive data is given. Our contribution in this research includes a new approximation algorithms with constant approximate factors for such problems in the distributed model, as well as improving the best available algorithm for capacitated k-center problem. Composable coreset as one of the most important techniques in distributed and streaming model is our primary tools in designing these algorithms. This technique gives the opportunity of solving... 

In the unit clustering problem, given a set of points on the plane, the goal is to group these points into minimum number of clusters of unit size. In the online version, the points arrive one by one and upon each point’s arrival, it must be assigned to some cluster. Another related problem is online unit covering in which moving clusters after opening them is not allowed. In this project, the online clustering and online unit covering problems are studied in two dimensional Euclidean space. An online algorithm with competitive ratio of 5 is presented for the online unit covering problem. In addition, lowerbounds of 2:5 and 4:66 are established for these problems  

With the emergence of massive datasets, storing all of the data in memory is not feasible for many problems. This fact motivated the introduction of new data processing models such as the streaming model. In this model, data points arrive one by one and the available memory is too small to store all of the data points. For many problems, more recent data points are more important than the old ones. The sliding window model captures this fact by trying to find the solution for the n most recent data points using only o(n) memory. The k-center problem is an important optimization problem in which given a graph, we are interested in labeling k vertices of the graph as centers such that the... 

In recent years, significant interest has been attracted towards using directional antennae in wireless and sensor networks due to it’s decreased energy consumption, increased security and decreased radio wave overlapping. Substituting omni-directional antennae with directional antennae should be performed in a way that not only guarantees connectivity, but also minimizes radius and latency in network nodes. The problem is not polynomially solvable in general and effort has been made in recent years to present approximation algorithms to solve this problem. The approximation algorithms have been made possible using unit disk
graphs, Euclidean minimum spanning trees, grouping of... 

In this thesis, we study the heterogeneous vehicles routing problem, which is a general-ization of the well-known travelling salesperson problem. In this problem, we are given a set of heterogeneous vehicles located in specific depots, along with a set of customers in the form of a graph. The vertices of the graph represent clients and depots, and the weight of edges represents the cost of travel between vertices of the graph. The goal is to find a subgraph for each vehicle so that the union of the subgraphs contain all customers and the total travel cost is minimized. The heterogeneity here means that the cost of traversing edges for each vehicle might be different. In this work, we briefly... 

In this thesis, we focus on a subset of geometric optimization problems (including k-center) in the Sliding Window model. The sliding window model is driven from the Data Stream model in which input points arrive one by one and the space is limited. The main diffrenece of these two models is that in the sliding window model we are interested in the N latest points not all of the arrived points. In this thesis, we study Minimum Enclosing Ball, 2-center, and Euclidean k-center in the Sliding Window model. We provide a (1 + ")-approximation algorithm for MEB in d-dimensions. To our knowledge there is no algorithm for MEB in d-dimensions where d >2. We also provide a (1 + ")-approximation... 

The k-center problem—covering a set of points using k congruent balls with minimum radius—is a well-known clustering model in computer science with a wide range of applications. The k-center is a well known NP-Hard problem. In this thesis, we focus on the k-center problem with outliers in high dimensional data streams. Due to increase in data size, we focus on the data stream model of the problem. Moreover, in real-world applications, where input points are noisy, it is very important to consider outliers. In this thesis, we study 1-center and 2-center with outliers in high dimensional data streams in Euclidean space. We provide a 1:7-approximation streaming algorithm for 1-center with z... 

In the competitive facility location problem, two service providers fight over winning consumers by establishing a series of well-located facilities on the line. The consumers seek their required services from the facility which is closest to them. The order in which service providers locate and build their facilities is as follows: the first service provider establishes k facilities after which the second service provider locates another k facilities and this goes on for m rounds. The payoff of each service provider equals the number of consumers that choose that provider’s facilities to seek service from. In this dissertation, we address different types of this problem and try to analyze... 

In this thesis, we study the max-min fair allocation of goods which represents one of the most basic problems in scheduling theory. Here we consider the online problem of allocating a set of indivisible items among agents whom each has her own preference list over the items. The goal is to maximize the minimum happiness (profit) of the agents.We present an algorithm with competitive ratio n 2 and prove that this competitive ratio is tight. We also consider this problem when each agent has a type and the number of types is constant and solve the problem where the number of types is 2 or 3. Furthermore, we study the problem with resource augmentation. We focus on doubling the number of items... 

In this research, we consider the problem of finding longest paths in some special classes of graphs. This problem is NP-hard in general case. It has been proven that there is no constant factor polynomial time approximation algorithm for this problem, unless P = NP. Therefore, the problem is usually solved for special classes of graphs.In this research, we focus on finding longest paths in solid grids, and propose a factor 3/4 approximation algorithm. Our algorithm improves the best current result which has an approximation factor of 2/ 3 . As a side result, we also propose a linear time algorithm for finding Hamiltonian cycle in a subset of Hamiltonian solid grids which improves the... 

recent years, by the huge increase in the number of components on a board, the bus routing problem and minimizing the number of layers of a board has emerged as a significant challenge in designing Printed Circuit Boards (PCBs). Two related theoretical problems to bus routing are the Rectangle Escape Problem (REP) and the maximum independent set of rectangles problem where both of these problems are NP-hard in the general form. In this thesis we study some variants of these problems and try to improve the running time or approximation factor of the best algorithms previously provided for solving them. The maximum disjoint set of boundary rectangles problem is a variant of the maximum... 

Local algorithm is a distributed algorithm which runs in constant number of rounds and its running time is independent of network size. Local approximation is a local algorithm which presents an approximation for optimal solution. In large networks (whether be natural or virtual), locality is a central subject. In networks like Internet, human society, human brain, and etc, nodes are confined to local information since accessing the whole topology of their graph is impossibile or expensive. Concerning ever-growing large networks like ad hoc and wireless sensor networks, having local algorithms for the tasks in these networks are very interested and becomes a necessity. Although there are... 

In this thesis, we study a rigidity problem for a 2-step nilmanifold such as Γ by some information about its geodesic flows, where is a simply connected 2-step nilpotent Lie group with a left invariant metric, and Γ is a discrete cocompact subgroup of . For the solution to this problem, first, we consider an algebraic aspect of it; since isometry groups of simply connected Riemannian manifolds can be characterized in a purely algebraic way, i.e., normalizers. Also, as we will show, proper and smooth actions of Lie groups and closed subgroups of isometries for smooth Riemannian structures can be regarded as the same topic. Then, in a generalized setting, when passing from the case of...