Sharif Digital Repository

Abstract The problem of competetive facility location can be defined as follows: There are a number of customers in the form of points in space, and two players arrange a number of facilities in the form of points in space, given some limitations, respectively. Each customer’s connection to each facility has a cost for the customer and an advantage for the facility, and each customer wants to be connected to at most one of the facilities which has the lowest cost for him. The goal is to find the strategy of placing the facilities and determining the cost which the facility receives from the customer, in such a way that the player’s profit is maximised.In this thesis, we first sought to... 

Let P be a set of points in the plane, each equipped with a directional antenna that can cover a sector of angle and range r. In the symmetric model of communication, two antennas u and v can communicate to each other, if and only if v lies in u’s coverage area and vice versa. In this paper, we introduce the concept of fault-tolerant spanners for directional antennas, which enables us to construct communication networks that retain their connectivity and spanning ratio even if a subset of antennas are removed from the network. We show how to orient the antennas with angle and range r to obtain a k-fault-tolerant spanner for any positive integer k. For ⩾ , we show that the range 13 for the... 

Because of growing amount of data which communicated across the world, analyzing massive data set becomes more important than ever. The graph structure as an important and broadly used data structure is much more important than others. In this thesis, besides introducing the sublinear time algorithm model for massive data as a very efficient and superfast method to deal with massive data, we use them to solve problems for colored graphs. As a fundamental property, we try to compute the average of colored degree. This problem is analyzed from different aspects with various limitation. Two algorithms with multiplicative approximation factor, based on the average degree of simple graphs are... 

Location observations from Global Positioning System (GPS) loggers are an important source of data for travel behavior researchers. Using GPS, requires a receiver, a digital map and a software that matches the user logged positions to the map. Identifying the location of the GPS data points on a coded map of the transportation network is not a trivial process. There is always an error in calculating the coordinates obtained from the GPS satellites, and also the available maps are not accurate enough. As a result, map-matching always carries some errors. Thus, different algorithms to do this mapping more precisely are presented. The output of the map-matching process is the identification of... 

A geometric network is an undirected weighted graph whose vertices are defined as points in a geometric space, such as the real d-dimensional space. The weight of each edge of the network is equal to the Euclidean distance between the two endpoints of that edge.Geometric networks are used to model many real systems such as roads, communication networks, etc. When designing a network, one may consider different criteria. For lots of applications, existence of a short path between any two points of the network is important. The direct connection between any two points of a network is the ideal lution; however, this not possible in many applications because of its high cost. In order to... 

The Minimization of multi output Boolean functions is a fundamental and classic problem in logical circuits. For single output functions Karnough Map and Quine-McCluskey algorithms are the most common ways; however, for multi output functions they can be used to every output, but it does not necessary give the optimal solution. In this thesis we are going to compute simple expressions (expression in which we have just one type of operators) with minimum operations. We assume that we can use intermediate results in calculating other expressions to reduce repeated operations. In this thesis, first we compare the optimal solution of ‘and’ and ‘or’ operator with the ‘xor’ operator, and after... 

This thesis introduces and investigates a new kind of scheduling model, in which the total amount of computational resources to be allocated at each moment by the processor is constrained and upper bounded by a constant. The time for a task to be executed is also dependent on the computational resources, dedicated to it by the processor, assuming that the multiplication of the execution time by the dedicated computational resources is a constant for each job. We investigate various objective functions, such as minimizing the number of tardy jobs, minimizing maximum latency and etc, under varying constraints. The investigated objective functions are: a) Minimize number of tardy jobs b)... 

Let S be a set of n regions of some fixed type (such as discs, axis-parallel rectangles, etc). A conflict-free coloring (CF-coloring) of S is an assignment of colors to each region such that for every point p covered by a non-empty subset of S, there is a region with a unique color among the regions containing p. We try to maintain a CF-coloring as the objects move. Regions are moving, so after an intersection, the coloring might not be conflict-free anymore and we need to recolor some regions. The goal is to recognize the intersections and to minimize both the total number of colors used and the number of recolorings needed. In this thesis we study kinetic CF-coloring for moving... 

Conflict-free coloring problem is one of the computational geometry problems that has been studied in the last two decades. The root of this problem comes from the frequency assignment to cellular networks problem. In that research, we have a cellular network containing several regions in two-dimensional plane and we want to assign a frequency to each of the regions so that if we look at any intersection of them, there is a region with a unique color compared to other regions of that intersection. In other words, a person’s cellular phone in the range of some regions (in the intersection of them) would be able to connect to one antenna with a unique frequency and function properly. This... 

Motion is an impartible component of our physical environment and the most obvious consequence of motion is collision. Diverse applications of collision detection algorithms in various real world simulated environments have led to numerous solutions to this problem from different aspects of computer science. Within computational geometry, kinetic data structure is the well-known framework to detect collisions of moving objects.This thesis will look into the problem of collision detection for a number of disjoint disks moving freely in 2D plane. To this end, we will provide a kinetic data structure (KDS) that uses O(n) storage and under the assumption of pseudo algebraic motion plans,... 

We study the problem of preclustering a set B of imprecise points in Rd. we wish to cluster the regions specifying the potential locations of the points such that, no matter where the points are located within their regions, the resulting clustering approximates the optimal clustering for those locations. We consider k-median clustering, and obtain the following results.First we present a (6k, 2)-preclustering in R2. Then we present a preclusterings for balls in Rd, including a (3k, α)-preclustering with α = 13.9 for the k-median problem. At last we show that we need at least 3k − 3 preclusters to obtain a bounded approximation ratio for this problem  

Let S be a set of n regions of some fixed type (such as discs, axis-parallel rectangles, etc). A conflict-free coloring (CF-coloring) of S is an assignment of colors to each region such that for every point p covered by a non-empty subset of S, there is a region with a unique color among the regions containing p. The goal is to use minimum number of colors. In this thesis we study online CF-coloring for axis-parallel rectangles in the plane which rectangles are on x-axis. We present a randomized algorithm with O(〖log〗^6 n) colors. We also prove that this algorithm works with high probability  

One of the ultimate goals in robotics is to design autonomous robots. Among other things, this means a robot has to be able to plan its own motion. To be able to plan a motion, a robot must have some knowledge about the environment in which it is moving. Some of this information can be provided by a floor plan. For other information the robot will have to rely on its sensors. In this thesis, we consider that the environment is unknown for the robot. So, the robot uses weak sensor to planning its motion. The sensor can distinguish between colors. The robot can follow the colors. Therefore, we need to specify the umber of colors in which the robot can navigate by them. We propose an efficient... 

In this thesis, we consider the problem of online conflict-free coloring for regions. First, we analyze online variant of the conflict-free coloring of intervals on the line, where each newly inserted interval must be assigned a color upon insertion such that the coloring remains conflict-free, i.e. for every point p in the union of the current intervals, there must be an interval I with a unique color among all intervals covering p. we present deterministic algorithm for achieving this goal which uses O(log^3⁡n) color. Next, we consider the online conflict-free coloring of nearly-equal axis-parallel rectangles in the plane and present a deterministic algorithms which uses O(log^12⁡n)... 

In this research, we introduce the concept of local spanners for planar point sets with respect to a family of regions and prove the existence of local spanners of small size for some families. For a geometric graph $G$ on a point set $\points$ and a region $R$ belonging to a family $\Re$, we define $G \cap R$ to be the part of the graph $G$ that is inside $R$. An $\Re$-local $t$-spanner is a geometric graph $G$ on $\points$ such that for any region $R$ in $\Re$, the graph $G\cap R$ is a t-spanner with respect to $G_{c}(\points) \cap R$, where $G_{c}(\points)$ is the complete geometric graph on $P$.For any set P of n points and any constant $\eps > 0$, we prove that $P$ admits local $(1 +... 

The conflict-free coloring is one of the computational geometry problems that has received attention in the last two decades. The root of this problem comes from the frequency allocation problem to telecommunications antennas, where we have several telecommunication antennas in two-dimensional space aiming to assign to each of them a frequency so that every point on the plane, which is at least inside one of the antenna ranges, stays in a range with a frequency different from the other antennas containing point.One of the generalizations of this problem is the k-strong case, where we have n regions (equivalent to the circular region of the antennas) that we want to color so that for each... 

In the general case of minimum color-spanning tree which is one of the color-spanning set problems, given a weighted graph with n vertices of k different colors, the goal is to find a subtree of minimum weight such that vertices of this subtree include all the colors in the graph. In the planar case, the input is a complete graph with n colored vertices on the plane and the weight of each edge is the Euclidean distance between its corresponding vertices. In this thesis we consider the problem of minimum color-spanning tree. To this end, first we present various color-spanning set problems and some other related problems like Steiner tree and we study the previous work on these problems. Then... 

A coloring of regions is conflict-free if for any covered point in the plane, there exist a region that cover it, with a unique color. In this thesis, we examine the problem of conflict-free coloring of regions in dynamic model, with minimum colors, and with the possibility of recoloring the optimal numbers of colored regions. That means, regions are inserted one by one on the plane, and by inserting any region, inserted regions shall be conflict-free. Dynamic model are divided into three categories: offline model, online with absolute positions model and online with relative positions model. The difference between these three categories is in the information contained in insertion of... 

A geometric network is a weighted undirected graph whose vertices are points in the plane and edge weights are equal to the Euclidean distance of its endpoints. Routing is one of the important problems in graph theory and if the underlying graph is a geometric network,it has applications in different fields including transportation, communication networks and robotics. In many applications the graph data is not accessible or finding the optimal path is costly. Therefore local routing is used. In local routing the goal is that at each vertex,the routing must be done only given the neighbours of the current vertex, the origin and the destination of the route. Geometric spanner is a subgraph of... 

In this thesis, we investigate some problems in Sublinear field. We designed an algorithm that approximate the minimum distance of a shape to be a disk. In the first algorithm, we assumed that its input is a matrix and 1s in the matrix represent black pixels and 0s in the matrix represent white pixels. Also, we designed another algorithm that its input is a data structure contains the positions of a convex shape and we can sample uniformly from its pixels. In this problem similar to the first problem we must approximate the minimum distance of a shape to be a disk. Finally, we designed an algorithm for counting disks in a matrix. The input is a matrix that it contains some disks which have...