Sharif Digital Repository

An interesting theoretical and practical set of problems in computer science is concerned with the study of spatial relations among objects in a geometric space. Examples of such problems for a set of points P are finding the closest pair of the points P, partitioning space into regions such that all points of a region have minimum distance to the same point in P, and computing the Euclidean minimum spanning tree on P. Moreover, we need mechanisms to efficiently update these properties when the points P are allowed to move or may be inserted or deleted. This is to avoid re-computation of these properties from scratch. Here, we consider the Euclidean minimum spanning tree (EMST) of a set of... 

Planar visibility computing is defined as determining the region of the plane that is visible from a specific observer. This concept has many applications in computer graphics, robotic and computer games. In this thesis we consider two visibility problems: the visibility testing problem which checks the inter-visibility of two objects and the visibility counting problem which counts the number of visible objects from an observer. For these problems we propose new algorithms which have some improvements compare to the previous ones  

A basic technique in data reduction is to approximate a collection of data by another collection of smaller size. Then, the resulted data are easier to be processed or maintained. An example of such large scale data is the ordered sequence of n points describing a path or a region boundary. We are given a sequence of points p , p , ..., p , and we consider the 1 2 n problem of approximating these points by a path with k < n line segments which error of this path is not greater than special value. Various criterions are defined to compute the path simplification error.This problem can be used in GIS, Image Processing and Computer Graphics problems. In this thesis, we consider special case... 

Moving data is becoming more and more important in different areas such as air traffic control, mobile communications, geographical information systems, GPS and other advanced technologies. In most cases, the path trajectory of such moving data is not known in advance. Moreover, they can change the direction and their speed of motion. Recently, a moving model, called black box model, has been proposed for such environments. In this model, there is a sequence of time stamps in which the position of each object is reported and the maximum displacement of each object in a time step is bounded by a known constant, dmax. In this thesis, we use this model to solve some computational... 

Separating colored points is one of the important problems in computational geometry. In separating colored points problems a set of colored points in Euclidean space are given that each color designates a set of certain data.Different problems can be defined in the colored points subject. Among them,separating colored points is studied in this thesis. It is supposed that two sets of blue and red points are given. It is desired to find the minimum number of rectangles that separate the blue points from the red points. It is demonstrated hat if P ̸= NP then there is no polynomial time algorithm for solving this problem. Then, a constant factor approximation algorithm is introduced and applied... 

3D-object modeling and its representation in computers are one of the interested fields in computer science and engineering and problems like object and environment modeling, representation, storage and physical interactions are some of the important problems in this field. Increasing the applications of the technologies like localization, machine vision and virtual reality made the 3D-object modeling and its related problems, like 3D-model extraction and reconstruction, a nowadays interested challenges and a variety of solutions such as time of flight sensors,
structured light, sonar sensors and multi-camera reconstruction are presented for it. Multi-camera solutions, just like the... 

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. [1] proposed an (O(log2n);O(log2n)) 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 [2] introduced a parameterized approximation algorithm with performance... 

Pattern recognition in data streams using bounded memory and bounded time is a difficult task. There are many techniques for recognizing patterns but when we talk about data streams these algorithms became useless since there are no enough memory to store all data. In data stream model the entire data is not available at any time and we don’t have enough time processing each data.
In this thesis we consider current methods for recognizing patterns from a data streams. The goal pattern in this study was the minimum total area of k convex polygons encloses all data  

One of the methods to detect an unknown environment, is using the visual information about the environment. In this problem, we have an unknown two-dimensional environment consists of a set of edges and vertices that the geometric positions of vertices is not specified and only the combinatorial structure of the environment is available. In addition, certain information such as visibility of two vertices in the environment is known. The goal is to recognize the geometric shape of the environment with this information.This problem has many applications in Robotics particularly in dynamic envi- ronment where objects are moving.
In this thesis, we examine this problem in different conditions... 

We design a new algorithm for clustering data streams in any fixed di- mension, that use the framework of core-set to summarize data, in order to reduce the complexity of computation. Clustering is to separate data into distinct sets called clusters, which objects in the same cluster has the most similarity and objects in the different clusters has the least similarity.This problem has many application in the areas such as: machine learning,image processing, financial and stock transactions. Data stream has recently emerged as an important concept because in many applications, data is inherently streaming over a network or the data base is extremely large and sequential access is way faster... 

Due to recent advancements and wide usage of location detection devices, huge amount of data are collected by GPS and cellular technologies which exhibits moving objects trajectories. Using this information, it is possible to track a set of objects over a long period of time, as happens for instance in studying animal migration. Always, in these situations it is undesirable or even impossible (due to process and storage limits) to store the complete stream of positioning data. Then, simplifying a trajectory as a data reduction technique is the option for resolving such problems. Moreover, there is an increasing interest in queries capturing ”aggregate” behavior of trajectories as groups like... 

Visibility graph is a graph that comprises information about visibility of a set of objects in 2 or 3-dimensional space. By constructing this graph with respect to conditions of environment one can answer questions like minimum Euclidean distance. Inverse of this procedure is required too; i.e. knowing geometrical structure of objects and information about their visibility we want to guess their location in space and, as the saying goes, reconstruct the environment. For example some methods to reconstruct specific polygons knowing which vertices each vertex sees is proposed. In this dissertation we try to reconstruct a 2-dimensional environment that only has two x-monotone paths (chains)  

A triangulated irregular network (TIN) is a data structure that is usually used for representing and storing a geographic surface. TIN is a set of points and some segments between them that are distributed in space such that projection of segments on the plane is a triangulation for projection of points. Visibility graph of points is a graph in which there is a vertex correspond to each point of TIN and there is an edge between two vertexes if corresponding points can see each other. TIN’s visibility graph has many applications; for example, finding some points in a geographic surface that they can see all of the surface together, they are good places for radars.in first section of this... 

The visibility graph of a simple polygon represents visibility relations between its vertices. Since each vertex in a polygon is visible from the two vertices adjacent to it on the boundary of the polygon, this boundary is analogous to a Hamiltonian cycle in the visibility graph of the polygon. Therefore, visibility graphs are Hamiltonian. Finding this Hamiltonian cycle can be of great help when solving visibility graph recognition problems, in which one should decide whether a given graph is a visibility graph of a simple polygon; and reconstruction problems, which include constructing the polygon corresponding to a given visibility graph. These problems have been solved for the special... 

The geometric deformation of the polygon naturally changes its Visibility Graph. Also, any change in the Visibility Graph, which includes adding or removing the edge, requires a change in the position of the polygon vertices. Of course, some changes in the graph may cause no polygon corresponding to such a graph.In this thesis, we want to study the necessary and sufficient conditions to change the Visibility Graph of a polygon, so that the new Visibility Graph corresponds to the graph obtained by adding the edge to the initial graph. We consider this problem in the case of spiral, funnel, and quasi-triangular input polygons  

Computing shortest paths and minimum spanning trees are basic and well-known problems in computer science and graph theory; also they have been considered as basic subroutines for many practical algorithms. There exists optimal algorithms for solving these problems when information about the basic graph is certain and specified. But, in real applications when the graph is obtained by using measurement tools which have limited computing precision, practically we are facing a graph in which locations of vertices or weights of edges are estimated.In such cases determining upper bounds or lower bounds for solutions of the shortest path and the minimum spanning tree are estimations of the... 

Finding optimize tours on a given sequence of objects has applications in robitic. A tour on a given sequence of objects is a path that touchs or cuts each of them, in order. In STOC′03 it is shown that finding such a shortest path for a sequence of convex polygons is polynomial solvable and it is NP-hard for non-convex polygons with intersections. The complexity of the problem for disjoint polygons is asked as the importest open peoblem. In 2008 an approximation algorithm is presented for this problem. We show that the problem is NP-hard in each Lp norm, even if each polygon consists of two unit line segments. Also, in 2003 the problem, with obstacles has been proposed as a future work. An... 

Geographical environments can be modeled through the medium of TINs, in which some discrete points are sampled as vertices, and the union of polygonal faces, particularly triangles, constitutes its surface. As regards the two-dimensional space, the TIN boundary is an x-monotone path and we focus on this case. If the number of vertices is too large to be stored or some sampled points should be removed due to a lack of vital information, the necessity of TIN simplification arises. On top of this situation, if a point observer locates on or above the TIN, the points visible to the observer have more priorities to be considered in the simplified TIN. For example, human eyes can see the closer or... 

Guarding terrains(or covering terrains) is a well known problem in the field of visibility and computational geometry. The aim of this problem is to select a minimal set of elements (such as points on terrain, line segment, points above terrain ,...),as guard set, such that every point of the terrain is visible from at least one member of the guard set.
This problem was introduced in 2-dimensions as art gallery problem. Various types of this problem have been defiend and studied in the literature. Guarding 3-dimensional terrains is an example of this problem. In 3-dimentional case, the guarded terrain can be modeled with triangulated irregullar network(TIN), as one of the surface models,... 

For a simple polygon in the plane, the visibility graph is a graph whose vertices are the vertices of the polygon and there is an edge between two vertices if and only if their corresponding vertices see each other in the polygon. Two points of the polygon see each other if the line segment connecting them lies completely inside (or on the boundary of) the polygon. Although computing the visibility graph of a polygon has been solved efficiently, its reverse problem after three decades is still an open problem. The reverse problem is known as recognition and reconstruction visibility graphs. Recognizing visibility graphs is to determine the necessary and sufficient conditions on a graph to be...