Sharif Digital Repository

We study the problem of preclustering a set $B$ of imprecise points in~$\Reals^d$: 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 $k$-center, $k$-median, and $k$-means clustering, and obtain the various results. Then we study the higher order Voronoi Tessellations as an important concept in clustering area. Generalizing Lee’s inductive argument for counting the cells of higher order Voronoi tessellations in $\Reals^2$ to $\Reals^3$, we get precise relations in terms of Morse theoretic quantities... 

Edit distance or Levenshtein distance is one of the most fundamental problems in computer science and engineering to measure the dissimilarity between strings.This problem is solvable in O(n^2) time using dynamic programming and in 2015 it was conditionally proven that no truly subquadratic time algorithm (\textit{i.e.}, O(n^{2-ε})) exists for edit distance.Since 2001, approximating edit distance within a constant factor was recognized as one of the biggest unsolved problems in the field of combinatorial pattern matching.This thesis designs seven approximation algorithms for edit distance and its related problems.Firstly, two quantum approximation algorithms are designed with constant... 

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... 

This thesis is concerned with a fundamental problem in computational geometry. This problem inspects covering a space with a set of points provided that each point of the space is visible to at least one point from the selected subset. Applications of these problems are in the areas such as geographic information systems (GIS), robotics,computer graphics and the military.The investigated space is a planar rectangular region which is partitioned into smaller rectangles by connected edges. The objective of this study is to find an optimized corridor; i.e., a connected subset of edges which contains at least one point from each rectangle. For the minimum diameter corridor problem, an exact... 

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... 

Approximate Agreement (AA) is one of the most important issues in distributed networks in which non-faulty processes execute a voting algorithm to reach the values that are very close to each other. AA for Partially Connected Networks (PCNs) is very limited because a node may not have a complete view of the whole network which makes it difficult to reach the convergence. This research proposes a convergent Fuzzy algorithm for synchronous distributed networks in the presence of faulty or Byzantine processes. When each node receives a new value from its neighbors,it runs the convergent Fuzzy algorithm. If the received value is not in the expected range of values, the processor can discard the... 

We study a class of optimization network problems on pairs of structures. In particular, given n pairs of points P = f(p1; q1); (p2; q2); : : : ; (pn; qn)g, in the plane or in general metric space, the goal is to color points of each pair, one red and the other blue, so as a given cost function on the red points and blue points is optimized. The cost functions studied previously are MST, TSP, and matching. In this work, we consider a new cost function; i.e., the diameter of each colored point set, and present efficient algorithms to solve the problem according to some objective functions  

Designing a rapid transit network which is capable of providing a convenient coverage for passengers is studied in thisdoctoral thesis. First, a mathematical model of rapid transit network design which is a mixed integer program is presented (C_RTND). Objective function of the model is maximization of acceptable transit coverage and the model is capable of considering different rapid transit technologies. Furthermore, for each technology capacity constraint is considered in the model. Second, a heuristic algorithm is developed and used to solve the proposed model. The algorithm consists of 3 main parts: candidate line generation, network evaluation and a search algorithm for selecting a good... 

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... 

Due to increasing quality of satellite images, volume of stored data significantly increased, so speed of statistical and computational processing decreased. For solving this problem, simplification
problem has been suggested. Surface simplification problem is a fundamental problem in computational geometry and it has many applications in other fields such as GIS, computer graphics, and image processing. Major goal of simplification problems is reducing stored information in any surface, Because it improves speed of processes. One of common types in this field is 3D terrain simplification while error of simplified surface be acceptable. Simplification is NP-Hard problem. In this project,... 

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)... 

Nowadays, we observe the growth in the number of mobile smart phones. These smart phones have had an impressive progress in software and hardware. Many studies have focused on utilizing this new platform. One of these areas of research is collecting information from environment.Two major challenges for a successful crowdsourcing system are engaging a sufficient number of workers and validating of their registered data, which make the system more reliable. Collecting data from different points of a territory can be defined as some tasks and should be assigned to users near those tasks properly. Some users want a monetary reward for the cost they incur. On the other hand, information... 

Problem of 2-colorability of hypergraphs was introduced by Erd?os in 1963, and there are many results about this problem until now. In this thesis we consider this problem from the algorithmic viewpoint. In all of the algorithms we assume that the given hypergraph is 2-colorable and we want to color it. By using a simple construction, it was shown that for every r = 3, the problem of coloring of hypergraphs is as hard as the problem of coloring graphs. If NP ?= ZPP12, it is impossible to approximate the chromatic number of r-uniform hyhpergraphs with n vertices by a factor of n1-? for any fixed ? > 0, in time polynomial of n. In reference [KS03] a coloring algorithm with approximation factor... 

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... 

Computational complexity of graph problems is an important branch in the-oretical computer science. We introduce to some of ideas for computing the complexity of graph problems with some kind and beautiful examples. Next, we show hardness and inapproximability of some problems. Representation number of graphs has been introduce by Pavel Erdos by Number theory. We prove n1−ϵ inapproximability of that. Lucky number η has been studied by Grytczuk et.al . We show for planar and 3-colorable graphs, it is NP-Complete whether η = 2. Note that since a conjecture, for those graphs, 2 ≤ η ≤ 3. Also for each k ≥ 2, we show NP-completeness of η ≤ k for the graphs. Proper orientation number −→ is a... 

Data clustering is simply the process of partitioning data in to non empty subsets, called clusters, such that in every clusters members are related due to an pre-defined objective function. Assume that there are n points in eulcidian d-dimensional space and the goal is to determine k points in space as a representive of each cluster so that in every cluster, members have the least value of an objective function which is related to their distances from representatives. From most important types of data clustering problem we may mention k-center, k-median and k-means problem in which the objective functions are respectively, minimizing the maximum, sum and sum of squares of distance of every... 

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  

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... 

In this study, we consider the fair division problem. In this problem, a heterogeneous resource must be fairly divided among a set of agents with different preferences.The resource can be either a divisible good (i.e., time, land), or a set of indivisible goods. When the resource is a single divisible good, the problem is commonly known as cake cutting. To measure fairness, several notions are defined, i.e., envy-freeness, proportionality, equability, maximin-share. In this thesis, we first give a formal definition of these notions. Next, we present our results for envy-freeness and maximin-share.First, we prove the existence of an allocation that guarantees each agent a factor 3/4 of his... 

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...