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#### Efficient Observer-Dependent Simplification in Polygonal Domains

, Article Algorithmica (New York) ; April , 2011 , Pages 1-21 ; 01784617 (ISSN) ; Ghodsi, M ; Sharif University of Technology
Abstract

In this paper, we consider a special version of the well-known line-simplification problem for simplifying the boundary of a region illuminated by a point light source q, or its visibility polygon VP(q). In this simplification approach, we should take the position of q as an essential factor into account to determine the quality of the resulting simplification. For this purpose, we redefine the known distance- and area-distortion error criteria as the main simplification criteria to take into account the distance between the observer q and the boundary of VP(q). Based on this, we propose algorithms for simplifying VP(q). More precisely, we propose simplification algorithms of O(n2) and...

#### Walking in streets with minimal sensing

, Article Journal of Combinatorial Optimization ; Volume 30, Issue 2 , 2014 , Pages 387-401 ; ISSN: 13826905 ; Ghodsi, M ; Sharif University of Technology
Abstract

We consider the problem of walking a robot in an unknown polygon called “street”, starting from a point (Formula presented.) to reach a target (Formula presented.). The robot is assumed to have minimal sensing capability in a way that cannot infer any geometric properties of the environment, such as its coordinates, angles or distances; but it is equipped with a sensor that can only detect the discontinuities in the depth information (or gaps). Our robot can also locate the target point (Formula presented.) as soon as it enters in robot’s visibility region. In addition, one pebble is assumed to be available to the robot to be used as an identifiable point and to mark any position of the...

#### Efficient visibility maintenance of a moving segment observer inside a simple polygon

, Article 19th Annual Canadian Conference on Computational Geometry, CCCG 2007, Ottawa, ON, 20 August 2007 through 22 August 2007 ; 2007 , Pages 249-252 ; Zarei, A ; Ghodsi, M ; Sharif University of Technology
2007

Abstract

In this paper we consider maintaining the visibility of a segment observer moving inside a simple polygon. A practical instance of this problem is to identify the regions of a planar scene illuminated by a fluorescent lamp while the lamp moves around. We consider both strong and weak visibility in this paper. Our method is based on the shortest path tree which builds a linear-sized data structure in O(n) time, where n is the number of the vertices of the underlying simple polygon P. We first compute VP(st̄), the initial view of the segment observer st̄. Then, as st̄ moves, each change of VP(st̄) can be computed in O(log2(|V P(st̄)|)) time when the observer is allowed to change its direction,...

#### Shortest paths in simple polygons with polygon-meet constraints

, Article Information Processing Letters ; Volume 91, Issue 4 , 2004 , Pages 171-176 ; 00200190 (ISSN) ; Ghodsi, M ; Sharif University of Technology
2004

Abstract

We study a constrained version of the shortest path problem in simple polygons, in which the path must visit a given target polygon. We provide a worst-case optimal algorithm for this problem and also present a method to construct a subdivision of the simple polygon to efficiently answer queries to retrieve the shortest polygon-meeting paths from a single-source to the query point. The algorithms are linear, both in time and space, in terms of the complexity of the two polygons. © 2004 Elsevier B.V. All rights reserved

####
Investigating Path Simplification Problems

,
M.Sc. Thesis
Sharif University of Technology
;
Zarei, Alireza
(Supervisor)
Abstract

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...

#### Minimum Link Navigation

, M.Sc. Thesis Sharif University of Technology ; Zarei, Ali Reza (Supervisor)
Abstract

Finding a path between two specified points is one of the most common problem in computational geometry. The minimum-link path problem is in this category. The minimum-link path between two points is the path between these points with the minimum number of segments. Minimum-link paths have application in areas such as computer graphic, geographic information systems, robotics, image processing, cryptography and VLSI.In this thesis we study variations of the minimum link path problem. Specifically we consider the constrained minimum-link paths in which the path must passes through some given

regions. We detected a flaw in a previously presented algorithm and propose a method to fix this...

regions. We detected a flaw in a previously presented algorithm and propose a method to fix this...

#### Shortest Path to View a Point in General Polygons

, M.Sc. Thesis Sharif University of Technology ; Ghodsi, Mohammad (Supervisor) ; Abam, Mohammad Ali (Co-Advisor)
Abstract

We consider two variations of shortest paths problem inside a poly-gon. In the first variation, we want to find a shortest path from asource to view a destination inside a polygon with holes. We pro-pose two algorithms for this problem with the time complexity of O (hn log n) and O(n log n) in which the second one is optimal. In the second variation, we want to place a center policeman inside a sim-ple polygon so that the longer of the two shortest paths from two existing guards to the visibility polygon of the center policeman is minimized. We propose an algorithm for this problem with the time complexity of O(n 3)

#### A Novel Approach for Reconstructing Paths Using Visibility Graph

, M.Sc. Thesis Sharif University of Technology ; Zarei, Alireza (Supervisor)
Abstract

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)

#### Recognition and Reconstruction of Visibility Gragh in Special Cases

, M.Sc. Thesis Sharif University of Technology ; Zarei, Alireza (Supervisor)
Abstract

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...

#### Clustering Data Streams using Core-Sets

, M.Sc. Thesis Sharif University of Technology ; Zarei, Alireza (Supervisor)
Abstract

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...

#### Guarding Orthogonal Polygons with Sliding Cameras

, M.Sc. Thesis Sharif University of Technology ; Ghodsi, Mohammad (Supervisor)
Abstract

In this research, we consider the problem of guarding orthogonal polygon with sliding cameras. This problem is originated from the well-known Art Gallery problem. In the Art Gallery problem the goal is to guard the polygon with minimum number of guards. Each guard is considered as a point and can guard all the points inside the polygon that are in its visibility area. In sliding camera version, each guard is modeld as a horizontal or vertical segment which is entirely in the polygon. The guarding area of each camera is defined by its segment and its power. In 0-transmitter model, the guarding area of each camera is limited by the edges of polygon, but in the k-transmitter model cameras can...

#### Separating Colored Points

, M.Sc. Thesis Sharif University of Technology ; Zarei, Alireza (Supervisor)
Abstract

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...

#### K-Strong Conflict Free Coloring of Regions with Respect to a Family of Points

, M.Sc. Thesis Sharif University of Technology ; Abam, Mohammad Ali (Supervisor)
Abstract

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...

#### Computing the smallest color-spanning axis-parallel square

, Article Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) ; Volume 8283 , 2013 , Pages 634-643 ; 03029743 (ISSN) ; 9783642450297 (ISBN) ; Mohades, A ; Abam, M. A ; Kazemi, M. R ; Sharif University of Technology
2013

Abstract

For a given set of n colored points with k colors in the plane, we study the problem of computing the smallest color-spanning axis-parallel square. First, for a dynamic set of colored points on the real line, we propose a dynamic structure with O(log2 n) update time per insertion and deletion for maintaining the smallest color-spanning interval. Next, we use this result to compute the smallest color-spanning square. Although we show there could be Ω(kn) minimal color-spanning squares, our algorithm runs in O(nlog2 n) time and O(n) space

#### Spanning colored points with intervals

, Article CCCG 2013 - 25th Canadian Conference on Computational Geometry ; 2013 , Pages 265-270 ; Mohades, A ; Abam, M. A ; Kazemi, M. R ; Sharif University of Technology
Canadian Conference on Computational Geometry
2013

Abstract

We study a variant of the problem of spanning colored objects where the goal is to span colored objects with two similar regions. We dedicate our attention in this paper to the case where objects are points lying on the real line and regions are intervals. Precisely, the goal is to compute two intervals together spanning all colors. As the main ingredient of our algorithm, we first introduce a kinetic data structure to keep track of minimal intervals spanning all colors. Then we present a novel algorithm using the proposed KDS to compute a pair of intervals which together span all the colors with the property that the largest one is as small as possible. The algorithm runs in O(n2 log n)...

#### Staying close to a curve

, Article Proceedings of the 23rd Annual Canadian Conference on Computational Geometry, CCCG 2011, 10 August 2011 through 12 August 2011 ; 2011 ; Sack, J. R ; Shahbaz, K ; Zarrabi Zadeh, H ; Sharif University of Technology
Abstract

Given a point set S and a polygonal curve P in Rd, we study the problem of finding a polygonal curve through S, which has minimum Fréchet distance to P. We present an efficient algorithm to solve the decision version of this problem in O(nk2) time, where n and k represent the sizes of P and S, respectively. A curve minimizing the Fréchet distance can be computed in O(nk2 log(nk)) time. As a by-product, we improve the map matching algorithm of Alt et al. by an O(log k) factor for the case when the map is a complete grap

#### Euclidean movement minimization

, Article Proceedings of the 23rd Annual Canadian Conference on Computational Geometry, CCCG 2011, 10 August 2011 through 12 August 2011 ; February , 2011 ; Fazli, M ; Ghodsi, M ; Khalilabadi, P. J ; Safari, M ; Sharif University of Technology
2011

Abstract

We consider a class of optimization problems called movement minimization on euclidean plane. Given a set of nodes on the plane, the aim is to achieve some spe- cific property by minimum movement of the nodes. We consider two specific properties, namely the connectiv- ity (Con) and realization of a given topology (Topol). By minimum movement, we mean either the sum of all movements (Sum) or the maximum movement (Max). We obtain several approximation algorithms and some hardness results for these four problems. We obtain an O(m)-factor approximation for ConMax and ConSum and an O( p m=OPT)-factor approximation for Con- Max. We also extend some known result on graphical grounds in [1, 2] and...

#### Weak visibility queries in simple polygons

, Article Proceedings of the 23rd Annual Canadian Conference on Computational Geometry, CCCG 2011 ; 2011 ; Ghodsi, M ; Sharif University of Technology
Abstract

In this paper, we consider the problem of computing the weak visibility (WV ) of a query line segment in- side a simple polygon. Our algorithm first preprocesses the polygon and creates data structures from which any WV query is answered efficiently in an output sensitive manner. In our solution, the preprocessing is performed in time O(n3 log n) and the size of the constructed data structure is O(n3). It is then possible to report the WV polygon of any query line segment in time O(log n+k), where k is the size of the output. Our algorithm im- proves the current results for this problem

#### A heuristic homotopic path simplification algorithm

, Article Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), 20 June 2011 through 23 June 2011 ; Volume 6784 LNCS, Issue PART 3 , June , 2011 , Pages 132-140 ; 03029743 (ISSN) ; 9783642219306 (ISBN) ; Ghodsi, M ; Sharif University of Technology
2011

Abstract

We study the well-known problem of approximating a polygonal path P by a coarse one, whose vertices are a subset of the vertices of P. In this problem, for a given error, the goal is to find a path with the minimum number of vertices while preserving the homotopy in presence of a given set of extra points in the plane. We present a heuristic method for homotopy-preserving simplification under any desired measure for general paths. Our algorithm for finding homotopic shortcuts runs in O( mlog(n + m) + nlogn log(nm) + k) time, where k is the number of homotopic shortcuts. Using this method, we obtain an O(n 2 + mlog(n + m) + nlogn log(nm)) time algorithm for simplification under the Hausdorff...

#### Visibility maintenance of a moving segment observer inside polygons with holes

, Article Proceedings of the 22nd Annual Canadian Conference on Computational Geometry, CCCG 2010, 9 August 2010 through 11 August 2010, Winnipeg, MB ; 2010 , Pages 117-120 ; Ghodsi, M ; Sharif University of Technology
2010

Abstract

We analyze how to efficiently maintain and update the visibility polygons for a segment observer moving in a polygonal domain. The space and time requirements for preprocessing are O(n2) and after preprocessing, visibil- ity change events for weak and strong visibility can be handled in O(log VP) and O(log(VP1 + VP2)) re- spectively, or O(log n) in which VP is the size of the line segment's visibility polygon and VP 1 and VP2 represent the number of vertices in the visibility poly- gons of the line segment endpoints