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Touring a sequence of disjoint polygons: Complexity and extension
, Article Theoretical Computer Science ; Vol. 556, Issue. C , October , 2014 , pp. 45-54 ; ISSN: 03043975 ; Mozafari, A ; Zarei, A ; Sharif University of Technology
Abstract
In the Touring Polygons Problem (TPP) there is a start point s, a sequence of simple polygons P=(P1,. . .,Pk) and a target point t in the plane. The goal is to obtain a path of minimum possible length that starts from s, visits in order each of the polygons in P and ends at t. This problem was introduced by Dror, Efrat, Lubiw and Mitchell in STOC '03. They proposed a polynomial time algorithm for the problem when the polygons in P are convex and proved its NP-hardness for intersecting and non-convex polygons. They asked as an open problem whether TPP is NP-hard when the polygons are pairwise disjoint. In this paper, we prove that TPP is also NP-hard when the polygons are pairwise disjoint in...
Touring disjoint polygons problem is NP-hard
, Article Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) Volume 8287 LNCS, 2013, Pages 351-360 ; Volume 8287 , 2013 , Pages 351-360 ; 03029743 (ISSN) ; 9783319037790 (ISBN) ; Mozafari, A ; Zarei, A ; Sharif University of Technology
2013
Abstract
In the Touring Polygons Problem (TPP) there is a start point s, a sequence of simple polygons P = (P1,...,Pk) and a target point t in the plane. The goal is to obtain a path of minimum possible length that starts from s, visits in order each of the polygons in P and ends at t. This problem has a polynomial time algorithm when the polygons in P are convex and is NP-hard in general case. But, it has been open whether the problem is NP-hard when the polygons are pairwise disjoint. In this paper, we prove that TPP is also NP-hard when the polygons are pairwise disjoint in any Lp norm even if each polygon consists of at most two line segments. This result solves an open problem from STOC '03 and...
Kinetic Euclidean minimum spanning tree in the plane
, Article Journal of Discrete Algorithms ; Volume 16 , October , 2012 , Pages 2-11 ; 15708667 (ISSN) ; Zarei, A ; Sharif University of Technology
Elsevier
2012
Abstract
This paper presents a kinetic data structure (KDS) for maintenance of the Euclidean minimum spanning tree (EMST) on a set of moving points in 2-dimensional space. For a set of n points moving in the plane we build a KDS of size O(n) in O(nlogn) preprocessing time by which the EMST is maintained efficiently during the motion. This is done by applying the required changes to the combinatorial structure of the EMST which is changed in discrete timestamps. We assume that the motion of the points, i.e. x and y coordinates of the points, are defined by algebraic functions of constant maximum degree. In terms of the KDS performance parameters, our KDS is responsive, local, and compact. The...
Computing polygonal path simplification under area measures
, Article Graphical Models ; Volume 74, Issue 5 , September , 2012 , Pages 283-289 ; 15240703 (ISSN) ; Ghodsi, M ; Zarei, A ; Sharif University of Technology
2012
Abstract
In this paper, we consider the restricted version of the well-known 2D line simplification problem under area measures and for restricted version. We first propose a unified definition for both of sum-area and difference-area measures that can be used on a general path of n vertices. The optimal simplification runs in O(n 3) under both of these measures. Under sum-area measure and for a realistic input path, we propose an approximation algorithm of O n2 time complexity to find a simplification of the input path, where is the absolute error of this algorithm compared to the optimal solution. Furthermore, for difference-area measure, we present an algorithm that finds the optimal...
Kinetic pie delaunay graph and its applications
, Article Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) ; Volume 7357 LNCS , 2012 , Pages 48-58 ; 03029743 (ISSN) ; 9783642311543 (ISBN) ; Rahmati, Z ; Zarei, A ; Sharif University of Technology
2012
Abstract
We construct a new proximity graph, called the Pie Delaunay graph, on a set of n points which is a super graph of Yao graph and Euclidean minimum spanning tree (EMST). We efficiently maintain the Pie Delaunay graph where the points are moving in the plane. We use the kinetic Pie Delaunay graph to create a kinetic data structure (KDS) for maintenance of the Yao graph and the EMST on a set of n moving points in 2-dimensional space. Assuming x and y coordinates of the points are defined by algebraic functions of at most degree s, the structure uses O(n) space, O(nlogn) preprocessing time, and processes O(n 2 λ 2s∈+∈2(n)β s + 2(n)) events for the Yao graph and O(n 2 λ 2s + 2(n)) events for the...
Kinetic Euclidean minimum spanning tree in the plane
, Article Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) ; Volume 7056 LNCS , 2011 , Pages 261-274 ; 03029743 (ISSN) ; 9783642250101 (ISBN) ; Zarei, A ; Sharif University of Technololgy
2011
Abstract
This paper presents the first kinetic data structure (KDS) for maintenance of the Euclidean minimum spanning tree (EMST) on a set of n moving points in 2-dimensional space. We build a KDS of size O(n) in O(nlogn) preprocessing time by which their EMST is maintained efficiently during the motion. In terms of the KDS performance parameters, our KDS is responsive, local, and compact
Visibility testing and counting
, Article Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), 28 May 2011 through 31 May 2011, Jinhua ; Volume 6681 LNCS , 2011 , Pages 343-351 ; 03029743 (ISSN) ; 9783642212031 (ISBN) ; Zarei, A ; Sharif University of Technology
2011
Abstract
For a set of n disjoint line segments S in R2, the visibility counting problem (VCP) is to preprocess S such that the number of visible segments in S from a query point p can be computed quickly. For this configuration, the visibility testing problem (VTP) is to test whether p sees a fixed segment s. These problems can be solved in logarithmic query time by using O(n4) preprocessing time and space. In this paper, we approximately solve this problem using quadratic preprocessing time and space. Our methods are superior to current approximation algorithms in terms of both approximation factor and preprocessing cost. In this paper, we propose a 2-approximation algorithm for the VCP using at...
Combinatorial changes of euclidean minimum spanning tree of moving points in the plane
, Article Proceedings of the 22nd Annual Canadian Conference on Computational Geometry, CCCG 2010, 9 August 2010 through 11 August 2010, Winnipeg, MB ; 2010 , Pages 43-45 ; Zarei, A ; Sharif University of Technology
2010
Abstract
In this paper, we enumerate the number of combinatorial changes of the the Euclidean minimum spanning tree (EMST) of a set of n moving points in 2- dimensional space. We assume that the motion of the points in the plane, is defined by algebraic functions of maximum degree s of time. We prove an upper bound of O(n3β2s(n2)) for the number of the combinatorial changes of the EMST, where βs(n)= λs(n)/n and λs(n) is the maximum length of Davenport-Schinzel sequences of order s on n symbols which is nearly linear in n. This result is an O(n) improvement over the previously trivial bound of O(n4)
When diameter matters: Parameterized approximation algorithms for bounded diameter minimum steiner tree problem
, Article Theory of Computing Systems ; Volume 58, Issue 2 , 2016 , Pages 287-303 ; 14324350 (ISSN) ; Zarei, A ; Sharif University of Technology
Springer New York LLC
Abstract
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. (J. Algoritm. 28(1), 142–171, 1998) proposed an (O(lnn),O(lnn)) 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 (Theory Comput. Syst. 41(4), 779–794, 2007)...
Visibility graphs of anchor polygons
, Article Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), 26 August 2015 through 28 August 2015 ; Volume 9541 , 2016 , Pages 72-89 ; 03029743 (ISSN) ; 9783319286778 (ISBN) ; Zarei, A ; Sharif University of Technology
Springer Verlag
2016
Abstract
Visibility graph of a polygon corresponds to its internal diagonals and boundary edges. For each vertex on the boundary of the polygon, we have a vertex in this graph and if two vertices of the polygon see each other there is an edge between their corresponding vertices in the graph. Two vertices of a polygon see each other if and only if their connecting line segment completely lies inside the polygon. Recognizing visibility graphs is the problem of deciding whether there is a simple polygon whose visibility graph is isomorphic to a given graph. Another important problem is to reconstruct such a polygon if there is any. These are well-known and well-studied, but yet open problems in...
Path simplification under difference area measure
, Article 2009 14th International CSI Computer Conference, CSICC 2009, 20 October 2009 through 21 October 2009, Tehran ; 2009 , Pages 276-279 ; 9781424442621 (ISBN) ; Zarei, A ; Ghodsi, M ; Sharif University of Technology
Abstract
In this paper, we consider path simplification problem under difference area (diff-area) measure. Diff-area measure is defined as AA(Q) AB(Q) , where AA(Q) is the area under Q and above P and AB(Q) is the area above Q and under P (sec Figure 1). Bose et al. [1] presented an approximation algorithm for finding a simplificd path with at most k verticcs that minimizes the cliff-area measure which only works on i-monotone paths. The constraint of being i-monotone is restrictive in some applications like tracking bird migration paths or map boundary simplification. Here, we extend the method of Rosc et al. [1] and present algorithms with the same time complexities as theirs for gcneral paths....
Touring convex polygons in polygonal domain fences
, Article 11th International Conference on Combinatorial Optimization and Applications, COCOA 2017, 16 December 2017 through 18 December 2017 ; Volume 10628 LNCS , 2017 , Pages 61-75 ; 03029743 (ISSN); 9783319711461 (ISBN) ; Mozafari, A ; Zarei, A ; Sharif University of Technology
Springer Verlag
2017
Abstract
In the touring polygons problem (TPP), for a given sequence (s= P0, P1, ⋯, Pk, t = Pk+1) of polygons in the plane, where s and t are two points, the goal is to find a shortest path that starts from s, visits each of the polygons in order and ends at t. In the constrained version of TPP, there is another sequence (F0, ⋯, Fk) of polygons called fences, and the portion of the path from Pi to Pi+1 must lie inside the fence Fi. TPP is NP-hard for disjoint non-convex polygons, while TPP and constrained TPP are polynomially solvable when the polygons are convex and the fences are simple polygons. In this work, we present the first polynomial time algorithm for solving constrained TPP when the...
Connecting guards with minimum Steiner points inside simple polygons
, Article Theoretical Computer Science ; Volume 775 , 2019 , Pages 26-31 ; 03043975 (ISSN) ; Zarei, A ; Sharif University of Technology
Elsevier B.V
2019
Abstract
“How many guards are required to cover an art gallery?” asked Victor Klee in 1973, initiated a deep and interesting research area in computational geometry. This problem, referred to as the Art Gallery Problem, has been considered thoroughly in the literature. A recent version of this problem, introduced by Sadhu et al. in CCCG'10, is related to the connectivity of the guards. In this version, for a given set of initial guards inside a given simple polygon, the goal is to obtain a minimum set of new guards, such that the new guards alongside the initial ones have a connected visibility graph. The visibility graph of a set of points inside a simple polygon is a graph whose vertices correspond...
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,...
Recognizing visibility graphs of triangulated irregular networks
, Article Fundamenta Informaticae ; Volume 179, Issue 4 , 2021 , Pages 345-360 ; 01692968 (ISSN) ; Ostovari, M ; Zarei, A ; Sharif University of Technology
IOS Press BV
2021
Abstract
A Triangulated Irregular Network (TIN) is a data structure that is usually used for representing and storing monotone geographic surfaces, approximately. In this representation, the surface is approximated by a set of triangular faces whose projection on the XY-plane is a triangulation. The visibility graph of a TIN is a graph whose vertices correspond to the vertices of the TIN and there is an edge between two vertices if their corresponding vertices on TIN see each other, i.e. the segment that connects these vertices completely lies above the TIN. Computing the visibility graph of a TIN and its properties have been considered thoroughly in the literature. In this paper, we consider this...
Visibility graphs of anchor polygons
, Article Journal of Graph Algorithms and Applications ; Volume 26, Issue 1 , 2022 , Pages 15-34 ; 15261719 (ISSN) ; Zarei, A ; Sharif University of Technology
Brown University
2022
Abstract
The visibility graph of a polygon corresponds to its internal diagonals and boundary edges. For each vertex on the boundary of the polygon, we have a vertex in this graph and if two vertices of the polygon see each other there is an edge between their corresponding vertices in the graph. Two vertices of a polygon see each other if and only if their connecting line segment completely lies inside the polygon. Recognizing visibility graphs is the problem of deciding whether there is a simple polygon whose visibility graph is isomorphic to a given graph. Another important problem is to reconstruct such a polygon if there is any. These problems are well known and well-studied, but yet open...
Visibility testing and counting
, Article Information Processing Letters ; Volume 115, Issue 9 , September , 2015 , Pages 649-654 ; 00200190 (ISSN) ; Ghodsi, M ; Zarei, A ; Pourreza, M ; Sharif University of Technology
Elsevier
2015
Abstract
For a set of n disjoint line segments S in R2, the visibility testing problem (VTP) is to test whether the query point p sees a query segment s∈S. For this configuration, the visibility counting problem (VCP) is to preprocess S such that the number of visible segments in S from any query point p can be computed quickly. In this paper, we solve VTP in expected logarithmic query time using quadratic preprocessing time and space. Moreover, we propose a (1+δ)-approximation algorithm for VCP using at most quadratic preprocessing time and space. The query time of this method is Oε (1/δ 2√n) where Oε (f(n))=O(f(n)nε) and ε>0 is an arbitrary constant number
Wall thickness optimization of thick-walled spherical vessel using thermo-elasto-plastic concept
, Article International Journal of Pressure Vessels and Piping ; Volume 82, Issue 5 , 2005 , Pages 379-385 ; 03080161 (ISSN) ; Rezai Zarei, A ; Darijani, H ; Sharif University of Technology
2005
Abstract
A study of thick-walled spherical vessels under steady-state radial temperature gradients using elasto-plastic analysis is reported. By considering a maximum plastic radius and using the thermal autofrettage method for the strengthening mechanism, the optimum wall thickness of the vessel for a given temperature gradient across the vessel is obtained. Finally, in the case of thermal loading on a vessel, the effect of convective heat transfer on the optimum thickness is considered, and a general formula for the optimum thickness and design graphs for several different cases are presented. © 2004 Elsevier Ltd. All rights reserved
A simple, faster method for kinetic proximity problems
, Article Computational Geometry: Theory and Applications ; Volume 48, Issue 4 , 2015 , Pages 342-359 ; 09257721 (ISSN) ; Abam, M. A ; King, V ; Whitesides, S ; Zarei, A ; Sharif University of Technology
Elsevier
2015
Abstract
For a set of n points in the plane, this paper presents simple kinetic data structures (KDSs) for solutions to some fundamental proximity problems, namely, the all nearest neighbors problem, the closest pair problem, and the Euclidean minimum spanning tree (EMST) problem. Also, the paper introduces KDSs for maintenance of two well-studied sparse proximity graphs, the Yao graph and the Semi-Yao graph. We use sparse graph representations, the Pie Delaunay graph and the Equilateral Delaunay graph, to provide new solutions for the proximity problems. Then we design KDSs that efficiently maintain these sparse graphs on a set of n moving points, where the trajectory of each point is assumed to be...
Streaming algorithms for line simplification
, Article Discrete and Computational Geometry ; Volume 43, Issue 3 , 2010 , Pages 497-515 ; 01795376 (ISSN) ; de Berg, M ; Hachenberger, P ; Zarei, A ; Sharif University of Technology
2010
Abstract
We study the following variant of the well-known line-simplification problem: we are getting a (possibly infinite) sequence of points p0,p1,p2,... in the plane defining a polygonal path, and as we receive the points, we wish to maintain a simplification of the path seen so far. We study this problem in a streaming setting, where we only have a limited amount of storage, so that we cannot store all the points. We analyze the competitive ratio of our algorithms, allowing resource augmentation: we let our algorithm maintain a simplification with 2k (internal) points and compare the error of our simplification to the error of the optimal simplification with k points. We obtain the algorithms...