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#### 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,...

#### Weak visibility queries of line segments in simple polygons and polygonal domains

, Article International Journal of Computer Mathematics ; 2017 , Pages 1-18 ; 00207160 (ISSN) ; Ghodsi, M ; Sharif University of Technology
Taylor and Francis Ltd
2017

Abstract

In this paper we consider the problem of computing the weak visibility polygon of a query line segment pq (or (Formula presented.)) inside a given polygon (Formula presented.). Our first algorithm runs in simple polygons and needs (Formula presented.) time and (Formula presented.) space in the preprocessing phase to report (Formula presented.) of any query line segment pq in time (Formula presented.). We also give an algorithm to compute the weak visibility polygon of a query line segment in a non-simple polygon with (Formula presented.) pairwise-disjoint polygonal obstacles with a total of n vertices. Our algorithm needs (Formula presented.) time and (Formula presented.) space in the...

#### Weak visibility queries of line segments in simple polygons and polygonal domains

, Article International Journal of Computer Mathematics ; Volume 95, Issue 4 , 2018 , Pages 721-738 ; 00207160 (ISSN) ; Ghodsi, M ; Sharif University of Technology
Taylor and Francis Ltd
2018

Abstract

In this paper we consider the problem of computing the weak visibility polygon of a query line segment pq (or WVP(pq)) inside a given polygon P. Our first algorithm runs in simple polygons and needs O(n3 log n) time and O(n3) space in the preprocessing phase to report WVP(pq) of any query line segment pq in time O(log n + |WVP(pq)|).. We also give an algorithm to compute the weak visibility polygon of a query line segment in a non-simple polygon with h ≥ 1 pairwise-disjoint polygonal obstacles with a total of n vertices. Our algorithm needs O(n2 log n) time and O(n2) space in the preprocessing phase and WVP(pq) in query time of O(nh’ log n + k), in which h’ is an output sensitive parameter...

#### 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

#### How to extend visibility polygons by mirrors to cover invisible segments

, Article 11th International Conference and Workshops on Algorithms and Computation, WALCOM 2017, 29 March 2017 through 31 March 2017 ; Volume 10167 LNCS , 2017 , Pages 42-53 ; 03029743 (ISSN); 9783319539249 (ISBN) ; Ghodsi, M ; Sharif University of Technology
Springer Verlag
2017

Abstract

Given a simple polygon P with n vertices, the visibility polygon (V P) of a point q (V P(q)), or a segment (formula present) (V P(pq)) inside P can be computed in linear time. We propose a linear time algorithm to extend V P of a viewer (point or segment), by converting some edges of P into mirrors, such that a given non-visible segment (formula present) can also be seen from the viewer. Various definitions for the visibility of a segment, such as weak, strong, or complete visibility are considered. Our algorithm finds every edge such that, when converted to a mirror, makes (formula present) visible to our viewer. We find out exactly which interval of (formula present) becomes visible, by...

#### 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...

#### Visibility extension via mirror-edges to cover invisible segments

, Article Theoretical Computer Science ; Volume 789 , 2019 , Pages 22-33 ; 03043975 (ISSN) ; Ghodsi, M ; Sharif University of Technology
Elsevier B.V
2019

Abstract

Given a simple polygon P with n vertices, the visibility polygon (VP) of a point q, or a segment pq‾ inside P can be computed in linear time. We propose a linear time algorithm to extend the VP of a viewer (point or segment), by converting some edges of P into mirrors, such that a given non-visible segment uw‾ can also be seen from the viewer. Various definitions for the visibility of a segment, such as weak, strong, or complete visibility are considered. Our algorithm finds every edge that, when converted to a mirror, makes uw‾ visible to our viewer. We find out exactly which interval of uw‾ becomes visible, by every edge middling as a mirror, all in linear time. In other words, in this...

####
A Parallel Algorithm for Visibility Problem on GPU

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

Finding visibility region of an observer is one of the old and well-known problems in computational geometry and computer science.In this Problem, considering an observer (point, line segment or a geometrical object)inside an environment, the goal is to find the visible portion of that environment from the observer. Today visibility is used in many fields of computer science including robotics, computer vision and graphics. Nowadays tending to parallelism due to increases in data size and the need for accurate and complicated computations is inevitable. The GPU is composed of many weak processing units, which makes it a cheap and reachable resource for high performance computing. Due to it’s...

#### 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...

#### Approximation algorithms for computing partitions with minimum stabbing number of rectilinear and simple polygons

, Article Proceedings of the Annual Symposium on Computational Geometry, 13 June 2011 through 15 June 2011 ; June , 2011 , Pages 407-416 ; 9781450306829 (ISBN) ; Aronov, B ; De Berg, M ; Khosravi, A ; Sharif University of Technology
2011

Abstract

Let P be a rectilinear simple polygon. The stabbing number of a partition of P into rectangles is the maximum number of rectangles stabbed by any axis-parallel line segment inside P. We present a 3-approximation algorithm for the problem of finding a partition with minimum stabbing number. It is based on an algorithm that finds an optimal partition for histograms. We also study Steiner triangulations of a simple (nonrectilinear) polygon P. Here the stabbing number is defined as the maximum number of triangles that can be stabbed by any line segment inside P. We give an O(1)-approximation algorithm for the problem of computing a Steiner triangulation with minimum stabbing number

#### Pseudo-triangulating a simple polygon from its visibility graph

, Article 2015 5th International Conference on Computer and Knowledge Engineering, ICCKE 2015, 29 October 2015 ; October , 2015 , Pages 106-111 ; 9781467392808 (ISBN) ; Sharif University of Technology
Institute of Electrical and Electronics Engineers Inc
2015

Abstract

Visibility graph of a simple polygon in a plain is a graph in which the number of its vertices corresponds with the number of vertices in the polygon and each of its edges corresponds with a pair of visible vertices in the polygon. Visibility graph reconstruction of a polygon is one of the old and important problems in computational geometry for which no algorithm has been offered yet. Considering that the problem of visibility graph reconstruction of a pseudo-triangle has been solved, we present an O(n2)-time algorithm for pseudo-triangulation of a simple polygon, using the visibility graph corresponding with the polygon (n is number of vertices of the polygon). To do so, first we present a...

#### Weak visibility counting in simple polygons

, Article Journal of Computational and Applied Mathematics ; Volume 288 , November , 2015 , Pages 215-222 ; 03770427 (ISSN) ; Daneshpajouh, S ; Alipour, S ; Ghodsi, M ; Sharif University of Technology
Elsevier
2015

Abstract

For a simple polygon P of size n, we define weak visibility counting problem (WVCP) as finding the number of visible segments of P from a query line segment pq. We present different algorithms to compute WVCP in sub-linear time. In our first algorithm, we spend O(n7) time to preprocess the polygon and build a data structure of size O(n6), so that we can optimally answer WVCP in O(logn) time. Then, we reduce the preprocessing costs to O(n4+ε) time and space at the expense of more query time of O(log5n). We also obtain a trade-off between preprocessing and query time costs. Finally, we propose an approximation method to reduce the preprocessing costs to O(n2) time and space and O(n1/2+ε) query...

#### Geometric spanners for points inside a polygonal domain

, Article 31st International Symposium on Computational Geometry, SoCG 2015, 22 June 2015 through 25 June 2015 ; Volume 34 , 2015 , Pages 186-197 ; 18688969 (ISSN) ; 9783939897835 (ISBN) ; Adeli, M ; Homapour, H ; Zafar-Asadollahpoor, P ; Sharif University of Technology
Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
2015

Abstract

Let P be a set of n points inside a polygonal domain D. A polygonal domain with h holes (or obstacles) consists of h disjoint polygonal obstacles surrounded by a simple polygon which itself acts as an obstacle. We first study t-spanners for the set P with respect to the geodesic distance function π where for any two points p and q, (p, q) is equal to the Euclidean length of the shortest path from p to q that avoids the obstacles interiors. For a case where the polygonal domain is a simple polygon (i.e., h = 0), we construct a (10 + ε)-spanner that has O(n log2 n) edges. For a case where there are h holes, our construction gives a (5 + ε)-spanner with the size of O(nh log2 n). Moreover, we...

#### GPU-based parallel algorithm for computing point visibility inside simple polygons

, Article Computers and Graphics (Pergamon) ; Volume 49 , 2015 , Pages 1-9 ; 00978493 (ISSN) ; Ghodsi, M ; Sharif University of Technology
Elsevier Ltd
2015

Abstract

Given a simple polygon P in the plane, we present a parallel algorithm for computing the visibility polygon of an observer point q inside P. We use chain visibility concept and a bottom-up merge method for constructing the visibility polygon of point q. The algorithm is simple and mainly designed for GPU architectures, where it runs in O(logn) time using O(n) processors. This is the first work on designing a GPU-based parallel algorithm for the visibility problem. To the best of our knowledge, the presented algorithm is also the first suboptimal parallel algorithm for the visibility problem that can be implemented on existing parallel architectures. We evaluated a sample implementation of...

#### Near optimal line segment queries in simple polygons

, Article Journal of Discrete Algorithms ; Volume 35 , November , 2015 , Pages 51-61 ; 15708667 (ISSN) ; Ghodsi, M ; Sharif University of Technology
Elsevier
2015

Abstract

This paper considers the problem of computing the weak visibility polygon (WVP) of any query line segment pq (or WVP(pq)) inside a given simple polygon P. We present an algorithm that preprocesses P and creates a data structure from which WVP(pq) is efficiently reported in an output sensitive manner. Our algorithm needs O(n2log n) time and O(n2) space in the preprocessing phase to report WVP(pq) of any query line segment pq in time O(|WVP(pq)|+log2 n+κlog2 (nκ)), where κ is an input and output sensitive parameter of at most |WVP(pq)|. We improve the preprocessing time and space of current results for this problem [11,6] at the expense of more query time

#### Spanners for geodesic graphs and visibility graphs

, Article Algorithmica ; 2017 , Pages 1-15 ; 01784617 (ISSN) ; Sharif University of Technology
Springer New York LLC
2017

Abstract

Let (Formula presented.) be a set of n points inside a polygonal domain (Formula presented.). A polygonal domain with h holes (or obstacles) consists of h disjoint polygonal obstacles surrounded by a simple polygon which itself acts as an obstacle. We first study t-spanners for the set (Formula presented.) with respect to the geodesic distance function (Formula presented.) where for any two points p and q, (Formula presented.) is equal to the Euclidean length of the shortest path from p to q that avoids the obstacles interiors. For a case where the polygonal domain is a simple polygon (i.e., (Formula presented.)), we construct a ((Formula presented.))-spanner that has (Formula presented.)...

#### 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...

#### Spanners for geodesic graphs and visibility graphs

, Article Algorithmica ; Volume 80, Issue 2 , February , 2018 , Pages 515-529 ; 01784617 (ISSN) ; Sharif University of Technology
Springer New York LLC
2018

Abstract

Let P be a set of n points inside a polygonal domain D. A polygonal domain with h holes (or obstacles) consists of h disjoint polygonal obstacles surrounded by a simple polygon which itself acts as an obstacle. We first study t-spanners for the set P with respect to the geodesic distance function π where for any two points p and q, π(p, q) is equal to the Euclidean length of the shortest path from p to q that avoids the obstacles interiors. For a case where the polygonal domain is a simple polygon (i.e., h= 0), we construct a (10+ϵ)-spanner that has O(nlog 2n) edges. For a case where there are h holes, our construction gives a (5 + ϵ)-spanner with the size of O(nhlog2n). Moreover, we study...

#### 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...

#### Computing boundary cycle of a pseudo-triangle polygon from its visibility graph

, Article 3rd IFIP WG 1.8 International Conference on Topics in Theoretical Computer Science, TTCS 2020, 1 July 2020 through 2 July 2020 ; Volume 12281 LNCS , 2020 , Pages 61-71 ; Farokhi, S ; Sharif University of Technology
Springer Science and Business Media Deutschland GmbH
2020

Abstract

Visibility graph of a simple polygon is a graph with the same vertex set in which there is an edge between a pair of vertices if and only if the segment through them lies completely inside the polygon. Each pair of adjacent vertices on the boundary of the polygon are assumed to be visible. Therefore, the visibility graph of each polygon always contains its boundary edges. This implies that we have always a Hamiltonian cycle in a visibility graph which determines the order of vertices on the boundary of the corresponding polygon. In this paper, we propose a polynomial time algorithm for determining such a Hamiltonian cycle for a pseudo-triangle polygon from its visibility graph. © 2020, IFIP...