Visibility queries in a polygonal region

Inkulu, R.; Kapoor, Sanjiv

doi:10.1016/j.comgeo.2009.02.004

Cited by 19 publications

(15 citation statements)

References 12 publications

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“…Zarei and Ghodsi [2005] presented an algorithm that determines V(q) in time O((1 + h ) log n+|V(q)|) (where h ≤ min(h, |V(q)|)) with O(n 3 log n) preprocessing time to construct a data structure of size O(n 3 ). Another solution to the query version is given by Inkulu and Kapoor [2009], who use a decomposition of the polygon into (simple) corridors and junctions and-depending on whether they use an approach from Aronov et al [2002] or from Hershberger and Suri [1995] as a subprocedure for computing the visibility polygon in a simple polygon-achieve O((1 + min(h, |V(q)|)) log 2 n + h + |V(q)|) or O(|V(q)| log n+ h) query time using O(n 2 log n) or O(T + |E| + n log n) preprocessing time and O(n 2 ) or O(min(|E|, hn) + n) space, respectively. (Here, |E| is the number of edges in the visibility graph, and O(T ) is the time to triangulate the given polygon).…”

Section: Geometric Aspects Of the Separation Problemmentioning

confidence: 99%

Exact solutions and bounds for general art gallery problems

Kröller

Baumgärtner

Fekete

et al. 2012

ACM J. Exp. Algorithmics

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The classical Art Gallery Problem asks for the minimum number of guards that achieve visibility coverage of a given polygon. This problem is known to be NP-hard, even for very restricted and discrete special cases. For the case of vertex guards and simple orthogonal polygons, Cuoto et al. have recently developed an exact method that is based on a set-cover approach. For the general problem (in which both the set of possible guard positions and the point set to be guarded are uncountable), neither constant-factor approximation algorithms nor exact solution methods are known.We present a primal-dual algorithm based on linear programming that provides lower bounds on the necessary number of guards in every step and-in case of convergence and integrality-ends with an optimal solution. We describe our implementation and give experimental results for an assortment of polygons, including nonorthogonal polygons with holes.

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Section: Geometric Aspects Of the Separation Problemmentioning

confidence: 99%

Exact solutions and bounds for general art gallery problems

Kröller

Baumgärtner

Fekete

et al. 2012

ACM J. Exp. Algorithmics

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“…This polygon helps in determining vertices of C that are visible to q when q is located in C. If q is not located in C, the other three simple polygons in P C , denoted by P 1 (S 1 ), P 2 (S 1 ), P 3 (S 2 ), each corresponding to a side of C, together help in determining vertices of C that are visible to q. In specific, two of these simple polygons P 1 (S 1 ), P 2 (S 1 ) correspond to one side S 1 of C, and P 3 (S 2 ) correspond to the other side S 2 of C. For more details, refer to [23].…”

Section: Preprocessing Algorithm and Data Structuresmentioning

confidence: 99%

“…As in [23], we traverse each of the visibility trees in depth-first order. At every node t, for each side S of C t , we determine vertices of V P P (q) that belong to S ∩ vc t , by applying the algorithm in [25] to each simple polygon in P Ct that corresponds to S, with q and vc t as the additional two parameters.…”

Section: Computing Visible Vertices At the Nodes Of T V Is Bmentioning

confidence: 99%

“…Zarei and Ghodsi [42] presented an algorithm that preprocesses the given polygonal domain in O(n 3 lg n) time to build data structures of size O(n 3 ), and answers each visibility polygon query in O((1 + h ) lg n + |V P (q)|) time, where h = min(h, |V P (q)|). The algorithm by Inkulu and Kapoor [23] preprocesses the input polygonal domain in O(n 2 lg n) time, builds data structures of size O(n 2 ), and answers visibility polygon query in O(min(h, |V P (q)|)(lg n) 2 + h + |V P (q)|) time. This paper also presented another algorithm with preprocessing time O(T + |V G| + n lg n), space O(min(|V G|, hn) + n), and query time O(|V P (q)| lg n + h).…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Visibility Polygon Queries Among Dynamic Polygonal Obstacles in Plane

Agrawal

Inkulu

2020

Lecture Notes in Computer Science

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We devise an algorithm for maintaining the visibility polygon of any query point in a dynamic polygonal domain, i.e., as the polygonal domain is modified with vertex insertions and deletions to its obstacles, we update the data structures that store the visibility polygon of the query point. After preprocessing the initial input polygonal domain to build a few data structures, our algorithm takes O(k(lg |V P P (q)|) + (lg n) 2 + h) (resp. O(k(lg n) 2 + (lg |V P P (q)|) + h)) worstcase time to update data structures that store visibility polygon V P P (q) of a query point q when any vertex v is inserted to (resp. deleted from) any obstacle of the current polygonal domain P. Here, n is the number of vertices in P , h is the number of obstacles in P , V P P (q) is the visibility polygon of q in P (|V P P (q)| is the number of vertices of V P P (q)), and k is the number of combinatorial changes in V P P (q) due to the insertion (resp. deletion) of v. As an application of the above algorithm, we also devise an algorithm for maintaining the visibility graph of a dynamic polygonal domain, i.e., as the polygonal domain is modified with vertex insertions and deletions to its obstacles, we update data structures that store the visibility graph of the polygonal domain. After preprocessing the initial input polygonal domain, our dynamic algorithm takes O(k(lg n) 2 + h) (resp. O(k(lg n) 2 + h)) worst-case time to update data structures that store the visibility graph when any vertex v is inserted to (resp. deleted from) any obstacle of the current polygonal domain P. Here, n is the number of vertices in P , h is the number of obstacles in P , and k is the number of combinatorial changes in the visibility graph of P due to the insertion (resp. deletion) of v.

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“…For example, using the algorithm of Zarei and Ghodsi [21], we need O(g s (1 + h ) log n + q∈Gs |V(q)|) time to construct the arrangement for the primal separation, with O(n 3 log n) preprocessing and O(n 3 ) space. Other options include the method of Pocchiola and Vegter [17], or by Inkulu and Kapoor [14]. These improvements are left for future work.…”

Section: Lemma 23 (Optimality)mentioning

confidence: 99%

Exact Solutions and Bounds for General Art Gallery Problems

Baumgärtner¹,

Fekete²,

Kröller³

et al. 2010

2010 Proceedings of the Twelfth Workshop on Algorithm Engineering and Experiments (ALENEX)

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The classical Art Gallery Problem asks for the minimum number of guards that achieve visibility coverage of a given polygon. This problem is known to be NPhard, even for very restricted and discrete special cases. For the case of vertex guards and simple orthogonal polygons, Cuoto et al. have recently developed an exact method that is based on a set cover approach. For the general problem (in which both the set of possible guard positions and the point set to be guarded are uncountable), neither constant-factor approximation algorithms nor exact solution methods are known.We present a primal-dual algorithm based on linear programming that provides lower bounds on the necessary number of guards in every step and-in case of convergence and integrality-ends with an optimal solution. We describe our implementation and give results for an assortment of polygons, including non-orthogonal polygons with holes.

show abstract

Visibility queries in a polygonal region

Cited by 19 publications

References 12 publications

Exact solutions and bounds for general art gallery problems

Exact solutions and bounds for general art gallery problems

Visibility Polygon Queries Among Dynamic Polygonal Obstacles in Plane

Exact Solutions and Bounds for General Art Gallery Problems

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