POLYGON DECOMPOSITION AND THE ORTHOGONAL ART GALLERY PROBLEM

2007 ◽  
Vol 17 (02) ◽  
pp. 105-138 ◽  
Author(s):  
CHRIS WORMAN ◽  
J. MARK KEIL
Keyword(s):  
Polynomial Time ◽  
Art Gallery ◽  
Minimum Cardinality ◽  
Decomposition Problem ◽  
Art Gallery Problem ◽  
Orthogonal Polygon ◽  
Polygon Decomposition

A decomposition of a polygon P is a set of polygons whose geometric union is exactly P. We study a polygon decomposition problem that is equivalent to the Orthogonal Art Gallery problem. Two points are r-visible if the orthogonal bounding rectangle for p and q lies within P. A polygon P is an r-star if there exists a point k ∈ P such that for each point q ∈ P, q is r-visible from k. In this problem we seek a minimum cardinality decomposition of a polygon into r-stars. We show how to compute the minimum r-star cover of an orthogonal polygon in polynomial time.

LINEAR-TIME 3-APPROXIMATION ALGORITHM FOR THE r-STAR COVERING PROBLEM

2012 ◽  
Vol 22 (02) ◽  
pp. 103-141 ◽  
Author(s):  
ANDRZEJ LINGAS ◽  
AGNIESZKA WASYLEWICZ ◽  
PAWEŁ ŻYLIŃSKI
Keyword(s):  
Approximation Algorithm ◽  
Linear Time ◽  
Covering Problem ◽  
The Novel ◽  
Art Gallery ◽  
Art Gallery Problem ◽  
Orthogonal Polygon ◽  
Polygon Decomposition ◽  
Orthogonal Polygons ◽  
Complexity Status

The complexity status of the minimum r-star cover problem for orthogonal polygons had been open for many years, until 2004 when Ch. Worman and J. M. Keil proved it to be polynomially tractable (Polygon decomposition and the orthogonal art gallery problem, IJCGA 17(2) (2007), 105-138). However, since their algorithm has Õ(n17)-time complexity, where Õ(·) hides a polylogarithmic factor, and thus it is not practical, in this paper we present a linear-time 3-approximation algorithm. Our approach is based upon the novel partition of an orthogonal polygon into so-called o-star-shaped orthogonal polygons.

The Art Gallery Problem is ∃ℝ-complete

2022 ◽  
Vol 69 (1) ◽  
pp. 1-70
Author(s):  
Mikkel Abrahamsen ◽  
Anna Adamaszek ◽  
Tillmann Miltzow
Keyword(s):  
Simple Polygon ◽  
Art Gallery ◽  
Arbitrary Degree ◽  
Minimum Cardinality ◽  
Art Gallery Problem ◽  
Algebraic Set ◽  
Classic Problem ◽  
Finite Set ◽  
Roots Of Polynomials ◽  
System Of Polynomial Equations

The Art Gallery Problem (AGP) is a classic problem in computational geometry, introduced in 1973 by Victor Klee. Given a simple polygon 풫 and an integer k , the goal is to decide if there exists a set G of k guards within 풫 such that every point p ∈ 풫 is seen by at least one guard g ∈ G . Each guard corresponds to a point in the polygon 풫, and we say that a guard g sees a point p if the line segment pg is contained in 풫. We prove that the AGP is ∃ ℝ-complete, implying that (1) any system of polynomial equations over the real numbers can be encoded as an instance of the AGP, and (2) the AGP is not in the complexity class NP unless NP = ∃ ℝ. As a corollary of our construction, we prove that for any real algebraic number α, there is an instance of the AGP where one of the coordinates of the guards equals α in any guard set of minimum cardinality. That rules out many natural geometric approaches to the problem, as it shows that any approach based on constructing a finite set of candidate points for placing guards has to include points with coordinates being roots of polynomials with arbitrary degree. As an illustration of our techniques, we show that for every compact semi-algebraic set S ⊆ [0, 1] 2 , there exists a polygon with corners at rational coordinates such that for every p ∈ [0, 1] 2 , there is a set of guards of minimum cardinality containing p if and only if p ∈ S . In the ∃ ℝ-hardness proof for the AGP, we introduce a new ∃ ℝ-complete problem ETR-INV. We believe that this problem is of independent interest, as it has already been used to obtain ∃ ℝ-hardness proofs for other problems.

Tight bounds for the rectangular art gallery problem

1992 ◽  
pp. 105-112 ◽  
Author(s):  
J. Czyzowicz ◽  
E. Rivera-Campo ◽  
N. Santoro ◽  
J. Urrutia ◽  
J. Zaks
Keyword(s):  
Art Gallery ◽  
Art Gallery Problem

Combining Polygon Schematization and Decomposition Approaches for Solving the Cavity Decomposition Problem

2021 ◽  
Vol 7 (4) ◽  
pp. 1-37
Author(s):  
Serafino Cicerone ◽  
Mattia D’emidio ◽  
Daniele Frigioni ◽  
Filippo Tirabassi Pascucci
Keyword(s):  
Printed Circuit Boards ◽  
Printed Circuit Board ◽  
Low Complexity ◽  
Circuit Board ◽  
Three Solutions ◽  
Decomposition Problem ◽  
Printed Circuit ◽  
Polygon Decomposition ◽  
Finite Set ◽  
Polygonal Elements

The cavity decomposition problem is a computational geometry problem, arising in the context of modern electronic CAD systems, that concerns detecting the generation and propagation of electromagnetic noise into multi-layer printed circuit boards. Algorithmically speaking, the problem can be formulated so as to contain, as sub-problems, the well-known polygon schematization and polygon decomposition problems. Given a polygon P and a finite set C of given directions, polygon schematization asks for computing a C -oriented polygon P ′ with “low complexity” and “high resemblance” to P , whereas polygon decomposition asks for partitioning P into a set of basic polygonal elements (e.g., triangles) whose size is as small as possible. In this article, we present three different solutions for the cavity decomposition problem, which are obtained by suitably combining existing algorithms for polygon schematization and decomposition, by considering different input parameters, and by addressing both methodological and implementation issues. Since it is difficult to compare the three solutions on a theoretical basis, we present an extensive experimental study, employing both real-world and random data, conducted to assess their performance. We rank the proposed solutions according to the results of the experimental evaluation, and provide insights on natural candidates to be adopted, in practice, as modules of modern printed circuit board design software tools, depending on the observed performance and on the different constraints on the desired output.

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