Art Gallery Problem Research Articles

Pano2Geo: An efficient and robust building height estimation model using street-view panoramas

Building height serves as a crucial parameter in characterizing urban vertical structure, which has a profound impact on urban sustainable development. The emergence of street-view data offers the opportunity to observe urban 3D scenarios from the human perspective, benefiting the estimation of building height. In this paper, we propose an efficient and robust building height estimation model, which we call the Pano2Geo model, by precisely projecting street-view panorama (SVP) coordinates to geospatial coordinates. Firstly, an SVP refinement stratagem is designed, incorporating NENO rules for observation quality assessment from four aspects: number of buildings, extent of the buildings, number of nodes, and orthogonal observations, followed by the application of the art gallery theorem to further refine the SVPs. Secondly, the Pano2Geo model is constructed, which provides a pixel-level projection transformation from SVP coordinates to 3D geospatial coordinates for locating the height features of buildings in the SVP. Finally, the valid building height feature points in the SVP are extracted based on a slope mutation test, and the 3D geospatial coordinates of the building height feature points are projected using the Pano2Geo model, so as to obtain the building height. The proposed model was evaluated in the city of Wuhan in China, and the results indicate that the Pano2Geo model can accurately estimate building height, with an average error of 1.85 m. Furthermore, compared with three state-of-the-art methods, the Pano2Geo model shows superior performance, with only 10.2 % of buildings have absolute errors exceeding 2 m, compared to the Map-image-based (27.2 %), Corner-based (16.8 %), and Single-view-based (13.9 %) height estimation methods. The SVP refinement method achieves optimal observation quality with less than 50 % of existing SVPs, leading to highly efficient building height estimation, particularly in areas of a high building density. Moreover, the Pano2Geo model exhibits robustness in building height estimation, maintaining errors within 2 m even as building shape complexity and occlusion degree increase within the SVP. Our source dataset and code are available at https://github.com/Giser317/Pano2Geo.git.

A Practical Algorithm with Performance Guarantees for the Art Gallery Problem

Given a closed simple polygon $P$, we say two points $p,q$ see each other if the segment $pq$ is fully contained in $P$. The art gallery problem seeks a minimum size set $G\subset P$ of guards that sees $P$ completely. The only currently correct algorithm to solve the art gallery problem exactly uses algebraic methods and is attributed to Sharir. As the art gallery problem is ER-complete, it seems unlikely to avoid algebraic methods, without additional assumptions. In this paper, we introduce the notion of vision stability. In order to describe vision stability consider an enhanced guard that can see "around the corner" by an angle of $\delta$ or a diminished guard whose vision is by an angle of $\delta$ "blocked" by reflex vertices. A polygon $P$ has vision stability $\delta$ if the optimal number of enhanced guards to guard $P$ is the same as the optimal number of diminished guards to guard $P$. We will argue that most relevant polygons are vision stable. We describe a one-shot vision stable algorithm that computes an optimal guard set for visionstable polygons using polynomial time and solving one integer program. It guarantees to find the optimal solution for every vision stable polygon. We implemented an iterative visionstable algorithm and show its practical performance is slower, but comparable with other state of the art algorithms. Our iterative algorithm is inspired and follows closely the one-shot algorithm. It delays several steps and only computes them when deemed necessary. Given a chord $c$ of a polygon, we denote by $n(c)$ the number of vertices visible from $c$. The chord-width of a polygon is the maximum $n(c)$ over all possible chords $c$. The set of vision stable polygons admits an FPT algorithm when parametrized by the chord-width. Furthermore, the one-shot algorithm runs in FPT time, when parameterized by the number of reflex vertices.

The dispersive art gallery problem

We introduce a new variant of the art gallery problem that comes from safety issues. In this variant we are not interested in guard sets of smallest cardinality, but in guard sets with largest possible distances between these guards. To the best of our knowledge, this variant has not been considered before. We call it the Dispersive Art Gallery Problem. In particular, in the dispersive art gallery problem we are given a polygon P and a real number ℓ, and want to decide whether P has a guard set such that every pair of guards in this set is at least a distance of ℓ apart.In this paper, we study the vertex guard variant of this problem for the class of polyominoes. We consider rectangular visibility and distances as geodesics in the L1-metric. Our results are as follows. We give a (simple) thin polyomino such that every guard set has minimum pairwise distances of at most 3. On the positive side, we describe an algorithm that computes guard sets for simple polyominoes that match this upper bound, i.e., the algorithm constructs worst-case optimal solutions. We also study the computational complexity of computing guard sets that maximize the smallest distance between all pairs of guards within the guard sets. We prove that deciding whether there exists a guard set realizing a minimum pairwise distance for all pairs of guards of at least 5 in a given polyomino is NP-complete.We also present an optimal dynamic programming approach that computes a guard set that maximizes the minimum pairwise distance between guards in tree-shaped polyominoes, i.e., computes optimal solutions. Because the shapes constructed in the NP-hardness reduction are thin as well (but have holes), this result completes the case for thin polyominoes.

Topological Art in Simple Galleries

Let P be a simple polygon, then the art gallery problem is looking for a minimum set of points (guards) that can see every point in P. We say two points a,b∈P\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$a,b\\in P$$\\end{document} can see each other if the line segment seg(a,b)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ ext {seg}} (a,b)$$\\end{document} is contained in P. We denote by V(P) the family of all minimum guard placements. The Hausdorff distance makes V(P) a metric space and thus a topological space. We show homotopy-universality, that is, for every semi-algebraic set S there is a polygon P such that V(P) is homotopy equivalent to S. Furthermore, for various concrete topological spaces T, we describe instances I of the art gallery problem such that V(I) is homeomorphic to T.

Lower Bound for Sculpture Garden Problem: Localization of IoT Devices

The purpose of the current study is to investigate a special case of art gallery problem, namely a sculpture garden problem. In this problem, for a given polygon P, the ultimate goal is to place the minimum number of guards (landmarks) to define the interior polygon P by applying a monotone Boolean formula composed of the guards. Using this problem, it can replace the operation-based method with time-consuming, pixel-based algorithms. So, the processing time of some problems in the fields of machine vision, image processing and gamification can be strongly reduced. The problem has also many applications in mobile device localization in the Internet of Things (IoT). An open problem in this regard is the proof of Eppstein’s conjecture, which has remained an open problem since 2007. According to his conjecture, in the worst case, n−2 vertex guards are required to describe any n-gon. In this paper, a lower bound is introduced for the special case of this problem (natural vertex guard), which shows that if a polygon can be defined with natural vertex guards, then n−2 is a lower bound.

Extensions of the Art Gallery Theorem

Several domination results have been obtained for maximal outerplanar graphs (mops). The classical domination problem is to minimize the size of a set S of vertices of an n-vertex graph G such that $$G - N[S]$$ , the graph obtained by deleting the closed neighborhood of S, contains no vertices. In the proof of the Art Gallery Theorem, Chvátal showed that the minimum size, called the domination number of G and denoted by $$\gamma (G)$$ , is at most n/3 if G is a mop. Here we consider a modification by allowing $$G - N[S]$$ to have a maximum degree of at most k. Let $$\iota _k(G)$$ denote the size of a smallest set S for which this is achieved. If $$n \le 2k+3$$ , then trivially $$\iota _k(G) \le 1$$ . Let G be a mop on $$n \ge \max \{5,2k+3\}$$ vertices, $$n_2$$ of which are of degree 2. Upper bounds on $$\iota _k(G)$$ have been obtained for $$k = 0$$ and $$k = 1$$ , namely $$\iota _{0}(G) \le \min \{\frac{n}{4},\frac{n+n_2}{5},\frac{n-n_2}{3}\}$$ and $$\iota _1(G) \le \min \{\frac{n}{5},\frac{n+n_2}{6},\frac{n-n_2}{3}\}$$ . We prove that $$\iota _{k}(G) \le \min \{\frac{n}{k+4},\frac{n+n_2}{k+5},\frac{n-n_2}{k+2}\}$$ for any $$k \ge 0$$ . For the original setting of the Art Gallery Theorem, the argument presented yields that if an art gallery has exactly n corners and at least one of every $$k + 2$$ consecutive corners must be visible to at least one guard, then the number of guards needed is at most $$n/(k+4)$$ . We also prove that $$\gamma (G) \le \frac{n - n_2}{2}$$ unless $$n = 2n_2$$ , $$n_2$$ is odd, and $$\gamma (G) = \frac{n - n_2 + 1}{2}$$ . Together with the inequality $$\gamma (G) \le \frac{n+n_2}{4}$$ , obtained by Campos and Wakabayashi and independently by Tokunaga, this improves Chvátal’s bound. The bounds are sharp.

Automatic Placement of Visual Sensors in a Smart Space to Ensure Required PPM Level in Specified Regions of Interest.

This work is devoted to a cost-effective method for the automatic placement of visual sensors within a smart room to ensure the requirements for its design. Various unique conditions make the process of manually placing sensors time consuming and can also lead to a decrease in system efficiency. To automate the design process, we solve a multi-objective optimization problem known as the art gallery problem in 3D, modified as follows. For the specified regions of interest within a smart room, the required pixels per meter level (PPM) should be ensured. The optimization criteria are visibility of the room and the cost of equipment. To meet these criteria, we describe a room model with doors, windows, and obstacles represented in such a way as to consider their impact on the field of view of the sensors. To model sensor placement, a genetic algorithm is used. The optimal solution is selected from the Pareto front by means of the technique for order of preference by similarity to ideal solution (TOPSIS). The developed method’s effectiveness has been tested on modeling real premises of various types. The method is flexible because of the assignment of weights to certain aspects when placing sensors. Further, it can be scalable to other types of sensors.

Line segment visibility with sidedness constraints

We study a family of line segment visibility problems, related to classical art gallery problems, which are motivated by monitoring requirements in commercial data centers. Given a collection of non-overlapping line segments in the interior of a rectangle, and a requirement to monitor the segments from one side or the other, we examine the problem of finding a minimal set of point guards. Guards may be placed anywhere in the interior of the rectangle but not on a line segment. We consider combinatorial bounds of problem variants where the problem solver gets to decide which side of the segments to guard or the problem poser gets to decide which side to guard, and many others. We show that virtually all variants are NP-Hard to solve exactly, and then provide heuristics and experimental results to give insight into the associated practical problems. Finally we describe a program for using experiments to guide the search for optimal combinatorial bounds.

Smoothing the Gap Between NP and ER

We study algorithmic problems that belong to the complexity class of the existential theory of the reals ($\exists \mathbb{R}$). A problem is \ensuremath\exists \mathbbR-complete if it is as hard as the problem existential theory of the reals (ETR) and if it can be written as an ETR formula. Traditionally, these problems are studied in the real random access machine (RAM), a model of computation that assumes that the storage and comparison of real-valued numbers can be done in constant space and time, with infinite precision. The complexity class $\exists \mathbb{R}$ is often called a real RAM analogue of NP, since the problem ETR can be viewed as the real-valued variant of SAT. The real RAM assumption that we can represent and in which we can compare arbitrary irrational values in constant space and time is not very realistic. Yet this assumption is vital, since some $\exists \mathbb{R}$-complete problems have an “exponential bit phenomenon,” where there exists an input for the problem, such that the witness of the solution requires geometric coordinates which need exponential word size when represented in binary. The problems that exhibit this phenomenon are NP-hard (since ETR is NP-hard) but it is unknown if they lie in NP. NP membership is often showed by using the famous Cook--Levin theorem, which states that the existence of a polynomial-time verification algorithm for the problem witness is equivalent to NP membership. The exponential bit phenomenon prohibits a straightforward application of the Cook--Levin theorem. In this paper we first present a result which we believe to be of independent interest: we prove a real RAM analogue to the Cook--Levin theorem which shows that $\exists \mathbb{R}$ membership is equivalent to having a verification algorithm that runs in polynomial-time on a real RAM. This gives an easy proof of \ensuremath\exists \mathbbR-membership, as verification algorithms on a real RAM are much more versatile than ETR formulas. We use this result to construct a framework to study $\exists \mathbb{R}$-complete problems under smoothed analysis. We show that for a wide class of $\exists \mathbb{R}$-complete problems, its witness can be represented with logarithmic input-precision by using smoothed analysis on its real RAM verification algorithm. This shows in a formal way that the boundary between NP and $\exists \mathbb{R}$ (formed by inputs whose solution witness needs high input-precision) consists of contrived input. We apply our framework to well-studied $\exists \mathbb{R}$-complete recognition problems which have the exponential bit phenomenon such as the recognition of realizable order types or the Steinitz problem in fixed dimension. Interestingly our techniques also generalize to problems with a natural notion of resource augmentation (geometric packing, the art gallery problem).

Facial Achromatic Number of Triangulations with Given Guarding Number

A (not necessarily proper) $k$-coloring $c : V(G) \rightarrow \{1,2,\dots,k\}$ of a graph $G$ on a surface is a {\em facial $t$-complete $k$-coloring} if every $t$-tuple of colors appears on the boundary of some face of $G$. The maximum number $k$ such that $G$ has a facial $t$-complete $k$-coloring is called a {\em facial $t$-achromatic number} of $G$, denoted by $\psi_t(G)$. In this paper, we investigate the relation between the facial 3-achromatic number and guarding number of triangulations on a surface, where a {\em guarding number} of a graph $G$ embedded on a surface, denoted by $\gd(G)$, is the smallest size of its {\em guarding set} which is a generalized concept of guards in the art gallery problem. We show that for any graph $G$ embedded on a surface, $\psi_{\Delta(G^*)}(G) \leq \gd(G) + \Delta(G^*) - 1$, where $\Delta(G^*)$ is the largest face size of $G$. Furthermore, we investigate sufficient conditions for a triangulation $G$ on a surface to satisfy $\psi_{3}(G) = \gd(G) + 2$. In particular, we prove that every triangulation $G$ on the sphere with $\gd(G) = 2$ satisfies the above equality and that for one with guarding number $3$, it also satisfies the above equality with sufficiently large number of vertices.

Can greedy-like heuristics be useful for solving the Weighted Orthogonal Art Gallery Problem under regular grid discretization?

In this paper we deal with the Weighted Orthogonal Art Gallery Problem. The task is to place guards on some vertices of an orthogonal polygon P, such that the total sum of prices assigned to the chosen vertices is minimal and all points from the polygon P are covered. The problem has applications in practice, for example, in installing cameras at the corners of a building such that all interior space is covered by at least one of the cameras but the price of installation is the smallest possible. To solve the problem, the regular grid discretization of the area of polygon is applied. We propose a novel greedy approach which is based on balancing the tradeoff between the total sum of guards' costs and the total number of not yet covered points from the discretization. This new approach and an existing greedy algorithm are further hybridized with the Integer Linear programming, which is originally formulated for the wellknown Minimum Set Cover problem. Our experimental results are conducted on two sets of polygons from the literature: one with small area and the other one with large area. They proved that the proposed greedy methods can achieve the optimal solutions in most cases for the class of large-area polygons, while in case of the small area polygons, they achieve solutions of reasonable quality within lower runtime than the exact algorithms.

The Art Gallery Problem is ∃ℝ-complete

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.

Novel Sensor/Access-Point Coverage-Area Maximization for Arbitrary Indoor Polygonal Geometries

Nowadays, sensor/access-point coverage is an essential problem for wireless communication and sensor systems, which will significantly impact the quality of access, monitoring, and surveillance. Indoor sensor/access-point placement still remains very challenging as the regions of interest (ROIs) or underlying geometries may be in an arbitrary polygonal shape. In this work, we would like to study how to place a sensor/access-point to maximize the coverage area within an arbitrary polygonal ROI. Our novel approach is based on finding the maximum-area clique over the visibility graph corresponding to the indoor geometry. According to many examples, our proposed optimal sensor/access-point scheme can lead to larger coverage efficiencies than the existing solution to the art gallery problem (AGP) and the conventional Delaunay triangulation method. Our new scheme is of great practical value as it can be applied for not only both convex and nonconvex simply-connected polygons but also both convex and nonconvex multiply-connected polygons with internal holes.

Novel Optimal Multisensor Placement for Indoor Rectilinear Line-of-Sight Coverage

Optimal multisensor placement/deployment for an arbitrary indoor geometry still remains very challenging nowadays. It is preferable to minimize the number of sensors which can still cover the entire indoor space with a balanced load. However, this multi-objective problem is quite complex in practice. In this work, we will study this critical multisensor placement problem mathematically and algorithmically and propose a systematic approach to tackle this problem. In this paper, we focus on the line-of-sight (LoS) coverage within a rectilinear indoor environment and the sensors’ locations are restricted on the perimeter. Our proposed new approach consists of two stages. First, a partitioning algorithm is designed to partition an arbitrary rectilinear geometry into a number (as few as possible) of feasible rectilinear subareas, each of which can be fully covered by a sensor located somewhere on its external perimeter. Second, the average squared Euclidean distance from a sensor to an arbitrary point within its coverage area is taken as an additional objective measure. By minimizing such average squared Euclidean distance, one can find the optimal sensor location. In our new approach, the first stage is related to solving the art gallery problem (AGP) based on discrete optimization while the second stage is based on continuous optimization. Therefore, our proposed new approach involves both continuous and discrete optimization schemes. By simulations, we also show that the number of sensors placed by our new algorithm is equal to or less than the theoretical upper-bound established in the existing AGP study for numerous rectilinear geometries. Moreover, we also show that by minimizing the average squared Euclidean distance, we maximize the average received signal strength indoors using the prevalent Feko wireless-channel emulator. Our proposed new systematic approach can also deal with practical scenarios involving internal obstacle(s) and sensing-range restriction.

Chromatic Art Gallery Problem with <i>r</i>-Visibility is NP-Complete

Chromatic Art Gallery Problem with <i>r</i>-Visibility is NP-Complete

Art Gallery Problem with Rook and Queen Vision

How many chess rooks or queens does it take to guard all squares of a given polyomino, the union of square tiles from a square grid? This question is a version of the art gallery problem in which the guards can “see” whichever squares the rook or queen attacks. We show that lfloor {frac{n}{2}} rfloor rooks or lfloor {frac{n}{3}} rfloor queens are sufficient and sometimes necessary to guard a polyomino with n tiles. We then prove that finding the minimum number of rooks or queens needed to guard a polyomino is NP-hard. These results also apply to d-dimensional rooks and queens on d-dimensional polycubes. Finally, we use bipartite matching theorems to describe sets of non-attacking rooks on polyominoes.

NP-completeness of chromatic orthogonal art gallery problem

The chromatic orthogonal art gallery problem is a well-known problem in the computational geometry. Two points in an orthogonal polygon P see each other if there is an axis-aligned rectangle inside P contains them. An orthogonal guarding of P is k-colorable, if there is an assignment between k colors and the guards such that the visibility regions of every two guards in the same color have no intersection. The purposes of this paper are discussing the time complexity of k-colorability of orthogonal guarding and providing algorithms for the chromatic orthogonal art gallery problem. The correctness of presented solutions is proved, mathematically. Herein, the heuristic method is used that leads us to an innovative reduction, some optimal and one approximation algorithms. The paper shows that deciding k-colorability of orthogonal guarding for P is NP-complete. First, we prove that deciding 2-colorability of P is NP-complete. It is proved by a reduction from planar monotone rectilinear 3-SAT problem. After that, a reduction from graph coloring implies this is true for every fixed integer $$k\ge 2$$ . In the third step, we present a 6-approximation algorithm for every orthogonal simple polygon. Also, an exact algorithm is provided for histogram polygons that finds the minimum chromatic number.

Connected Wireless Camera Network Deployment with Visibility Coverage

System deployments of IoT systems have drawn research attention, because it is very challenging to meet both physical and cyber constraints in real systems. In this article, we consider the problem of deploying wireless camera networks inside a complex indoor setting for surveillance applications. We formulate the problem of the minimum connected guarding network whose objective is to place a minimum number of cameras satisfying both visual coverage of the domain and wireless network connectivity. We prove that finding the minimum connected guarding network is NP-hard in both the geometric and discrete settings. We also give a 2-approximation algorithm to the geometric minimum guarding network problem. Motivated by the connection of this problem with the watchman tour problem and the art gallery problem, we developed two algorithms to calculate the locations of camera deployment. By deploying a prototype testbed, we verify the feasibility of the system design. Using simulations on 20 real floor plans, we demonstrate that our solutions reduce the number of cameras by up to 28%, and reduce the number of relay nodes by up to 47%.

Covering orthogonal polygons with sliding k-transmitters

In this paper, we consider a new variant of covering in an orthogonal art gallery problem where each guard is a sliding k-transmitter. Such a guard can travel back and forth along an orthogonal line segment, say s, inside the polygon. A point p is covered by this guard if there exists a point q∈s such that pq‾ is a line segment normal to s, and has at most k intersections with the boundary walls of the polygon. The objective is to minimize the sum of the lengths of the sliding k-transmitters to cover the entire polygon. In other words, the goal is to find the minimum total length of trajectories on which the guards can travel to cover the entire polygon. We prove that this problem is NP-hard when k=2, and present a 2-approximation algorithm for any fixed k≥2. The proposed algorithm also works well for an orthogonal polygon where the edges have thickness.

Tight Bounds for Illuminating and Covering of Orthotrees with Vertex Lights and Vertex Beacons

We consider two variants of the Art Gallery Problem: illuminating orthotrees with a minimum set of vertex lights, and covering orthotrees with a minimum set of vertex beacons. An orthotree P is a simply connected orthogonal polyhedron that is the union of a set S of cuboids glued face to face such that the graph whose vertices are the cuboids of S, two of which are adjacent if they share a common face, is a tree. A point p illuminates a point $$q \in P$$ if the line segment $$\ell$$ joining them is contained in P. A beacon b is a point in P that pulls other points in P towards itself similarly to the way a magnet attracts ferrous particles. We say that a beacon bcoversp if when b starts pulling p, p does not get stuck at a point of P before it reaches b. This happens, for instance if p reaches a point $$p'$$ such that there is an $$\epsilon >0$$ such that any point in P at distance at most $$\epsilon$$ from $$p'$$ is farther away from $$p'$$ than q (there is another pathological case that we will not detail in this abstract). In this paper we prove that any orthotree P with n vertices can be illuminated using at most $$\lfloor n/8 \rfloor$$ light sources placed at vertices of P, and that all of the points in P can always be covered with at most $$\lfloor n/12 \rfloor$$ vertex beacons. Both bounds are tight.