Improved Bounds for Wireless Localization

Video cameras are the most common and affordable means of security. What is captured by the camera lens is transmitted to the video surveillance monitor screens in the security room. It is important to reduce the number of the monitor screens to a minimum and place the surveillance cameras to cover the entire protected areas. Reducing the number of video cameras helps significantly reduce the price of the entire surveillance system. A series of problems on the protection of an art gallery are devoted to the optimization of security systems. Currently, art gallery security problems are fairly well-studied visibility problems in computational geometry. It can be seen as the problem of using the least number of surveillance cameras to provide video coverage of all the rooms and halls of an art gallery. In the twodimensional case, the gallery layout is represented as a simple polygon, with the guard as a point inside it. This paper considers two variations of the problem: the guard route problem and the problem of guarding an art gallery on the surface of a convex polyhedron. These problems were considered in the works of many mathematicians. The paper provides a description of algorithms for solving these problems and pseudocodes of the main procedures necessary to implement these algorithms in the Python programming language.

We prove that the art gallery problem is equivalent under polynomial time reductions to deciding whether a system of polynomial equations over the real numbers has a solution. The art gallery problem is a classic problem in computational geometry, introduced in 1973 by Victor Klee. Given a simple polygon P and an integer k, the goal is to decide if there exists a set G of k guards within P such that every point p ∈ P is seen by at least one guard g∈ G. Each guard corresponds to a point in the polygon P, and we say that a guard g sees a point p if the line segment pg is contained in P. The art gallery problem has stimulated extensive research in geometry and in algorithms. However, the complexity status of the art gallery problem has not been resolved. It has long been known that the problem is NP-hard, but no one has been able to show that it lies in NP. Recently, the computational geometry community became more aware of the complexity class ∃ ℝ, which has been studied earlier by other communities. The class ∃ ℝ consists of problems that can be reduced in polynomial time to the problem of deciding whether a system of polynomial equations with integer coefficients and any number of real variables has a solution. It can be easily seen that NP ⊆ ∃ ℝ. We prove that the art gallery problem is ∃ ℝ-complete, implying that (1) any system of polynomial equations over the real numbers can be encoded as an instance of the art gallery problem, and (2) the art gallery problem is not in the complexity class NP unless NP=∃ ℝ. As an illustration of our techniques, we can show that for every compact semi-algebraic set S⊂ [0,1]2 there exists a polygon with rational coordinates that enforces one of the guards to be at any position p∈ S, in any optimal guarding. As a corollary of our construction, we prove that for any real algebraic number α there is an instance of the art gallery problem 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. In the ∃ ℝ-hardness proof for the art gallery problem we introduce a new ∃ ℝ-complete problem ETR-INV. We believe that this problem is of independent interest, as it can be used to obtain ∃ ℝ-hardness proofs for other problems. In particular, ETR-INV has been very recently used to prove ∃ ℝ-hardness of other geometric problems.

In chromatic variants of the art gallery problem, simple polygons are guarded with point guards that are assigned one of k colors each. We say these guards cover the polygon. Here we consider the conflict-free chromatic art gallery problem, first studied by Bartschi and Suri (Algorithmica 2013): A covering of the polygon is conflict-free if each point of the polygon is seen by some guard whose color appears exactly once among the guards visible to that point. We are interested in the smallest number k(n) of colors that ensure such a covering for every n-vertex polygon. It is known that k(n) is O(log n) for orthogonal and for monotone polygons, and O(log2 n) for arbitrary simple polygons. Our main contribution in this paper is an improvement of the upper bound on k(n) to O(log n) for simple polygons. The bound is achieved through a partitioning of the polygon into weak visibility subpolygons, which is known as a window partition. In a weak visibility polygon, there is a boundary edge e such that each point of the polygon is seen by some point on e. We show for the first time for this special class of polygons an upper bound of O(log n). For the subpolygons of the window partition we prove a novel concept of independence that allows to reuse colors in independent subpolygons. Combining these results leads to the upper bound of O(log n) for arbitrary simple polygons.

Routing in a polygonal terrain with the shortest beacon watchtower

We use the concept of pointed pseudo-triangulations to establish new upper and lower bounds on a well known problem from the area of art galleries: What is the worst case optimal number of vertex π-guards that collectively monitor a simple polygon with n vertices? Our results are as follows: (1) Any simple polygon with n vertices can be monitored by at most \lfloor n/2 \rfloor general vertex π-guards. This bound is tight up to an additive constant of 1. (2) Any simple polygon with n vertices, k of which are convex, can be monitored by at most \lfloor (2n – k)/3 \rfloor edge-aligned vertexπ-guards. This is the first non-trivial upper bound for this problem and it is tight for the worst case families of polygons known so far.

We devise a polynomial-time algorithm for partitioning a simple polygon P into a minimum number of star-shaped polygons. The question of whether such an algorithm exists has been open for more than four decades [Avis and Toussaint, Pattern Recognit., 1981] and it has been repeated frequently, for example in O’Rourke’s famous book [Art Gallery Theorems and Algorithms, 1987]. In addition to its strong theoretical motivation, the problem is also motivated by practical domains such as CNC pocket milling, motion planning, and shape parameterization.

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.

We use the concept of pointed pseudo-triangulations to establish new upper and lower bounds on a well known problem from the area of art galleries: What is the worst case optimal number of vertex π-guards that collectively monitor a simple polygon with n vertices?Our results are as follows:1. Any simple polygon with n vertices can be mon- itored by at most [n/2] general vertex π-guards. This bound is tight up to an additive constant of 1.2. Any simple polygon with n vertices, k of which are convex, can be monitored by at most [(2n -- k)/3] edge-aligned vertex π-guards. This is the first non- trivial upper bound for this problem and it is tight for the worst case families of polygons known so far.

Two points in a polygon are called if the straight line between them lies entirely inside the polygon. The art gallery problem for a polygon P is to find a minimum set of points G in P such that every point in P is visible from some point of G. The author provides an introduction to art gallery theorems and surveys the recent results of the field. The emphasis is on the results rather than the techniques. Several new problems that have the same geometric flavor as art gallery problems are also examined.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Ruler of the Plane is a set of games illustrating concepts from combinatorial and computational geometry. The games are based on the art gallery problem, ham-sandwich cuts, the Voronoi game, and geometric network connectivity problems like the Euclidean minimum spanning tree and traveling salesperson problem.

We prove that for various impurity models, in both classical and quantum settings, the self-energy matrix is a sparse matrix with a sparsity pattern determined by the impurity sites. In the quantum setting, such a sparsity pattern has been known since Feynman. Indeed, it underlies several numerical methods for solving impurity problems, as well as many approaches to more general quantum many-body problems, such as the dynamical mean field theory. The sparsity pattern is easily motivated by a formal perturbative expansion using Feynman diagrams. However, to the extent of our knowledge, a rigorous proof has not appeared in the literature. In the classical setting, analogous considerations lead to a perhaps less-known result, i.e., that the precision matrix of a Gibbs measure of a certain kind differs only by a sparse matrix from the precision matrix of a corresponding Gaussian measure. Our argument for this result mainly involves elementary algebraic manipulations and is in particular non-perturbative. Nonetheless, the proof can be robustly adapted to various settings of interest in physics, including quantum systems (both fermionic and bosonic) at zero and finite temperature, non-equilibrium systems, and superconducting systems.

We present an O(nrG) time algorithm for computing and maintaining a minimum length shortest watchman tour that sees a simple polygon under monotone visibility in direction \(\theta \), while \(\theta \) varies in \([0,180^{\circ })\), obtaining the directions for the tour to be the shortest one over all tours, where n is the number of vertices, r is the number of reflex vertices, and \(G\le r\) is the maximum number of gates of the polygon used at any time in the algorithm.

The art gallery problem is aiming at finding the minimum number of guards to cover a gallery. In this paper, we consider the problem of covering a gallery with holes. The guards are required to cover a gallery that is represented as a simple polygon with n vertices and h holes. We present a robust and fast algorithm to compute a small number of vertex guards in such polygons, which runs in O(nlogn) time and uses O(n) storage. The proposed algorithm is not only offering a better performance in terms of computational cost but also ease in implementation. Simulation results demonstrate the efficiency, robustness, and potential of the proposed algorithm in motion planning systems

Given a simple polygon P on n vertices, two points x, y in P are said to be visible to each other if the line segment between x and y is contained in P. The Point Guard Art Gallery problem asks for a minimum-size set S such that every point in P is visible from a point in S. The set S is referred to as guards. Assuming integer coordinates and a specific general position on the vertices of P, we present the first O(log OPT)-approximation algorithm for the point guard problem. This algorithm combines ideas in papers of Efrat and Har-Peled and Deshpande et al. We also point out a mistake in the latter.

High-dimensional integrals occur in a variety of areas, including mathematical finance. In the classical settings, multivariate integration problems suffer from the curse of dimensionality. To vanquish the curse of dimensionality, one may shrink the function class. Here, we use weighted function spaces, in which groups of variables are associated with weights, in order to capture the different importance of each group of variables. For practical applications, the principal difficulty is then in choosing the ‘right’ weights for a given problem or a class of problems. We work in weighted reproducing kernel Hilbert spaces (with ‘general’ rather than ‘product’ weights). We first present a principle to find the good weights for a given problem. This general approach is then applied to a simplified high-dimensional problem from finance, in which the dimension is the number of discrete timesteps for a price of a risky asset which follows geometric Brownian motion. The second focus of this paper is on the dimension reduction techniques, such as the Brownian bridge (BB) and the principal component analysis (PCA). It turns out that the behaviour of the model problem is dramatically improved when quasi-Monte Carlo method is used in conjunction with the dimension reduction techniques: if the right weights are used in every case, then the integration error can be bounded independently of the dimension, whereas without BB or PCA the error bound depends exponentially on the dimension. Finally, for this model problem we show how to construct shifted lattice rules which, when used in conjunction with BB or PCA, yield integration errors converging as O(n−3/4+δ) or O(n−1+δ) (for arbitrary small δ > 0), respectively, independently of the dimension. Thus, in both cases well-designed algorithms can avoid the curse of dimensionality, with PCA having an advantage over BB with respect to the proved order of convergence.