Guarding curvilinear art galleries with edge or mobile guards via 2-dominance of triangulation graphs

AbstractIn this paper we consider the problem of monitoring an art gallery modeled as a polygon, the edges of which are arcs of curves, with edge or mobile guards. Our focus is on piecewise-convex polygons, i.e., polygons that are locally convex, except possibly at the vertices, and their edges are convex arcs.We transform the problem of monitoring a piecewise-convex polygon to the problem of 2-dominating a properly defined triangulation graph with edges or diagonals, where 2-dominance requires that every triangle in the triangulation graph has at least two of its vertices in its 2-dominating set. We show that: (1) ⌊n+13⌋ diagonal guards are always sufficient and sometimes necessary, and (2) ⌊2n+15⌋ edge guards are always sufficient and sometimes necessary, in order to 2-dominate a triangulation graph. Furthermore, we show how to compute: (1) a diagonal 2-dominating set of size ⌊n+13⌋ in linear time and space, (2) an edge 2-dominating set of size ⌊2n+15⌋ in O(n2) time and O(n) space, and (3) an edge 2-dominating set of size ⌊3n7⌋ in O(n) time and space.Based on the above-mentioned results, we prove that, for piecewise-convex polygons, we can compute: (1) a mobile guard set of size ⌊n+13⌋ in O(nlogn) time, (2) an edge guard set of size ⌊2n+15⌋ in O(n2) time, and (3) an edge guard set of size ⌊3n7⌋ in O(nlogn) time. All space requirements are linear. Finally, we show that ⌊n3⌋ mobile or ⌈n3⌉ edge guards are sometimes necessary.When restricting our attention to monotone piecewise-convex polygons, the bounds mentioned above drop: ⌈n+14⌉ edge or mobile guards are always sufficient and sometimes necessary; such an edge or mobile guard set, of size at most ⌈n+14⌉, can be computed in O(n) time and space

Improved Bounds for Wireless Localization

We consider a novel class of art gallery problems inspired by wireless localization that has recently been introduced by Eppstein, Goodrich, and Sitchinava. Given a simple polygonP, place and orient guards each of which broadcasts a unique key within a fixed angular range. In contrast to the classical art gallery setting, broadcasts are not blocked by the edges ofP. At any point in the plane one must be able to tell whether or not one is located inside P only by looking at the set of keys received. In other words, the interior of the polygon must be described by a monotone Boolean formula composed from the keys. We improve both upper and lower bounds for the general problem where guards may be placed anywhere by showing that the maximum number of guards to describe any simple polygon on n vertices is between roughly $\frac{3}{5}n$ and $\frac{4}{5}n$ . Aguarding that uses at most $\frac{4}{5}n$ guards can be obtained in O(nlog n) time. For the natural setting where guards may be placed aligned to one edge or two consecutive edges ofP only, we prove that n−2 guards are always sufficient and sometimes necessar

Art Gallery Theorems

Some important results about art gallery theorems are proposed, starting from Chvátal’s essay, using also polygon triangulations and orthogonal polygons

Monitoring maximal outerplanar graphs

In this paper we define a new concept of monitoring the elements of triangulation graphs by faces. Furthermore, we analyze this, and other monitoring concepts (by vertices and by edges), from a combinatorial point of view, on maximal outerplanar graphs

Online Algorithms with Discrete Visibility - Exploring Unknown Polygonal Environments

The context of this work is the exploration of unknown polygonal environments with obstacles. Both the outer boundary and the boundaries of obstacles are piecewise linear. The boundaries can be nonconvex. The exploration problem can be motivated by the following application. Imagine that a robot has to explore the interior of a collapsed building, which has crumbled due to an earthquake, to search for human survivors. It is clearly impossible to have a knowledge of the building's interior geometry prior to the exploration. Thus, the robot must be able to see, with its onboard vision sensors, all points in the building's interior while following its exploration path. In this way, no potential survivors will be missed by the exploring robot. The exploratory path must clearly reflect the topology of the free space, and, therefore, such exploratory paths can be used to guide future robot excursions (such as would arise in our example from a rescue operation)

The three-dimensional art gallery problem and its solutions

This thesis addressed the three-dimensional Art Gallery Problem (3D-AGP), a version of the art gallery problem, which aims to determine the number of guards required to cover the interior of a pseudo-polyhedron as well as the placement of these guards. This study exclusively focused on the version of the 3D-AGP in which the art gallery is modelled by an orthogonal pseudo-polyhedron, instead of a pseudo-polyhedron. An orthogonal pseudopolyhedron provides a simple yet effective model for an art gallery because of the fact that most real-life buildings and art galleries are largely orthogonal in shape. Thus far, the existing solutions to the 3D-AGP employ mobile guards, in which each mobile guard is allowed to roam over an entire interior face or edge of a simple orthogonal polyhedron. In many realword applications including the monitoring an art gallery, mobile guards are not always adequate. For instance, surveillance cameras are usually installed at fixed locations. The guard placement method proposed in this thesis addresses such limitations. It uses fixedpoint guards inside an orthogonal pseudo-polyhedron. This formulation of the art gallery problem is closer to that of the classical art gallery problem. The use of fixed-point guards also makes our method applicable to wider application areas. Furthermore, unlike the existing solutions which are only applicable to simple orthogonal polyhedra, our solution applies to orthogonal pseudo-polyhedra, which is a super-class of simple orthogonal polyhedron. In this thesis, a general solution to the guard placement problem for 3D-AGP on any orthogonal pseudo-polyhedron has been presented. This method is the first solution known so far to fixed-point guard placement for orthogonal pseudo-polyhedron. Furthermore, it has been shown that the upper bound for the number of fixed-point guards required for covering any orthogonal polyhedron having n vertices is (n3/2), which is the lowest upper bound known so far for the number of fixed-point guards for any orthogonal polyhedron. This thesis also provides a new way to characterise the type of a vertex in any orthogonal pseudo-polyhedron and has conjectured a quantitative relationship between the numbers of vertices with different vertex configurations in any orthogonal pseudo-polyhedron. This conjecture, if proved to be true, will be useful for gaining insight into the structure of any orthogonal pseudo-polyhedron involved in many 3-dimensional computational geometrical problems. Finally the thesis has also described a new method for splitting orthogonal polygon iv using a polyline and a new method for splitting an orthogonal polyhedron using a polyplane. These algorithms are useful in applications such as metal fabrication

Online Algorithms with Discrete Visibility - Exploring Unknown Polygonal Environments

The context of this work is the exploration of unknown polygonal environments with obstacles. Both the outer boundary and the boundaries of obstacles are piecewise linear. The boundaries can be nonconvex. The exploration problem can be motivated by the following application. Imagine that a robot has to explore the interior of a collapsed building, which has crumbled due to an earthquake, to search for human survivors. It is clearly impossible to have a knowledge of the building's interior geometry prior to the exploration. Thus, the robot must be able to see, with its onboard vision sensors, all points in the building's interior while following its exploration path. In this way, no potential survivors will be missed by the exploring robot. The exploratory path must clearly reflect the topology of the free space, and, therefore, such exploratory paths can be used to guide future robot excursions (such as would arise in our example from a rescue operation)

AUTOMATED 3D CAMERA PLACEMENT

This project aims to find the minimum number of cameras needed to observe given three-dimensional (3D) environment and the cameras placement. This report traces the structure of the algorithm used to find the optimal number and placement of the cameras, the deployment of the algorithm in camera placement system, and presents the results of testing done on the system. This study related to the well known Art Gallery Problem (AGP) that addressed the problem of finding minimum number of guards necessary to guard the art gallery. Since this problem was posed, much research has been done on solving the problem from two dimensional (2D) perspectives. Not much research is done from 3D perspective and only recently more researchers are interested to study this problem in 3D environment.

Algorithms for Optimizing Search Schedules in a Polygon

In the area of motion planning, considerable work has been done on guarding problems, where "guards", modelled as points, must guard a polygonal space from "intruders". Different variants of this problem involve varying a number of factors. The guards performing the search may vary in terms of their number, their mobility, and their range of vision. The model of intruders may or may not allow them to move. The polygon being searched may have a specified starting point, a specified ending point, or neither of these. The typical question asked about one of these problems is whether or not certain polygons can be searched under a particular guarding paradigm defined by the types of guards and intruders. In this thesis, we focus on two cases of a chain of guards searching a room (polygon with a specific starting point) for mobile intruders. The intruders must never be allowed to escape through the door undetected. In the case of the two guard problem, the guards must start at the door point and move in opposite directions along the boundary of the polygon, never crossing the door point. At all times, the guards must be able to see each other. The search is complete once both guards occupy the same spot elsewhere on the polygon. In the case of a chain of three guards, consecutive guards in the chain must always be visible. Again, the search starts at the door point, and the outer guards of the chain must move from the door in opposite directions. These outer guards must always remain on the boundary of the polygon. The search is complete once the chain lies entirely on a portion of the polygon boundary not containing the door point. Determining whether a polygon can be searched is a problem in the area of visibility in polygons; further to that, our work is related to the area of planning algorithms. We look for ways to find optimal schedules that minimize the distance or time required to complete the search. This is done by finding shortest paths in visibility diagrams that indicate valid positions for the guards. In the case of the two-guard room search, we are able to find the shortest distance schedule and the quickest schedule. The shortest distance schedule is found in O(n^2) time by solving an L_1 shortest path problem among curved obstacles in two dimensions. The quickest search schedule is found in O(n^4) time by solving an L_infinity shortest path problem among curved obstacles in two dimensions. For the chain of three guards, a search schedule minimizing the total distance travelled by the outer guards is found in O(n^6) time by solving an L_1 shortest path problem among curved obstacles in two dimensions

Connectivity Constraints in Network Analysis

This dissertation establishes mathematical foundations of connectivity requirements arising in both abstract and geometric network analysis. Connectivity constraints are ubiquitous in network design and network analysis. Aside from the obvious applications in communication and transportation networks, they have also appeared in forest planning, political distracting, activity detection in video sequences and protein-protein interaction networks. Theoretically, connectivity constraints can be analyzed via polyhedral methods, in which we investigate the structure of (vertex)-connected subgraph polytope (CSP). One focus of this dissertation is on performing an extensive study of facets of CSP. We present the first systematic study of non-trivial facets of CSP. One advantage to study facets is that a facet-defining inequality is always among the tightest valid inequalities, so applying facet-defining inequalities when imposing connectivity constraints can guarantee good performance of the algorithm. We adopt lifting techniques to provide a framework to generate a wide class of facet-defining inequalities of CSP. We also derive the necessary and sufficient conditions when a vertex separator inequality, which plays a critical role in connectivity constraints, induces a facet of CSP. Another advantage to study facets is that CSP is uniquely determined by its facets, so full understanding of CSP's facets indicates full understanding of CSP itself. We are able to derive a full description of CSP for a wide class of graphs, including forest and several types of dense graphs, such as graphs with small independence number, s-plex with small s and s-defective cliques with small s. Furthermore, we investigate the relationship between lifting techniques, maximum weight connected subgraph problem and node-weight Steiner tree problem and study the computational complexity of generation of facet-defining inequalities. Another focus of this dissertation is to study connectivity in geometric network analysis. In geometric applications like wireless networks and communication networks, the concept of connectivity can be defined in various ways. In one case, connectivity is imposed by distance, which can be modeled by unit disk graphs (UDG). We create a polytime algorithm to identify large 2-clique in UDG; in another case when connectivity is based on visibility, we provide a generalization of the two-guard problem