Meeting in a Polygon by Anonymous Oblivious Robots

The Meeting problem for $k\geq 2$ searchers in a polygon $P$ (possibly with holes) consists in making the searchers move within $P$, according to a distributed algorithm, in such a way that at least two of them eventually come to see each other, regardless of their initial positions. The polygon is initially unknown to the searchers, and its edges obstruct both movement and vision. Depending on the shape of $P$, we minimize the number of searchers $k$ for which the Meeting problem is solvable. Specifically, if $P$ has a rotational symmetry of order $\sigma$ (where $\sigma=1$ corresponds to no rotational symmetry), we prove that $k=\sigma+1$ searchers are sufficient, and the bound is tight. Furthermore, we give an improved algorithm that optimally solves the Meeting problem with $k=2$ searchers in all polygons whose barycenter is not in a hole (which includes the polygons with no holes). Our algorithms can be implemented in a variety of standard models of mobile robots operating in Look-Compute-Move cycles. For instance, if the searchers have memory but are anonymous, asynchronous, and have no agreement on a coordinate system or a notion of clockwise direction, then our algorithms work even if the initial memory contents of the searchers are arbitrary and possibly misleading. Moreover, oblivious searchers can execute our algorithms as well, encoding information by carefully positioning themselves within the polygon. This code is computable with basic arithmetic operations, and each searcher can geometrically construct its own destination point at each cycle using only a compass. We stress that such memoryless searchers may be located anywhere in the polygon when the execution begins, and hence the information they initially encode is arbitrary. Our algorithms use a self-stabilizing map construction subroutine which is of independent interest.Comment: 37 pages, 9 figure

Searchable Sky Coverage of Astronomical Observations: Footprints and Exposures

Sky coverage is one of the most important pieces of information about astronomical observations. We discuss possible representations, and present algorithms to create and manipulate shapes consisting of generalized spherical polygons with arbitrary complexity and size on the celestial sphere. This shape specification integrates well with our Hierarchical Triangular Mesh indexing toolbox, whose performance and capabilities are enhanced by the advanced features presented here. Our portable implementation of the relevant spherical geometry routines comes with wrapper functions for database queries, which are currently being used within several scientific catalog archives including the Sloan Digital Sky Survey, the Galaxy Evolution Explorer and the Hubble Legacy Archive projects as well as the Footprint Service of the Virtual Observatory.Comment: 11 pages, 7 figures, submitted to PAS

Robot graphic simulation testbed

The objective of this research was twofold. First, the basic capabilities of ROBOSIM (graphical simulation system) were improved and extended by taking advantage of advanced graphic workstation technology and artificial intelligence programming techniques. Second, the scope of the graphic simulation testbed was extended to include general problems of Space Station automation. Hardware support for 3-D graphics and high processing performance make high resolution solid modeling, collision detection, and simulation of structural dynamics computationally feasible. The Space Station is a complex system with many interacting subsystems. Design and testing of automation concepts demand modeling of the affected processes, their interactions, and that of the proposed control systems. The automation testbed was designed to facilitate studies in Space Station automation concepts

Privacy-preserving proximity detection with secure multi-party computational geometry

Over the last years, Location-Based Services (LBSs) have become popular due to the global use of smartphones and improvement in Global Positioning System (GPS) and other positioning methods. Location-based services employ users' location to offer relevant information to users or provide them with useful recommendations. Meanwhile, with the development of social applications, location-based social networking services (LBSNS) have attracted millions of users because the geographic position of users can be used to enhance the services provided by those social applications. Proximity detection, as one type of location-based function, makes LBSNS more flexible and notifies mobile users when they are in proximity. Despite all the desirable features that such applications provide, disclosing the exact location of individuals to a centralized server and/or their social friends might put users at risk of falling their information in wrong hands, since locations may disclose sensitive information about people including political and religious affiliations, lifestyle, health status, etc. Consequently, users might be unwilling to participate in such applications. To this end, private proximity detection schemes enable two parties to check whether they are in close proximity while keeping their exact locations secret. In particular, running a private proximity detection protocol between two parties only results in a boolean value to the querier. Besides, it guarantees that no other information can be leaked to the participants regarding the other party's location. However, most proposed private proximity detection protocols enable users to choose only a simple geometric range on the map, such as a circle or a rectangle, in order to test for proximity. In this thesis, we take inspiration from the field of Computational Geometry and develop two privacy-preserving proximity detection protocols that allow a mobile user to specify an arbitrary complex polygon on the map and check whether his/her friends are located therein. We also analyzed the efficiency of our solutions in terms of computational and communication costs. Our evaluation shows that compared to the similar earlier work, the proposed solution increases the computational efficiency by up to 50%, and reduces the communication overhead by up to 90%. Therefore, we have achieved a significant reduction of computational and communication complexity

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