Mobile vs. point guards

We study the problem of guarding orthogonal art galleries with horizontal mobile guards (alternatively, vertical) and point guards, using "rectangular vision". We prove a sharp bound on the minimum number of point guards required to cover the gallery in terms of the minimum number of vertical mobile guards and the minimum number of horizontal mobile guards required to cover the gallery. Furthermore, we show that the latter two numbers can be calculated in linear time.Comment: This version covers a previously missing case in both Phase 2 &

Engineering Art Galleries

The Art Gallery Problem is one of the most well-known problems in Computational Geometry, with a rich history in the study of algorithms, complexity, and variants. Recently there has been a surge in experimental work on the problem. In this survey, we describe this work, show the chronology of developments, and compare current algorithms, including two unpublished versions, in an exhaustive experiment. Furthermore, we show what core algorithmic ingredients have led to recent successes

On k-Convex Polygons

We introduce a notion of $k$-convexity and explore polygons in the plane that have this property. Polygons which are \mbox{$k$-convex} can be triangulated with fast yet simple algorithms. However, recognizing them in general is a 3SUM-hard problem. We give a characterization of \mbox{$2$-convex} polygons, a particularly interesting class, and show how to recognize them in \mbox{$O(n \log n)$} time. A description of their shape is given as well, which leads to Erd\H{o}s-Szekeres type results regarding subconfigurations of their vertex sets. Finally, we introduce the concept of generalized geometric permutations, and show that their number can be exponential in the number of \mbox{$2$-convex} objects considered.Comment: 23 pages, 19 figure

Searching Polyhedra by Rotating Half-Planes

The Searchlight Scheduling Problem was first studied in 2D polygons, where the goal is for point guards in fixed positions to rotate searchlights to catch an evasive intruder. Here the problem is extended to 3D polyhedra, with the guards now boundary segments who rotate half-planes of illumination. After carefully detailing the 3D model, several results are established. The first is a nearly direct extension of the planar one-way sweep strategy using what we call exhaustive guards, a generalization that succeeds despite there being no well-defined notion in 3D of planar "clockwise rotation". Next follow two results: every polyhedron with r>0 reflex edges can be searched by at most r^2 suitably placed guards, whereas just r guards suffice if the polyhedron is orthogonal. (Minimizing the number of guards to search a given polyhedron is easily seen to be NP-hard.) Finally we show that deciding whether a given set of guards has a successful search schedule is strongly NP-hard, and that deciding if a given target area is searchable at all is strongly PSPACE-hard, even for orthogonal polyhedra. A number of peripheral results are proved en route to these central theorems, and several open problems remain for future work.Comment: 45 pages, 26 figure

Guarding and Searching Polyhedra

Guarding and searching problems have been of fundamental interest since the early years of Computational Geometry. Both are well-developed areas of research and have been thoroughly studied in planar polygonal settings. In this thesis we tackle the Art Gallery Problem and the Searchlight Scheduling Problem in 3-dimensional polyhedral environments, putting special emphasis on edge guards and orthogonal polyhedra. We solve the Art Gallery Problem with reflex edge guards in orthogonal polyhedra having reflex edges in just two directions: generalizing a classic theorem by O'Rourke, we prove that r/2 + 1 reflex edge guards are sufficient and occasionally necessary, where r is the number of reflex edges. We also show how to compute guard locations in O(n log n) time. Then we investigate the Art Gallery Problem with mutually parallel edge guards in orthogonal polyhedra with e edges, showing that 11e/72 edge guards are always sufficient and can be found in linear time, improving upon the previous state of the art, which was e/6. We also give tight inequalities relating e with the number of reflex edges r, obtaining an upper bound on the guard number of 7r/12 + 1. We further study the Art Gallery Problem with edge guards in polyhedra having faces oriented in just four directions, obtaining a lower bound of e/6 - 1 edge guards and an upper bound of (e+r)/6 edge guards. All the previously mentioned results hold for polyhedra of any genus. Additionally, several guard types and guarding modes are discussed, namely open and closed edge guards, and orthogonal and non-orthogonal guarding. Next, we model the Searchlight Scheduling Problem, the problem of searching a given polyhedron by suitably turning some half-planes around their axes, in order to catch an evasive intruder. After discussing several generalizations of classic theorems, we study the problem of efficiently placing guards in a given polyhedron, in order to make it searchable. For general polyhedra, we give an upper bound of r^2 on the number of guards, which reduces to r for orthogonal polyhedra. Then we prove that it is strongly NP-hard to decide if a given polyhedron is entirely searchable by a given set of guards. We further prove that, even under the assumption that an orthogonal polyhedron is searchable, approximating the minimum search time within a small-enough constant factor to the optimum is still strongly NP-hard. Finally, we show that deciding if a specific region of an orthogonal polyhedron is searchable is strongly PSPACE-hard. By further improving our construction, we show that the same problem is strongly PSPACE-complete even for planar orthogonal polygons. Our last results are especially meaningful because no similar hardness theorems for 2-dimensional scenarios were previously known

An algorithmic definition of the axial map

The fewest-line axial map, often simply referred to as the 'axial map, is one of the primary tools of space syntax. Its natural language definition has allowed researchers to draw consistent maps that present a concise description of architectural space; it has been established that graph measures obtained from the map are useful for the analysis of pedestrian movement patterns and activities related to such movement: for example, the location of services or of crime. However, the definition has proved difficult to translate into formal language by mathematicians and algorithmic implementers alike. This has meant that space syntax has been criticised for a lack of rigour in the definition of one of its fundamental representations. Here we clarify the original definition of the fewest-line axial map and show that it can be implemented algorithmically. We show that the original definition leads to maps similar to those currently drawn by hand, and we demonstrate that the differences between the two may be accounted for in terms of the detail of the algorithm used. We propose that the analytical power of the axial map in empirical studies derives from the efficient representation of key properties of the spatial configuration that it captures