589 research outputs found
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) ân+13â diagonal guards are always sufficient and sometimes necessary, and (2) â2n+15â 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 ân+13â in linear time and space, (2) an edge 2-dominating set of size â2n+15â in O(n2) time and O(n) space, and (3) an edge 2-dominating set of size â3n7â 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 ân+13â in O(nlogn) time, (2) an edge guard set of size â2n+15â in O(n2) time, and (3) an edge guard set of size â3n7â in O(nlogn) time. All space requirements are linear. Finally, we show that ân3â mobile or ân3â edge guards are sometimes necessary.When restricting our attention to monotone piecewise-convex polygons, the bounds mentioned above drop: ân+14â edge or mobile guards are always sufficient and sometimes necessary; such an edge or mobile guard set, of size at most ân+14â, can be computed in O(n) time and space
Polygon Exploration with Time-Discrete Vision
With the advent of autonomous robots with two- and three-dimensional scanning
capabilities, classical visibility-based exploration methods from computational
geometry have gained in practical importance. However, real-life laser scanning
of useful accuracy does not allow the robot to scan continuously while in
motion; instead, it has to stop each time it surveys its environment. This
requirement was studied by Fekete, Klein and Nuechter for the subproblem of
looking around a corner, but until now has not been considered in an online
setting for whole polygonal regions.
We give the first algorithmic results for this important algorithmic problem
that combines stationary art gallery-type aspects with watchman-type issues in
an online scenario: We demonstrate that even for orthoconvex polygons, a
competitive strategy can be achieved only for limited aspect ratio A (the ratio
of the maximum and minimum edge length of the polygon), i.e., for a given lower
bound on the size of an edge; we give a matching upper bound by providing an
O(log A)-competitive strategy for simple rectilinear polygons, using the
assumption that each edge of the polygon has to be fully visible from some scan
point.Comment: 28 pages, 17 figures, 2 photographs, 3 tables, Latex. Updated some
details (title, figures and text) for final journal revision, including
explicit assumption of full edge visibilit
Online Searching with an Autonomous Robot
We discuss online strategies for visibility-based searching for an object
hidden behind a corner, using Kurt3D, a real autonomous mobile robot. This task
is closely related to a number of well-studied problems. Our robot uses a
three-dimensional laser scanner in a stop, scan, plan, go fashion for building
a virtual three-dimensional environment. Besides planning trajectories and
avoiding obstacles, Kurt3D is capable of identifying objects like a chair. We
derive a practically useful and asymptotically optimal strategy that guarantees
a competitive ratio of 2, which differs remarkably from the well-studied
scenario without the need of stopping for surveying the environment. Our
strategy is used by Kurt3D, documented in a separate video.Comment: 16 pages, 8 figures, 12 photographs, 1 table, Latex, submitted for
publicatio
Minimum Vertex Guard problem for orthogonal polygons: a genetic approach
The problem of minimizing the number of guards placed on vertices needed to guard a given simple polygon (MINIMUM VERTEX GUARD problem) is NP-hard. This computational complexity opens two lines of investigation: the development of algorithms that determine approximate solutions and the determination of optimal solutions for special classes of simple polygons. In this paper we follow the first line of investigation proposing an approximation algorithm based on the general met heuristic Genetic Algorithms to solve the MINIMUM VERTEXGUARD problem
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