1,098 research outputs found
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
Covering the Boundary of a Simple Polygon with Geodesic Unit Disks
We consider the problem of covering the boundary of a simple polygon on n
vertices using the minimum number of geodesic unit disks. We present an O(n
\log^2 n+k) time 2-approximation algorithm for finding the centers of the
disks, with k denoting the number centers found by the algorithm
Two-Dimensional Pursuit-Evasion in a Compact Domain with Piecewise Analytic Boundary
In a pursuit-evasion game, a team of pursuers attempt to capture an evader.
The players alternate turns, move with equal speed, and have full information
about the state of the game. We consider the most restictive capture condition:
a pursuer must become colocated with the evader to win the game. We prove two
general results about pursuit-evasion games in topological spaces. First, we
show that one pursuer has a winning strategy in any CAT(0) space under this
restrictive capture criterion. This complements a result of Alexander, Bishop
and Ghrist, who provide a winning strategy for a game with positive capture
radius. Second, we consider the game played in a compact domain in Euclidean
two-space with piecewise analytic boundary and arbitrary Euler characteristic.
We show that three pursuers always have a winning strategy by extending recent
work of Bhadauria, Klein, Isler and Suri from polygonal environments to our
more general setting.Comment: 21 pages, 6 figure
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