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WADS '99 Proceedings of the 6th International Workshop on Algorithms and Data Structures
A New Visibility Partition for Affine Pattern Matching
DGCI '00 Proceedings of the 9th International Conference on Discrete Geometry for Computer Imagery
Efficient visibility queries in simple polygons
Computational Geometry: Theory and Applications
Efficient computation of query point visibility in polygons with holes
SCG '05 Proceedings of the twenty-first annual symposium on Computational geometry
Randomized Algorithms for Minimum Distance Localization
International Journal of Robotics Research
Query point visibility computation in polygons with holes
Computational Geometry: Theory and Applications
Visibility queries in a polygonal region
Computational Geometry: Theory and Applications
Planar visibility: testing and counting
Proceedings of the twenty-sixth annual symposium on Computational geometry
Algorithms and theory of computation handbook
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SWAT'12 Proceedings of the 13th Scandinavian conference on Algorithm Theory
A pseudopolynomial time O(log n)-approximation algorithm for art gallery problems
WADS'07 Proceedings of the 10th international conference on Algorithms and Data Structures
Space/query-time tradeoff for computing the visibility polygon
Computational Geometry: Theory and Applications
Fast vertex guarding for polygons with and without holes
Computational Geometry: Theory and Applications
Computational Geometry: Theory and Applications
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We consider the following problem: given a simple polygon ${\cal P}$ and a star-shaped polygon ${\cal V}$, find a point (or the set of points) in ${\cal P}$ from which the portion of ${\cal P}$ that is visible is translation-congruent to ${\cal V}$. The problem arises in the localization of robots equipped with a range finder and a compass---${\cal P}$ is a map of a known environment, ${\cal V}$ is the portion visible from the robot's position, and the robot must use this information to determine its position in the map. We give a scheme that preprocesses ${\cal P}$ so that any subsequent query ${\cal V}$ is answered in optimal time O(m + log n + A), where m and n are the number of vertices in ${\cal V}$ and ${\cal P}$ and A is the number of points in ${\cal P}$ that are valid answers (the output size). Our technique uses O(n5) space and preprocessing in the worst case; within certain limits, we can trade off smoothly between the query time and the preprocessing time and space. In the process of solving this problem, we also devise a data structure for output-sensitive determination of the visibility polygon of a query point inside a polygon ${\cal P}$. We then consider a variant of the localization problem in which there is a maximum distance to which the robot can "see"---this is motivated by practical considerations, and we outline a similar solution for this case. We finally show that a single localization query ${\cal V}$ can be answered in time O(mn) with no preprocessing.