Computational geometry: an introduction
Computational geometry: an introduction
Making data structures persistent
Journal of Computer and System Sciences - 18th Annual ACM Symposium on Theory of Computing (STOC), May 28-30, 1986
Finding the upper envelope of n line segments in O(n log n) time
Information Processing Letters
Davenport-Schinzel sequences and their geometric applications
Davenport-Schinzel sequences and their geometric applications
Facility Location Constrained to a Polygonal Domain
LATIN '02 Proceedings of the 5th Latin American Symposium on Theoretical Informatics
Finding Largest Empty Circles with Location Constraints
Finding Largest Empty Circles with Location Constraints
A Survey on Obnoxious Facility Location Problems
A Survey on Obnoxious Facility Location Problems
Optimal placement of convex polygons to maximize point containment
Computational Geometry: Theory and Applications
Localized geometric query problems
Computational Geometry: Theory and Applications
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The Largest Empty Circle problem seeks the largest circle centered within the convex hull of a set P of n points in R^2 and devoid of points from P. In this paper, we introduce a query version of this well-studied problem. In our query version, we are required to preprocess P so that when given a query line Q, we can quickly compute the largest empty circle centered at some point on Q and within the convex hull of P. We present solutions for two special cases and the general case; all our queries run in O(logn) time. We restrict the query line to be horizontal in the first special case, which we preprocess in O(n@a(n)logn) time and space, where @a(n) is the slow growing inverse of Ackermann's function. When the query line is restricted to pass through a fixed point, the second special case, our preprocessing takes O(n@a(n)^O^(^@a^(^n^)^)logn) time and space. We use insights from the two special cases to solve the general version of the problem with preprocessing time and space in O(n^3logn) and O(n^3) respectively.