Computational geometry: an introduction
Computational geometry: an introduction
The ultimate planar convex hull algorithm
SIAM Journal on Computing
On k-hulls and related problems
SIAM Journal on Computing
Epsilon geometry: building robust algorithms from imprecise computations
SCG '89 Proceedings of the fifth annual symposium on Computational geometry
Applications of random sampling in computational geometry, II
Discrete & Computational Geometry - Selected papers from the fourth ACM symposium on computational geometry, Univ. of Illinois, Urbana-Champaign, June 6 8, 1988
Computational Geometry: Theory and Applications
An optimal algorithm for intersecting line segments in the plane
Journal of the ACM (JACM)
Reporting points in halfspaces
Computational Geometry: Theory and Applications
A simple randomized sieve algorithm for the closest-pair problem
Information and Computation
Davenport-Schinzel sequences and their geometric applications
Davenport-Schinzel sequences and their geometric applications
On range reporting, ray shooting and k-level construction
SCG '99 Proceedings of the fifteenth annual symposium on Computational geometry
Data structures for mobile data
Journal of Algorithms
Lower bounds for algebraic computation trees
STOC '83 Proceedings of the fifteenth annual ACM symposium on Theory of computing
A Method for Proving Lower Bounds for Certain Geometric Problems
A Method for Proving Lower Bounds for Certain Geometric Problems
Kinetic sorting and kinetic convex hulls
Computational Geometry: Theory and Applications
Computational Geometry: Algorithms and Applications
Computational Geometry: Algorithms and Applications
Triangulating input-constrained planar point sets
Information Processing Letters
Preprocessing Imprecise Points and Splitting Triangulations
ISAAC '08 Proceedings of the 19th International Symposium on Algorithms and Computation
Computational Complexity: A Modern Approach
Computational Complexity: A Modern Approach
Computing hereditary convex structures
Proceedings of the twenty-fifth annual symposium on Computational geometry
Discrete & Computational Geometry - Special Issue: 24th Annual Symposium on Computational Geometry
Introduction to Algorithms, Third Edition
Introduction to Algorithms, Third Edition
Delaunay triangulation of imprecise points in linear time after preprocessing
Computational Geometry: Theory and Applications
Instance-Optimal Geometric Algorithms
FOCS '09 Proceedings of the 2009 50th Annual IEEE Symposium on Foundations of Computer Science
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Self-improving algorithms for coordinate-wise maxima
Proceedings of the twenty-eighth annual symposium on Computational geometry
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Motivated by the desire to cope with data imprecision, we study methods for preprocessing a set of line-segments (or just lines) in the plane such that whenever we are given a set of points, each of which lies on a distinct object, we can compute their convex hull more efficiently than in "standard settings" (that is, without preprocessing). In particular, we study the following problem: given a set L of n lines in the plane, we wish to preprocess L such that later, upon receiving a set P of n points, each of which lies on a distinct line of L, we can construct the convex hull of P efficiently. We show that in quadratic time and space it is possible to construct a data structure on L that enables us to compute the convex hull of any such point set P in O(n alpha(n) log* n) expected time. If we further assume that the points are "oblivious" with respect to the data structure, the running time improves to O(n alpha(n)). The analysis applies almost verbatim when L is a set of line-segments, and yields similar asymptotic bounds. We present several extensions, including a trade-off between space and query time and an output-sensitive algorithm. We also study the "dual problem" where we show how to efficiently compute the ( We complement our results by Omega(n log n) lower bounds under the algebraic computation tree model for several related problems, including sorting a set of points (according to, say, their x-order), each of which lies on a given line known in advance. Therefore, the convex hull problem under our setting is easier than sorting, contrary to the "standard" convex hull and sorting problems, in which the two problems require Theta(n log n) steps in the worst case (under the algebraic computation tree model).