An O(n log n) algorithm for the all-nearest-neighbors problem
Discrete & Computational Geometry
Maintaining the minimal distance of a point set in polylogarithmic time
Discrete & Computational Geometry
Davenport-Schinzel sequences and their geometric applications
Davenport-Schinzel sequences and their geometric applications
Data structures for mobile data
Journal of Algorithms
Kinetic binary space partitions for intersecting segments and disjoint triangles
Proceedings of the ninth annual ACM-SIAM symposium on Discrete algorithms
Cylindrical static and kinetic binary space partitions
Computational Geometry: Theory and Applications
Simplified kinetic connectivity for rectangles and hypercubes
SODA '01 Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms
Static and kinetic geometric spanners with applications
SODA '01 Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms
Separation Sensitive Kinetic Separation Structures for Convex Polygons
JCDCG '00 Revised Papers from the Japanese Conference on Discrete and Computational Geometry
ESA '02 Proceedings of the 10th Annual European Symposium on Algorithms
Divide-and-conquer in multidimensional space
STOC '76 Proceedings of the eighth annual ACM symposium on Theory of computing
Journal of Computer and System Sciences - Special issue on PODS 2000
Kinetic collision detection between two simple polygons
Computational Geometry: Theory and Applications
Kinetic collision detection with fast flight plan changes
Information Processing Letters
Kinetic and dynamic data structures for convex hulls and upper envelopes
Computational Geometry: Theory and Applications
Computational Geometry: Algorithms and Applications
Computational Geometry: Algorithms and Applications
SFCS '75 Proceedings of the 16th Annual Symposium on Foundations of Computer Science
Kinetic stable Delaunay graphs
Proceedings of the twenty-sixth annual symposium on Computational geometry
A kinetic triangulation scheme for moving points in the plane
Proceedings of the twenty-sixth annual symposium on Computational geometry
A kinetic triangulation scheme for moving points in the plane
Computational Geometry: Theory and Applications
Kinetic Euclidean minimum spanning tree in the plane
Journal of Discrete Algorithms
Kinetic and stationary point-set embeddability for plane graphs
GD'12 Proceedings of the 20th international conference on Graph Drawing
Kinetic data structures for all nearest neighbors and closest pair in the plane
Proceedings of the twenty-ninth annual symposium on Computational geometry
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We present simple, fully dynamic and kinetic data structures, which are variants of a dynamic two-dimensional range tree, for maintaining the closest pair and all nearest neighbors for a set of n moving points in the plane; insertions and deletions of points are also allowed. If no insertions or deletions take place, the structure for the closest pair uses O(n log n) space, and processes O(n2βs+2(n)log n) critical events, each in O(log2n) time. Here s is the maximum number of times where the distances between any two specific pairs of points can become equal, βs(q) = λs(q)/q, and λs(q) is the maximum length of Davenport-Schinzel sequences of order s on q symbols. The dynamic version of the problem incurs a slight degradation in performance: If m ≥ n insertions and deletions are performed, the structure still uses O(n log n) space, and processes O(mnβs+2(n)log3 n) events, each in O(log3n) time. Our kinetic data structure for all nearest neighbors uses O(n log2 n) space, and processes O(n2β2s+2(n)log3 n) critical events. The expected time to process all events is O(n2βs+22(n) log4n), though processing a single event may take Θ(n) expected time in the worst case. If m ≥ n insertions and deletions are performed, then the expected number of events is O(mnβ2s+2(n) log3n) and processing them all takes O(mnβ2s+2(n) log4n). An insertion or deletion takes O(n) expected time.