Proceedings ArticleDOI
Dynamic half-space reporting, geometric optimization, and minimum spanning trees
Pankaj K. Agarwal,David Eppstein,Jiří Matoušek +2 more
- pp 80-89
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TLDR
Using dynamic data structures for half-space range reporting and for maintaining the minima of a decomposable function, the authors obtain efficient dynamic algorithms for a number of geometric problems, including closest/farthest neighbor searching, fixed dimension linear programming, bi-chromatic closest pair, diameter, and Euclidean minimum spanning tree.Abstract:
The authors describe dynamic data structures for half-space range reporting and for maintaining the minima of a decomposable function. Using these data structures, they obtain efficient dynamic algorithms for a number of geometric problems, including closest/farthest neighbor searching, fixed dimension linear programming, bi-chromatic closest pair, diameter, and Euclidean minimum spanning tree. >read more
Citations
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Efficient indexing methods for probabilistic threshold queries over uncertain data
TL;DR: This paper develops two index structures and associated algorithms to efficiently answer Probabilistic Threshold Queries (PTQs), and establishes the difficulty of this problem by mapping one-dimensional intervals to a two-dimensional space, and shows that the problem of intervals indexing with probabilities is significantly harder than interval indexing which is considered a well-studied problem.
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Minkowski-Type Theorems and Least-Squares Clustering
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Dynamic Half-Space Range Reporting and Its Application
K P Agarwal,Jirí Matousek +1 more
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Dynamic Euclidean minimum spanning trees and extrema of binary functions
TL;DR: This work maintains the minimum spanning tree of a point set in the plane subject to point insertions and deletions, in amortized timeO(n1/2 log2n) per update operation, and uses a novel construction, theordered nearest neighbor path of a set of points.
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Proximity problems on moving points
TL;DR: A kinetic data structure for the maintenance of a multidimensional range search tree is introduced and is used as a building block to obtain kinetic data structures for two classical geometric proximity problems in arbitrary dlmensions.
References
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Book
Introduction to Algorithms
TL;DR: The updated new edition of the classic Introduction to Algorithms is intended primarily for use in undergraduate or graduate courses in algorithms or data structures and presents a rich variety of algorithms and covers them in considerable depth while making their design and analysis accessible to all levels of readers.
Computational geometry. an introduction
TL;DR: This book offers a coherent treatment, at the graduate textbook level, of the field that has come to be known in the last decade or so as computational geometry.
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Computational Geometry: An Introduction
TL;DR: In this article, the authors present a coherent treatment of computational geometry in the plane, at the graduate textbook level, and point out the way to the solution of the more challenging problems in dimensions higher than two.
Proceedings ArticleDOI
Applications of random sampling in computational geometry, II
TL;DR: Asymptotically tight bounds for a combinatorial quantity of interest in discrete and computational geometry, related to halfspace partitions of point sets, are given.
Journal ArticleDOI
Linear-Time Algorithms for Linear Programming in $R^3 $ and Related Problems
TL;DR: A linear-time algorithm is given for the classical problem of finding the smallest circle enclosing n given points in the plane, which disproves a conjecture by Shamos and Hoey that this problem requires Ω(n log n) time.