Computational geometry: an introduction
Computational geometry: an introduction
A unified approach to approximation algorithms for bottleneck problems
Journal of the ACM (JACM)
Ray shooting and parametric search
SIAM Journal on Computing
Dealing with higher dimensions: the well-separated pair decomposition and its applications
Dealing with higher dimensions: the well-separated pair decomposition and its applications
Euclidean spanners: short, thin, and lanky
STOC '95 Proceedings of the twenty-seventh annual ACM symposium on Theory of computing
Algorithms for proximity problems in higher dimensions
Computational Geometry: Theory and Applications
Faster algorithms for some geometric graph problems in higher dimensions
SODA '93 Proceedings of the fourth annual ACM-SIAM Symposium on Discrete algorithms
Approximate distance oracles for geometric graphs
SODA '02 Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms
Introduction to Algorithms
Approximating the Stretch Factor of Euclidean Graphs
SIAM Journal on Computing
Computing the Maximum Detour and Spanning Ratio of Planar Paths, Trees, and Cycles
STACS '02 Proceedings of the 19th Annual Symposium on Theoretical Aspects of Computer Science
Approximating Geometric Bottleneck Shortest Paths
STACS '03 Proceedings of the 20th Annual Symposium on Theoretical Aspects of Computer Science
Approximating geometric bottleneck shortest paths
Computational Geometry: Theory and Applications
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The stretch factor of a Euclidean graph is the maximum ratio of the distance in the graph between any two points and their Euclidean distance. The following problem is considered. Preprocess a set S of n points in Rd into a data structure that supports the following queries: Given an arbitrary query value b 0, compute a constant-factor approximation of the stretch factor of the graph Gb, which is the graph on S containing all edges of length at most b. We give a data structure for this problem having size O(log n) and query time O(log log n). Even though there could be up to (n 2) different stretch factors in the collection {Gb : b 0} of graphs, we show that this data structure can be constructed in subquadratic time.Our algorithms use techniques from computational geometry, such as minimum spanning trees, well-separated pairs, data structures for the nearest-neighbor problem, and algorithms for selecting and ranking distances.