Dynamic algorithms for geometric spanners of small diameter: Randomized solutions

Let S be a set of n points in R d and let t>1 be a real number. A t-spanner for S is a directed graph having the points of S as its vertices, such that for any pair p and q of points there is a path from p to q of length at most t times the Euclidean distance between p and q. Such a path is calle...

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Bibliographic Details
Published in:Computational geometry : theory and applications Vol. 13; no. 2; pp. 91 - 107
Main Authors: Arya, Sunil, Mount, David M., Smid, Michiel
Format: Journal Article
Language:English
Published: Elsevier B.V 01.06.1999
Subjects:
Computational geometry
Dynamic data structures
Proximity problems
Randomization
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Computational geometry
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Randomization
Dynamic data structures
Proximity problems
ISSN:0925-7721
Online Access:Get full text
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Summary:Let S be a set of n points in R d and let t>1 be a real number. A t-spanner for S is a directed graph having the points of S as its vertices, such that for any pair p and q of points there is a path from p to q of length at most t times the Euclidean distance between p and q. Such a path is called a t-spanner path. The spanner diameter of such a spanner is defined as the smallest integer D such that for any pair p and q of points there is a t-spanner path from p to q containing at most D edges. A randomized algorithm is given for constructing a t-spanner that, with high probability, contains O( n) edges and has spanner diameter O(log n). A data structure of size O( nlog d n) is given that maintains this t-spanner in O(log d nloglog n) expected amortized time per insertion and deletion, in the model of random updates, as introduced by Mulmuley.
ISSN:0925-7721
DOI:10.1016/S0925-7721(99)00014-0