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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| Published in: | Computational geometry : theory and applications Vol. 13; no. 2; pp. 91 - 107 |
|---|---|
| Main Authors: | , , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Elsevier B.V
01.06.1999
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| Subjects: | |
| ISSN: | 0925-7721 |
| Online Access: | Get full text |
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| Abstract | 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. |
|---|---|
| AbstractList | 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. |
| Author | Mount, David M. Smid, Michiel Arya, Sunil |
| Author_xml | – sequence: 1 givenname: Sunil surname: Arya fullname: Arya, Sunil email: arya@cs.ust.hk organization: Department of Computer Science, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong – sequence: 2 givenname: David M. surname: Mount fullname: Mount, David M. email: mount@cs.umd.edu organization: Department of Computer Science and Institute for Advanced Computer Studies, University of Maryland, College Park, MD, USA – sequence: 3 givenname: Michiel surname: Smid fullname: Smid, Michiel email: michiel@isg.cs.uni-magdeburg.de organization: Department of Computer Science, University of Magdeburg, D-39106 Magdeburg, Germany |
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| Snippet | 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... |
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| SubjectTerms | Computational geometry Dynamic data structures Proximity problems Randomization Skip lists |
| Title | Dynamic algorithms for geometric spanners of small diameter: Randomized solutions |
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