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Max-stretch reduction for tree spanners

Iwama, Kazuo; Lingas, Andrzej LU and Okita, Masaki (2008) In Algorithmica 50(2). p.223-235
Abstract
A tree t-spanner T of a graph G is a spanning tree of G whose max-stretch is t, i.e., the distance between any two vertices in T is at most t times their distance in G. If G has a tree t-spanner but not a tree (t−1)-spanner, then G is said to have max-stretch of t. In this paper, we study the Max-Stretch Reduction Problem: for an unweighted graph G=(V,E), find a set of edges not in E originally whose insertion into G can decrease the max-stretch of G. Our results are as follows: (i) For a ring graph, we give a linear-time algorithm which inserts k edges improving the max-stretch optimally. (ii) For a grid graph, we give a nearly optimal max-stretch reduction algorithm which preserves the structure of the grid. (iii) In the general case, we... (More)
A tree t-spanner T of a graph G is a spanning tree of G whose max-stretch is t, i.e., the distance between any two vertices in T is at most t times their distance in G. If G has a tree t-spanner but not a tree (t−1)-spanner, then G is said to have max-stretch of t. In this paper, we study the Max-Stretch Reduction Problem: for an unweighted graph G=(V,E), find a set of edges not in E originally whose insertion into G can decrease the max-stretch of G. Our results are as follows: (i) For a ring graph, we give a linear-time algorithm which inserts k edges improving the max-stretch optimally. (ii) For a grid graph, we give a nearly optimal max-stretch reduction algorithm which preserves the structure of the grid. (iii) In the general case, we show that it is $mathcal{NP}$ -hard to decide, for a given graph G and its spanning tree of max-stretch t, whether or not one-edge insertion can decrease the max-stretch to t−1. (iv) Finally, we show that the max-stretch of an arbitrary graph on n vertices can be reduced to s′≥2 by inserting O(n/s′) edges, which can be determined in linear time, and observe that this number of edges is optimal up to a constant. (Less)
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author
organization
publishing date
type
Contribution to journal
publication status
published
subject
in
Algorithmica
volume
50
issue
2
pages
223 - 235
publisher
Springer
external identifiers
  • wos:000252190700004
  • scopus:38149111914
ISSN
0178-4617
DOI
10.1007/s00453-007-9058-x
project
VR 2005-4085
language
English
LU publication?
yes
id
9521794f-f6ba-4f82-8fc8-6cef334ce517 (old id 629545)
date added to LUP
2007-11-27 14:39:31
date last changed
2017-01-01 05:04:56
@article{9521794f-f6ba-4f82-8fc8-6cef334ce517,
  abstract     = {A tree t-spanner T of a graph G is a spanning tree of G whose max-stretch is t, i.e., the distance between any two vertices in T is at most t times their distance in G. If G has a tree t-spanner but not a tree (t−1)-spanner, then G is said to have max-stretch of t. In this paper, we study the Max-Stretch Reduction Problem: for an unweighted graph G=(V,E), find a set of edges not in E originally whose insertion into G can decrease the max-stretch of G. Our results are as follows: (i) For a ring graph, we give a linear-time algorithm which inserts k edges improving the max-stretch optimally. (ii) For a grid graph, we give a nearly optimal max-stretch reduction algorithm which preserves the structure of the grid. (iii) In the general case, we show that it is $mathcal{NP}$ -hard to decide, for a given graph G and its spanning tree of max-stretch t, whether or not one-edge insertion can decrease the max-stretch to t−1. (iv) Finally, we show that the max-stretch of an arbitrary graph on n vertices can be reduced to s′≥2 by inserting O(n/s′) edges, which can be determined in linear time, and observe that this number of edges is optimal up to a constant.},
  author       = {Iwama, Kazuo and Lingas, Andrzej and Okita, Masaki},
  issn         = {0178-4617},
  language     = {eng},
  number       = {2},
  pages        = {223--235},
  publisher    = {Springer},
  series       = {Algorithmica},
  title        = {Max-stretch reduction for tree spanners},
  url          = {http://dx.doi.org/10.1007/s00453-007-9058-x},
  volume       = {50},
  year         = {2008},
}