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Max-stretch reduction for tree spanners

Iwama, K ; Lingas, Andrzej LU and Okita, M (2005) 9th International Workshop, WADS 2005 3608. p.122-133
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... (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 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
; and
organization
publishing date
type
Chapter in Book/Report/Conference proceeding
publication status
published
subject
host publication
Algorithms and Data Structures / Lecture Notes in Computer Science
volume
3608
pages
122 - 133
publisher
Springer
conference name
9th International Workshop, WADS 2005
conference location
Waterloo, Canada
conference dates
2005-08-15 - 2005-08-17
external identifiers
  • wos:000231873500012
  • scopus:26844486244
ISSN
1611-3349
0302-9743
ISBN
978-3-540-28101-6
DOI
10.1007/11534273
project
VR 2002-4049
language
English
LU publication?
yes
id
9057d95e-64b8-4fb9-88c9-75809aadded9 (old id 220496)
date added to LUP
2016-04-01 12:08:15
date last changed
2021-03-24 05:04:07
@inproceedings{9057d95e-64b8-4fb9-88c9-75809aadded9,
  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 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, K and Lingas, Andrzej and Okita, M},
  booktitle    = {Algorithms and Data Structures / Lecture Notes in Computer Science},
  isbn         = {978-3-540-28101-6},
  issn         = {1611-3349},
  language     = {eng},
  pages        = {122--133},
  publisher    = {Springer},
  title        = {Max-stretch reduction for tree spanners},
  url          = {http://dx.doi.org/10.1007/11534273},
  doi          = {10.1007/11534273},
  volume       = {3608},
  year         = {2005},
}