TY - GEN

T1 - Max-stretch reduction for tree spanners

AU - Iwama, K

AU - Lingas, Andrzej

AU - Okita, M

PY - 2005

Y1 - 2005

N2 - 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.

AB - 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.

U2 - 10.1007/11534273

DO - 10.1007/11534273

M3 - Paper in conference proceeding

SN - 978-3-540-28101-6

VL - 3608

SP - 122

EP - 133

BT - Algorithms and Data Structures / Lecture Notes in Computer Science

PB - Springer

T2 - 9th International Workshop, WADS 2005

Y2 - 15 August 2005 through 17 August 2005

ER -