Max-stretch reduction for tree spanners

Summary, in English

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: K Iwama; Andrzej Lingas; M Okita
• Department/s: Computer Science
• Publishing year: 2005
• Language: English
• Pages: 122-133
• Publication/Series: Algorithms and Data Structures / Lecture Notes in Computer Science
• Volume: 3608
• Links: http://dx.doi.org/10.1007/11534273
• Document type: Conference paper
• Publisher: Springer
• Topic: Computer Science
• Conference name: 9th International Workshop, WADS 2005
• Conference date: 2005-08-15 - 2005-08-17
• Conference place: Waterloo, Canada
• Status: Published
• Project: VR 2002-4049
• ISBN/ISSN/Other: ISSN: 1611-3349; ISSN: 0302-9743; ISBN: 978-3-540-28101-6