Max-stretch reduction for tree spanners

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Max-stretch reduction for tree spanners. / Iwama, Kazuo; Lingas, Andrzej; Okita, Masaki.

I: Algorithmica, Vol. 50, Nr. 2, 2008, s. 223-235.

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Iwama, Kazuo ; Lingas, Andrzej ; Okita, Masaki. / Max-stretch reduction for tree spanners. I: Algorithmica. 2008 ; Vol. 50, Nr. 2. s. 223-235.

RIS

TY - JOUR

T1 - Max-stretch reduction for tree spanners

AU - Iwama, Kazuo

AU - Lingas, Andrzej

AU - Okita, Masaki

PY - 2008

Y1 - 2008

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 $mathcal{NP}$ -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.

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 $mathcal{NP}$ -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.

U2 - 10.1007/s00453-007-9058-x

DO - 10.1007/s00453-007-9058-x

M3 - Article

VL - 50

SP - 223

EP - 235

JO - Algorithmica

JF - Algorithmica

SN - 0178-4617

IS - 2

ER -