## Tree spanners (1995)

Venue: | SIAM J. Discrete Math |

Citations: | 57 - 1 self |

### BibTeX

@ARTICLE{Cai95treespanners,

author = {Leizhen Cai and Derek Corneil},

title = {Tree spanners},

journal = {SIAM J. Discrete Math},

year = {1995},

volume = {8},

pages = {359--387}

}

### Years of Citing Articles

### OpenURL

### Abstract

A tree t-spanner T of a graph G is a spanning tree in which the distance between every pair of vertices is at most t times their distance in G. This notion is motivated by applications in communication networks, distributed systems, and network design. This paper studies graph theoretic, algorithmic and complexity issues about tree spanners. It is shown that a tree 1-spanner, if it exists, in a weighted graph with m edges and n vertices is a minimum spanning tree and can be found in O(m log β(m, n)) time, where β(m, n) = min{i | log (i) n ≤ m/n}. On the other hand, for any fixed t> 1, the problem of determining the existence of a tree t-spanner in a weighted graph is proven to be NP-complete. For unweighted graphs, it is shown that constructing a tree 2-spanner takes linear time, whereas determining the existence of a tree t-spanner is NP-complete for any fixed t ≥ 4. A theorem which captures the structure of tree 2-spanners is presented for unweighted graphs. For digraphs, an O((m+n)α(m, n)) algorithm is provided for